To: Researchers working on El Naschie E-infinity Cantorian-fractal spacetime theory of quantum high energy physics.
Dear All,
Even the most gullible, unsuspecting and least inclined to a conspiracy theory must be asking themselves by now why all this viscous attack not only against E-infinity and fractal spacetime as a theory but far worse against Prof. Ji-Huan He, L. Marek-Crnjac, Prof. Mohamed El Naschie, Gerardo Iovane and in fact every single member of the E-infinity group. Who is behind all that? Who is behind El Naschie Watch and the daily hundreds of perverted viscous comments aimed at any and everybody who has anything good to say about us, particularly Prof. El Naschie and Prof. He? Is it the establishment and what kind of establishment is this which is scared to death from a couple of simple equations and the word fractal spacetime? If I did not know better I would think that a Cantor set is where the devil himself lives or it is a code name similar to D-day or Dessert Storm only directed towards theoretical physics. In what follows I would like to explain to you in some detail that we have achieved a great deal with our theory. The work of Goldfain, Mohamed El Naschie and before them Nottale and Ord and Richard Feynman were all not in vain. I would like to explain that we are on the verge of a great truth and paradigm shift in physics. Some unteachable elements of the establishment are resorting to methods far away from science which failed in the past to prevent the inevitable. Others more cunning are working hard to translate our terminology to another terminology. The famous plagiarism which took place in Scientific American is only one of the very early and visible examples of this cunningness which is in reality lack of scientific honor as aptly described by one of us in a letter to the people concerned.
Let me start by first counting the unshakable final results which we have achieved on a few points and then we will move from there to discuss the wider picture and the experimental facts which are coming in daily. In brief we found the following results which will endure any future discoveries and could be counted as the absolutely secured part of what we have done.
1. The geometry of micro spacetime is best described by a fractal. This is the result of the work of Garnet Ord and Laurent Nottale following the pioneering idea of Richard Feynman and his path integral method.
2. The building blocks of spacetime are elementary random Cantor sets. The Hausdorff dimension of these elementary random Cantor sets is the golden mean. By varying the resolution you can obtain everything you want from an infinite collection of these Cantor sets. This is the essence of the work of Mohamed El Naschie, L. Crnjac, Ji-Huan He and also G. Iovane.
3. The expectation value for the Hausdorff dimension is 4 plus the golden mean to the power of three. This is 4.23606799. The expectation value for the topological dimension is exactly 4. The formal dimensionality is however infinite. This will bring us nearer to the theory of multiverse as I will explain later.
4. The most fundamental symmetry groups are the exceptional Lie symmetry groups. These are 8 in all forming a family. The most important member of this family is E8. What is important however is that the sum over all exceptional Lie symmetry group leads to a probability measure which is consistent with the random Cantor sets and its golden mean dimension. The sum of the dimensions of all eight groups was shown by El Naschie to give a total dimension equal 4 where = 137.
5. You can extend summing over exceptional Lie group to compact and non-compact exceptional Lie groups and find 17 of them. The sum of all dimensions was shown by El Naschie to be This theory led to speculation about an even larger symmetry group, namely E12 which is more important than the recently discovered E10 and E11 but I will not consider this part of secured knowledge and I will stop here, mentioning only that Ray Munroe was the first to find E12 before El Naschie.
As for the experimental verification we now have a few extremely important ones:
1. Indication of a Cantorian spacetime and a fractal spacetime coming from an analysis of the cosmic rays and microwave background radiation. The expert on the first is Goldfain and on the second Mohamed El Naschie and you can consult their publications on this. However there are many results independent of our group confirming the same and it would be great if Goldfain could write a report for us all on this for internal use on our blog.
2. The discovery of E8 in nanostructures and the golden mean in quantum mechanics which was recently made public by the Helmholtz Center in Germany is the most definite result confirming Cantorian spacetime geometry experimentally. I say this is the tip of the iceberg. From now on you will see the golden mean mushrooming everywhere in quantum mechanics and high energy physics.
Under these circumstances many people became worried and anxious that a group like ours, not considered to be specialists in mathematical physics and high energy particle physics should have made such a major step forwards and been able to predict the masses of elementary particles and the value of fundamental constants with such precision and ease. The frustration is to a certain extent understandable and the reason is the following.
1. Garnet Ord and Nottale did not use set theory per se. Mohamed El Naschie was also not the first to propose set theory in quantum mechanics and high energy physics. The first impulse came from somewhere completely different. They came from David Finkelstein and Carl Friedrich von Weizsaker. The two great scientists were not interested in details. However Stanley Gudder in the USA and his school as well as Fay Dowker in England and her collaborators felt that partially ordered sets could solve the problem of quantum mechanics. They were rather near but not quite there because they had no simple way of performing real quantitative calculations. Far better suited to quantum mechanics are random Cantor sets. When you use them you have the golden mean. And when you use the golden mean then you have at your hand an unrivalled number system which can handle any complex computation with unheard of simplicity. This was Mohamed El Naschie’s good luck or misfortune. By pure accident or providence El Naschie stumbled on a basic problem in fundamental mathematics. Basing your number system on the irrational number and the irrational golden mean system you can see the world with new eyes with unheard of simplicity. The recent book by Alexey Stakhov published in World Scientific under the title The Mathematics of Harmony is a profound meditation on this theme. Chaos, Solitons & Fractals had the honor of publishing the larger part of Prof. Stakhov’s work. I think when certain elements in the establishment realize that we surpassed everybody else, they panicked.
2. When certain elements of the establishment panicked, they published the paper in Scientific American in 2008. To overcome the problem of not having the golden mean they used the most powerful existing new generation of computers and pretended to use their expression, that they found the holy grail of theoretical physics by calculating the four dimensionalities of spacetime from first principles using a desktop computer. One of our associates joked about them by calculating the same dimension using a pocket calculator. In fact using the golden mean you can find everything by counting on your fingers, that is if you know the rules of the golden mean arithmetic. The rest is history and you can read it on the comments of this work in Scientific American.
3. An exceptional member of the mainstream who does not normally work in quantum mechanics is Prof. Tim Palmer of the University of Oxford. This professor realized the importance of fractals for quantum mechanics. His first paper did not refer to our group at all. Later on he revised his stance and he referred to Garnet Ord, Laurent Nottale and Mohamed El Naschie on the first page of his revised ArXived paper. Later on when this paper appeared in the Royal Society, the three names were relegated to the very rear of the paper. Never the less, the man at least had sufficient objectivity to acknowledge our priority. Of course he should have noticed that there is no difference between our work using Rene Thom/El Naschie VAK and his proposal. Spacetime and phase space are exchangeable at this high energy where time is spatialized. Anyway this was at least one of the establishment acknowledging that we were there first. On a personal level we have the greatest respect for Prof. Tim Palmer who is an exceptional meteorological scientist.
4. There is at least one earlier attempt to use elements out of our work and overlook mentioning our group. The first which comes to my mind is that of Dr. Garrett Lisi. He is not a mainstream scientist at all but he was supported by some people from Perimeter Inst. in Canada. Needless to say, most of what Lisi wrote about E8 was well known to us long ago and was published in Chaos, Solitons & Fractals years before Lisi’s paper. Of course the establishment in general neither likes Lisi’s nor our work and thus we and Lisi were equal although we were more equal than Lisi in being disliked by the establishment for reasons which have no scientific basis.
5. Here I must now mention the discovery of the important utility of the multiverse. The objective of this comment is really to talk about this subject. What I have written so far was just a summary of the past. I would like to show clearly that the multiverse theory is nothing but a new label for our Cantorian spacetime theory particularly when we couple it with the holographic principle of ‘tHooft.
ON THE EQUIVALENCE BETWEEN THE MULTIVERSE THEORY OF RAPHAEL BOUSSO AND E-INFINITY THEORY
A multiverse is a universe with an infinite number of pocket universes. Please note that the most important thing here is infinity. Fractal spacetime in the E-infinity version is an infinite dimensional universe. To avoid contradictions and inconsistencies, Bousso introduced causal patch measures. This measure corresponds to the relative topological probability used in E-infinity theory. However the really interesting point comes when you apply to both theories the holographic principle. You recall from E-infinity theory that the holographic manifold of E-infinity is Klein’s modular curve. This curve has 336 triangles. These 336 correspond to SL 2,7 which constitutes 336 isometries in two dimensions being the surface. They also correspond to 336 kissing points of 10 dimensional spheres. This in turn corresponds to 336 quantum curvatures in 8 dimensions. You can think of the 336 of Klein’s modular curve as cutting the 10 dimensional spaces of the kissing spheres and flattening them to a surface. You see know the equivalence between particle like states and isometries. Since the kissing points are the points of interactions, they can be regarded as messenger particles or massless gauge bosons. In fact following the holographic principle, they represent all that is going to be particle physics of the standard model later on at lower energy. To see the connection you just compare the 496 of E8 E8 of super string theory to the sum of the electromagnetism as represented by the 137 alpha bar when you add to it Einstein’s gravity in four dimensions which is 20 and the 336 of particle physics. Now you see that it does not completely add up. There are 3 missing. This 3 can be taken care of in two ways. Either you think of them as the only electromagnetic photons which are massive, namely W minus, W plus and Z zero. Thus we have 3 plus 336 equal 339. Plus 137 plus 20 which exactly matches the 496 of super strings as should be. But there is a better geometrical way to look at that which agrees exactly with the holographic multiverse interpretation of Raphael Bousso. Remember that we have to compactify Klein’s modular curve to come to a picture similar to that of Escher’s devil and angel. This is the hyperbolic structure well known from hyperbolic geometry. To reach the boundary we could walk for ever. If there is something like an outside observer he will find that we are nearing the boundary but becoming slower and slower and never ever reach the boundary which lies at infinity. Thus although we have a finite area, we have infinite distance to the boundary. If we identify this infinite distance with infinite time then our theory becomes identical to that of Bousso. The famous chaos scientist Otto Rössler compared the situation to a pseudo sphere of a certain cosmological model. In other words, Bousso’s theory, probably unknowingly, adopts words for word our theory and there is a one to one correspondence between our terminology and the new terminology. E-infinity is a multiverse theory. It always was and it will always be. You see we are at the cutting edge of everything in theoretical physics. In addition we can calculate things and not only philosophize about it. That is why some find El Naschie more than irritating and are extremely upset that we have been supporting him because quite honestly, without this help, he could not have achieved anything. In fact without our moral support he would not have survived the last operation which he had in London.
We must think about all these things and develop them further and keep each other informed. Let us, following Leo Tolstoy, try to forgive our opponents and wish them peace of their soul so that they can leave us in peace to complete our work.
Best wishes to everyone,
E-Infinity Communication
E-infinity communication No. 2
Why the golden mean?
Dear All,
Encouraged by the balanced and civilized remark of Dr. Munroe we will attempt to update you on anything new in E-infinity which comes to our attention and answer any reasonable well posed question which anyone of you has as far as we can. We hope that arguments and tone will remain within what you would do in a scientific conference or a discussion in a learned journal. By all means you can make the odd witty remark or polite joke. Just consider that you are not anonymous and that you are responsible for what you are saying. Science is a worthwhile and respectable endeavor. Even non religious people do not commit scandalous actions in a house of worship whether it is a church, synagogue, mosque or a budhist temple. Science is a kind of temple for us scientists and even if you do not believe in that, please out of respect to other believers, leave this site for science and let us discuss science without resorting to personal hidden agendas, irrational hate or jealousy and inferiority complexes. If you would love to see a trial you will have a long one in the High Court of London. Here we talk only about science. Thank you for understanding and now I can move to the next item.
I would like to give here a plausibility explanation of why the golden mean popped up in quantum mechanics. To understand our point of view you must know a little bit of nonlinear dynamics. The most important thing which you have to know in that respect is the KAM theorem. For instance you could consult a book by Heinz Georg Schuster called Deterministic Chaos, published by VCH Verlag, 1989 but any other book on nonlinear dynamics would do. KAM is an acrynomn relating to the name Kolomogorov, Arnold and Moser, the three mathematicians who developed it. Loosely speaking it states that the last periodic orbit which can be destroyed by perturbation is the one which has the most irrational winding number. You can think of a winding number as the ratio of two frequencies as in resonance. The more irrational the winding number is, the more stable the orbit is. Since there is nothing more irrational than the golden mean because it is the least well approximated by a rational as you can see from continued fraction expansion it follows then that this orbit is the most stable. The stationary states which can be observed experimentally is therefore connected as close as possible to the golden mean. You can think of elementary particles as a stationary state of quantum mechanics. When you connect quantum mechanics to KAM then you are effectively making Rene Thom’s VAK hypothesis. Rene Thom conjectured that the Hamiltonian quantum mechanics has a form of attraction although it is non-dissipative and conservative. This strange attraction has a vague resemblance to strange attractors of dissipative systems. That is why it was called the vague attractor of Kologomorov by Rene Thom. El Naschie’s work and the subsequent experimental confirmation of the golden mean in quantum mechanics in the Helmholtz Centre is the proof that the VAK conjecture is correct. You see this is all mathematically perfect and correct but of course we are extending mathematics to physics. When ever you extend mathematics to physics you leave the secure ground of absolute logic and enter into the messy realm of reality. But that is why theoretical physics is for us far more interesting than pure mathematics. In theoretical physics you need not only maths but something more difficult to pin down given to man by God if I may say so and this thing is called intuition. That is the reason why the golden mean will keep coming out in every measurement of quantum mechanical phenomena. Thank you for your patience and we await your questions if you have any before we move to the next point.
E-infinity communication.
E-Infinity Communication No. 3
Further reasons why the golden mean?
Dear Ray
There may be a slight misunderstanding here. Irrationality of Phi is expressed by the fact that the fraction expansion involves only unity. In other words this is 1 divided by 1 plus 1 and 1 is divided again by 1 plus 1 and so on indefinitely. At infinity the result is the golden mean. Some call the inverse of the golden mean the golden mean. This is only semantics and totally irrelevant. We should also not loose sight of what we want to talk about. Whether the Fibonacci progression is more fundamental or the golden mean may be a question of interest to a number of theoreticians involved in a learned discourse. It is also irrelevant that John Baez had a vested interest to draw attention to the work involving the golden mean published by a Russian scientist in a far more restrictive area as compared to the fundamental generalization of E-Infinity. Important are only the facts that a fundamental theorem about stationary states relates elementary particles to the golden mean. The theorem is the VAK which as we said an extension of KAM to quantum mechanics. The experimental verification is a fact. Scientists engage in an honest historical analysis of science will show at some time who was first and who was not. Now let me go back to the fundamental question of why the golden mean?
A Slovenian scientist and mathematician following Mohamed El Naschie expand the idea of mechanical oscillators. Many papers have been published on this subject by L. Marek-Crnjac. Take a two degree of freedom oscillator. Two masses connected by two linear springs. Write the equation of motion. Set the value for the masses as well as the spring constants equal unity. The secular equation is then simply a quadratic equation. The Eigen values are golden mean related. The only positive real Eigen value is the golden mean. Imagine now that you have infinitely many such oscillators connected together. Consequently you can estimate the Eigen value using two well known theorems on Eigen values. These are the Southwell theorem and the Dunkerly theorem. They correspond to what we have studied in school about joining electrical resistance of Ome’s law. When they are successive you add the inverses and when they are parallel you add them. Eigen values are frequencies. Frequencies are energy and energy is mass. Extrapolating the whole thing to quantum mechanics as argued by El Naschie and Marek-Crnjac you have another plausibility explanation for why the golden mean will pop up in any accurate measurement in quantum mechanics phenomenon.
You can see all this theorem in any good book on Mechanical Vibration. There are of course many other ways to argue the appearance of the golden mean which I will discuss next as soon as you have made your comment.
Best regards,
E-Infinity communication
E-Infinity Communication No 4
Mathematical reason for the golden mean in quantum physics
Dear Ray
You are right. But like all of us you are right to a point. We and science exist because philosophy exists. We lose track of things for the same reason. Do not fall in love in general in eloquent formulations no matter how beautiful a sentence is. Reality is indeed fractal. But then you can lose track of reality when you gloss over. You do that when your observation is inaccurate. Your observation ergo reality is as good as the resolution of your instrument. The whole idea of random cantor set as a building block of spacetime is that it is the reality which we were not able to observe directly until now. But it is there. For the first time it manifested itself in quantum mechanics indirectly through the golden mean. Let me first give you some more demanding reasons why the golden mean is there.
We will have to make a jump. It is not systematic. It does not follow from what I said in previous communications. Please accept it for now on its face value. The most fundamental thing which we have for the whole shebang is Kline modular curve in the compactified version. I am not sure of this great man if this great man Roger Penrose knew that he rediscovered the same thing from a far more fundamental viewpoint as far as quantum mechanics is concerned. The entire world of quantum mechanics is encoded in Penrose universe. I know that theoretical physicists can be quite impatient with Penrose and secretly they curse him. I know that his twistor program has come to a halt. And he sometimes said something to the effect that it has failed. But that does not apply to his fractal tiling. Not at all. There is another mathematician who is interested in physics and who is truly an exceptional man. Not as general as Penrose but he runs in the same direction. The work of Alain Connes on non-commutative geometry is best illustrated by Penrose Universe. The work of both men derives its essence from Von Neumann Continuous Geometry. For the sake of this communication, I will refer to page, verse and chapter of Alain Connes Noncommutative Geometry published by Academic Press, copyright 1994. Please look at page 89. Examine figure II.3 entitled Penrose Tilings. Move to page 90. When you read the second paragraph your mind will lit. He says x is a quantum space and then he says that the entire thing is described as tiling. Then he gives an unheard of elementary equation which he calls dimensional function. The dimensional function is z plus the inverse golden mean multiplied by z. It was El Naschie who noticed the profound meaning of this equation and realized that applying some elementary matrix analysis to it, he obtained his bijection formula. You recall that the bijection formula relates the Hausdorff dimension for an n topological space to the backbone or Hausdorff dimension of a zero dimensional space. It is difficult to explain these things without mathematics on this blog. Let me give you a small example: a random cantor set has a golden mean as a Hausdorff dimension. The Menger Uhryson dimension which is nothing but an extended topological dimension for this set is zero. To find the dimension for n = 4, we take the inverse golden mean to the power of 4 – 1 equal three. This means the Hausdorff dimension is 4.23606799 or 4 plus the golden mean to the power of three. You met this famous formula before. But this time we are driving it from the work of Alain Connes. The same thing can be done using von Neumann’s Continuous Geometry which is the basis of Alain Connes work. However nothing can rival the simplicity of Penrose Fractal Tiling. This fractal tiling is the holographic boundary of the theory of everything. And now comes the unexpected expected result. It is impossible to have a Penrose Universe without golden mean proportionality. Penrose kit and dart inside kit and dart inside kit and dart and so on indefinitely could not be designed without the golden mean. In terms of the mathematics of Hamiltonian system, we say we could not have smooth tiling without gaps or overlapping unless we have golden mean proportionalities. Please read the elementary but fascinatingly beautiful Penrose Tilings. Penrose did not stop at that. He has driven a fantastic formula called the isomorphic length. The isomorphic lens was popularized by Martin Gardner. Believe it or not, when you multiply this length by 2 what do you get? You get exactly 4.23606799. In other words you get half of the Hausdorff dimension of the expectation value of El Naschie E-Infinity space. Remember this Hausdorff dimension is found in the most elementary fashion by putting a 4 dimensional cube inside another dimensional cube and so on indefinitely. To get the value you write a continued fraction. That is 4 plus one over four plus one over four and so on indefinitely in the familiar fashion and the final result after infinitely many iteration is 4 plus the golden mean to the power of 3. I will repeat again, this is double as much as we have with a Penrose isomorphic length. The isomorphic length is a wonder which is not a wonder. You stand anywhere in a Penrose universe holographic boundary. You look around yourself and see the world. You close your eyes and move a distance. At a distance not larger than the isomorphic length, you open your eyes again and you will think you have not moved at all. You have a recurrence of the whole universe around you giving you the illusion you have not moved. That is where infinity and finiteness become exchangeable. That is what Mohamed El Naschie observed and used. All very simple. This is the wonder of the hyperbolic geometry in compactified form. You have used related ideas in driving your E12 Exceptional Lie group. I have looked at your paper in Chaos, Solitons & Fractals. I have seen some of your figures. This is all thanks to the work of von Neumann’s noncommutative algebra and Alain Connes noncommutative geometry and Penrose incredible geometrical intuition and knowledge. Mohamed El Naschie, the engineer gave all these subjects the time they needed to digest and reproduce them in his own language in what we call today E-Infinity. I know this communication is far more complex than the previous one. There may be many gaps in my explanation. But I will come back to everything once again and my next communication will be far more comprehensive and far more elementary. But I would like to draw your attention to the last informal review which Mohamed El Naschie wrote on the subject. It is called: The theory of Cantorian spacetime and high energy physics (an informal review), published in Chaos, Solitons & Fractals 41(2009) 2635-2646. As for the golden mean and the symplictic character of quantum vacuum, I advise you to read a short and very simple and beautiful paper by Mohamed El Naschie relating the whole thing to the Banach-Tarski paradox and the no squeezing theorem. The paper is titled: New elementary particles as a possible product of a disintegrating symplictic vacuum published in Chaos, Solitons & Fractals 20(2004) 905-913.
I hope I was of some help and I advise to experiment yourself by doing some elementary calculations yourself. E-Infinity is hands on mathematics. We bring everything we know to bear on the problem. We start from absolutely abstract mathematics and go down without any problem to numerics as well as plausibility explanation. We need all the help we can get. Anything goes as long as it is logical and helpful to our understanding.
E-Infinity communication
E-infinity communication No. 5
Deriving the basic equations of E-infinity
Part I – Elementary introduction
Ignoring nonsense and concentrating on science I would like to explain how to derive the basic dimensional equations of E-infinity. I will start for simplicity with what is more familiar to people not acquainted with E-infinity as a starting point. Later on we will give the accurate, exact derivation and finally we will look into the correspondence to noncommutative geometry and other mathematical theory.
Dimension is the most important topological invariant we have. It is more fundamental than the Euler number or what have you. We start with topology to construct what Wheeler rather loosely calls pre-geometry. We imagine an infinite number of Cantor sets to be joined together statistically to form a pre-geometry and we use for this purpose a Gausian-like distribution which is called gamma distribution. There is some contradiction here because we are joining discrete sets but we use continuous distribution. Later on we will introduce the correct discrete gamma distribution. OK. What we are distributing now is not the sets but the Hausdorff dimension of the sets. We distribute them like Wheeler’s bucket of dust. You can check that in the literature. What is the expectation value of a Hausdorff dimension for the whole collection of this dust? Wheeler calls it Borel mix. I do not think Wheeler had exactly a Borel set in mind, but never mind. To find this average or expectation dimension you have to know the multiplier lambda and a shape factor r. The expectation value is given by r divided by the natural logarithm of lambda. A little contemplation will show that lambda must be the inverse golden mean because we are taking only random elementary Cantor sets in our Borel mix. Taking the simplest camel hump shape we see that r must be equal to two. You can find that in any text book on statistics and probability theory for instance the book of Pitman. Divide now two by ℓn the inverse golden mean. This gives you 4.156174. Please do this yourself using a pocket calculator. You will remember the exact value should have been 4.236067977. Well this is the price in accuracy which you have to pay for using the continuous gamma distribution. If you want to have the exact result then you have to get rid of the spurious nonlinear terms in the expansion of the natural logarithm. Mohamed El Naschie pointed out that the linear terms of the expansion correspond to the exact discrete distribution. If you do that and make no mistakes then the two divided by the logarithm of lambda changes to 1 plus the golden mean all divided by 1 minus the golden mean. When you do this you will find the exact solution which is 4.236067977.
Let me show how we get that from first principles exactly. You assume you have a random Cantor set with a golden mean Hausdorff dimension. You add to that all the Cantor sets in the world, that is to say infinitely many, each having a Hausdorff dimension equal to the golden mean to the power of n, where n goes from zero to infinity. This would give you a sum which when summed correctly then it is equal to 1 divided by 1 minus phi, namely the golden mean. To compare that with the initial Cantor set which you started with, you have to divide the whole sum again by phi. The result is the well known expectation value of the Hausdorff dimension of E-infinity. Suppose we do not know that phi is the golden mean. Leave it be just a formula for the expectation value of the Hausdorff dimension. This formula now reads 1 ϕ. Remember we had another formula from the gamma distribution which says (1 + ϕ) . Now comes the most important condition we have. The first dimension is the Hausdorff dimension, namely an expectation value. The second dimension is an expectation value of a topological dimension. To have a space worth the name space you should have no gaps and no overlapping. The requirement for that is that both preceding dimensions should be equal. Equating both formulas you find a quadratic equation for ϕ. Solving this equation you find that ϕ is indeed the golden mean, namely 0.618033989. This is space filling condition which Mohamed El Naschie introduced to derive this formula. Inserting the value for ϕ in any of the two formulas you always find that the dimension is equal to 4.236067977 as should be.
You can find the detailed derivation in many papers of El Naschie as well as in the work of Marek-Crnjac and others. We see that E-infinity pre-geometry is described by more than one dimension. We have a topological dimension on the average which is equal to the Hausdorff dimension on the average equal to 4 + the golden mean to the power of 3. In addition you have exactly four dimensional topologically speaking. However formally you have been adding infinitely many Cantor sets so that you really have infinitely many dimensions. To understand that the topological dimension is exactly four, I have to refer you to the bijection formula which is connecting El Naschie’s work with noncommutative geometry. In this particular case the formula says that the correction dimension is obtained from raising the inverse golden mean to the power of n minus one. To get the correct result, namely 4.236067977 you have to have n equals 4. When n is equal n then n minus one is equal 3 and hey presto, you see that the inverse golden mean to the power of 3 gives indeed the correct expected dimension, namely our familiar 4.236067977. I urge every reader to do it himself using the pocket calculator. Working yourself through all these little calculations you will start having a feel for the golden mean symphony which is governing quantum mechanics and high energy physics. In the next communication we will go deeper into all the subjects once more from a higher view point. This was just an elementary taste of what it is all about.
E-infinity communication No. 6
Derivation of the fundamental equations of E-infinity Part II
As Confuscious says, ignore perturbation and like Mohamed says, may peace be upon all of you. Let us continue with our scientific discussion.
We would like to tie up some loose ends and unintentional omission of some important aspects of the main equation as discussed so far in communication No. 5.
El Naschie draws attention to an important interpretation of the approximate formula of the expectation value of the gamma distribution of E-infinity. You recall that it was 2 divided by ln lambda. Taking lambda to be the inverse golden mean then ln lambda will have an important interpretation in terms of an entropy which mathematicians call topological entropy. The topological entropy for a certain map turns out ln 1.6180333 equal 0.481212. If you interpret the 2 as being the Hausdorff dimension of a quantum path following Abbot and Wise famous paper, then dividing 2 by this topological entropy is giving you the entropic content of a quantum path. In other words the approximate solution for the expectation value of the Hausdorff dimension, namely 4.156174 is effectively an entropic content. We see the important interplay between different notions which are not different at all when seen from a higher level. Those familiar with string theory and Brane theory may find an analogy for this in the fact that P-Brane and D-Brane which were thought to be different are in fact not different at all. The difference is in the eye of the beholder.
In this context we should apply the same interpretation but this time using a different lambda of a different map. The value of this lambda is itself our 2. Consequently we have 2 divided by ln2. This is a different entropic content using again a topological entropy definition. The value in this case is 2.885390. Try to remember this for later use because this value approximates the missing part needed to compactify chi 7. I mean the part you need in order to make 336 come nearer to 339 by compactification. Of course I am running ahead of my story but please remember this bit for later on.
Now let me give you the exact E-infinity derivation of our 4.23606799 dimensions which we can obtain without making any reference to anything except the pre-geometry we are constructing and the center of gravity theorem of probability theory. Let us take all integers from 0 to infinity. Let us regard this as infinite dimensions. Now we give each dimension a weight. This weight is equal to the golden mean to the power of n where n runs from 0 to infinity. What then is the center of gravity of the whole thing? This is a very simple problem in probability theory which mechanical engineers like to think of as searching for the resultant of many forces. It is now very easy to guess that the resultant which acts at the center of gravity will be given by the sum from 0 to infinity of n multiplied with the golden mean to the power of n and all multiplied with n again, where the last n is now the arm of the force while the force is n multiplied with the golden mean to the power of n. You have to divide this now by the sum of all forces which is sum from 0 to infinity of n multiplied with the golden mean to the power of n. In other words you have a sum of moments divided by the sum of forces. This gives you the expectation value of the average arm of the resultant. This arm is the expectation value of our dimension of the space made of these infinitely many weighted topological dimensions. These infinite series can be summed exactly as shown by El Naschie and it is an elementary matter to show that the final sum is simply 1 plus the golden mean all divided by minus the golden mean which all comes exactly to the well known and familiar value of 4 plus the golden mean to the power of 3. In sum pure mathematical communication Mohamed El Naschie was able to show that this is a well known result in the theory of the Coxeter groups. You can find this paper in Elsevier’s Science Direct.
In a forthcoming communication we promise to show you that all these results are derivable from the theory of subfactors as well as Alan Connes’ noncommutative geometry.
Now to close this 6th communication I would like to give you the exact derivation of the dimension of the holographic boundary of E8 E8 in the case of complete compactification. You recall that we started with 336. Then we keep adding triangles indefinitely in a hyperbolic way. To understand how to find a finite value for this compactification to infinity, you have to be familiar with the theory of Klein’s modular curve. You have to know that there is an important orbit with 42 points in the 336 curve. This 42 can be thought of as 4 and 0.2 lifted to 10 dimensions by taking 10 copies. That is to say it is 4.2 multiplied with 10. A little contemplation will make it evident that 4.2 is just an approximation to 4.23606799. Taking 10 copies of that you get 42.3606799. Now this is the exact coupling constant of unification in the absence of super symmetry. Thus in the ordinary case of non-compactified Klein’s modular curve, you get 336 from 42 multiplied with 8. In the exact transfinite case you get the correct result by multiplying 8 with 42.3606799. When you do that you get 338.885438. This is exactly 2.885438. You see now how close this is to what I told you earlier, namely 2 divided by ln2 equals 2.885390. Very close but not completely correct. The exact expression is well known from the theory of transfinite corrections as developed by Mohamed El Naschie following a similar theory dealing with operators due to Fritz John. The exact expression is 16k. Here k is equal to the golden mean to the power of 3 multiplied with 1 minus the golden mean to the power of 3. When you do that you find that it is 0.180340. Now you take 16 copies of it and you have your exact result 2.885438. Now let us derive the exact theoretical inverse electromagnetic constant using Mohamed El Naschie’s fundamental equation and the previous exact transfinite value. The number connected with Einstein’s gravity Reimannian tensor will remain 20. However the dimension of E8 is no longer exactly 248 but 247.983739. This is 248 minus k square divided by 2. Now our equation should read E8 E8 minus holographic boundary minus Einstein gravity equal inverse electromagnetic constant. Taking the appropriate dimension we have 495.967478 minus 338.885438 minus 20 equals 137.082039325. This is the exact theoretical value as promised. It is the integer value 137 plus a transfinite correction equal to k o where this new constant is equal to the golden mean power 5 multiplied with 1 minus the golden mean power 5 equals 0.082039325. You see the extent of our precision and how the constants of nature are obtained as a probability resulting from summing over infinitely many states but I think it is enough for now and promise to come back in our next communication in due course.
E-infinity communication No. 7
Derivation of the fundamental equations of E-infinity Part III
For the benefit of those who appreciate scientific discussion as opposed to defamatory allegations and those who would like to learn something about E-infinity as opposed to hearsay and parrot repetitions of misconceptions, we continue our discussion by giving literature which we omitted to mention in detail in Communication No. 6.
Pre-geometry and the main idea of Borel mix as used by Wheeler can be found on page 1205 of his monumental book Gravitation, by Charles W. Misner, Kip S. Thorne and John Archibald Wheeler, published by Freeman, New York 1973. You can find everything about gamma distribution in the book Probability by Jim Pitman, published by Springer, Berlin 1993. The expectation value of the gamma and other distributions are conveniently summarized in the table on page 476-477.
Now let me reminisce about Mohamed El Naschie and his lectures when I was just about to finish my Ph.D. He said something very remarkable about the Cantor sets. He said it is something which is there and yet not there. In his youth Mohamed was a friend and follower of Jean Paul Sartre. He was greatly influenced by Sartre’s major book Being and Nothingness (original title L’Etre et le neant: Essai d’ontologie phenomenologique). Mohamed read the book in the German translation, having German as quasi his mother tongue (Das Sein und das Nichts). He was even acquainted with it at the tender age of 12 in the Arabic translation by Abdul Rahman Badawi, the famous Egyptian philosopher at that time. In fact Mohamed’s father who was an army general was quite concerned and worried that his young son is reading these things of which he could not make out head or tail. Education is never in vain, even if it is philosophy. The Cantor set made quite an impression on Mohamed and he instantly connected it with what he had read in Sartre’s book. He even mentioned the work of Sartre as well as Martin Heidegger in one of his most profound very early papers on the subject. This particular paper was cited by many pure mathematicians thinking that Mohamed is a pure mathematician. In his review of E-infinity theory Mohamed likened the Hausdorff dimension of the Cantor set with the spirit of a body which has long decayed. He likened it with this famous ghost story by Oscar Wilde. This particular review which is highly cited and although it is only five years old and despite all the negative propaganda artificially created in certain quarters, A Review of E-infinity Theory is found on Google Scholar to have been cited 328 times.
I recall something similar which was said by Leibnitz about imaginary numbers. Leibnitz said there are amphibian between being and not being. This is a remarkable formulation which covers almost exactly Mohamed’s notion of a Cantor set. It is an amphibian between being and nothingness to use the terminology which he borrowed from Jean Paul Sartre. There is no shame in borrowing things from famous as well as lesser mortals. What is shameful is to attempt to hijack other people’s ideas and deliberately calling them your own.
Now I would like to go back to the derivation of the fundamental equations. I must warn from the outset however from a major mistake which some could easily commit because they judge in haste. There is no element in Mohamed El Naschie’s work which is numerology. We agree fully with what Dr. Ray Munroe said on certain blogs that numerology is a pattern of numbers for which we have not yet discovered the real underlying reality, physics or theory. However nothing is further from the truth when it comes to E-infinity theory. If you still suffer from this delusion then you have not yet understood the theory. That would be a real pity because the theory is not difficult at all when you free yourself from all prejudice.
Now we give yet another derivation of the expectation value of the Hausdorff dimension. You have to start with the well known dimension of noncommutative geometry. The dimension is given as index or dim of N and M. It is written however as [M:N]. This is given by 1 divided by L plus 1 divided by 1 minus L. Here L is trace of E. From the theory of subfactors we have a very similar formula but this time giving [M:N] by 1 divided 2d-1. Now set L equal d and you will find a quadratic equation for d or L showing that it is equal to the golden mean or minus the inverse golden mean. For the positive value we get our 4 plus the golden mean to the power of 3. I leave it to you to have fun with Alain Connes and the great mathematician of the theory of subfactors. I leave it to you now to work out for yourself the four dimensional fusion algebra which has 1, 1, golden mean and golden mean again as a solution. It sounds like a puzzle but the next communication will make it clearer. Let me end in the same way that some of the rather pleasant commenters on the Munroe blog end their comments by saying, have fun and I add try to ignore perturbation.
E-infinity
E-infinity communication No. 8
The E-infinity counterpart of calculus – and Wyle scaling
It is my intention to start by making specific reference to the literature concerning the last communication which was omitted. The important noncommutative dimension which we used was given in detail in Connes marvelous book Noncommutative Geometry, Academic Press, 1994. The formula is given on page 506. He uses slightly different notation but everything is explained lucidly. You can understand subfactors from the wonderful book by V. Jones and V.S. Sunder called Introduction to Subfactors by Cambridge University Press, 1997. You may recall that Jones is the man who found the remarkable relation between knot theory and statistical mechanics. Much of what Mohamed El Naschie did could be reinterpreted in terms of Jones’ work. The relevant formula may be found on page 31 of this book. Several other important formulas may be found on page 143 and you can move from there to study how quantum field theory can be derived from subfactors. Believe me it all sounds far more complex than it is. Oh and there is an important paper by Mohamed El Naschie which has much of this stuff published in the Int. Journal of Theoretical Physics, Vol. 37, No. 12, 1998 entitled Superstrings, Knots and Noncommutative Geometry in E-infinity Space. The Editor in Chief of this journal at the time was no one less than the legendary David Finkelstein. David you will recall is one of the main people responsible for introducing set theory in quantum mechanics. Following Heisenberg and Finkelstein, Prof. Heinrich Saller excelled in developing a highly mathematical theory starting from a combination of symmetry groups and set theory. We warn however that this is very demanding mathematically for the average physicist. However if you would at least have a glance at the work of Saller, you will see high energy physics and quantum mechanics from a profoundly important and different view point which is not that dissimilar from that of what El Naschie is doing using what is comparatively humble mathematical tools.
Some of our members think I should move now to explaining the main tools of computation which we have in E-infinity theory. This would be Wyle scale. Other members think that we should no introduce the transfinite theory of dimensions which is the mathematical founding of E-infinity theory. A sizeable minority think we should outline the connection to wild topology and a few other disciplines used frequently in E-infinity theory. We will have to do all of that in the few coming communications. However I am inclined to think that it is time to introduce at least one simple example of the scaling analysis.
If you look to the beginning of the work of Nottale you realize that this eminent and great French scientist agonized about differentiation. He was slightly on the loose side when he equated fractals with non-differentiability. That brought him into conflict with fractal experts like the great Israeli scientist Itamar Proccacia. However the dispute is merely a misunderstanding when temperament takes over reasoning. Nottale like Ord and later on El Naschie realized the need for calculus machinery but he also realized that differentiation and similarly integration covers up all the interesting phenomena and leads us in the wrong direction. Ord in particular realized that taking the limit is the source of all contradictions in quantum mechanics. That is how Nottale decided for a compromise, albeit an ingenious one. Nottale does not give up continuity. For him a Cantor set is excluded a priori as the basis of the calculation. Therefore he used non-standard analysis in his calculations. That is how he came to most of his excellent results. Garnet Ord somehow managed to avoid taking the limit of his difference equations. In a manner of speech and in a way which Ord does not particularly like, you could say that he invented his own quantum calculus. Of course Garnet sees it completely different nowadays and he has grown more sophisticated about these things but for the purposes of this communication, it is sufficient to think of it in this way.
In noncommutative geometry Alain Connes was faced with similar problems like the pioneers of E-infinity. Being one of the greatest pure and applied mathematicians in the history of mathematics he had of course a sophisticated solution. Without going into the detail, let me give you a very short dictionary of the noncommutative solution. Alain Connes introduced a quantized calculus for which the following short dictionary applies. First infinitesimal is replaced by compact operators. Second integral is replaced by a Dixmier trace which is incidentally also used by El Naschie. The table for classical quantum correspondence in noncommutative geometry is given on page 20 of Alain Connes book which we mentioned earlier.
El Naschie as well as Ji-Huan He and Marek-Crnjac and occasionally Goldfain use something else. They return to Wyle scaling. Fractals has a marvelous character expressed in Bidenharn conjecture. The conjecture says the obvious that there are no a priori scales, man or God given, in a fractal spacetime. Therefore the counter argument of Einstein against Wyle’s idea does not apply in a fractal spacetime. The history of the whole controversy and its solution in E-infinity theory is duly explained in several papers by Mohamed El Naschie as well as a review by L. Marek-Crnjac. I will give you the exact reference later on. The final result is disarmingly simple. You are more or less scaling down when you want to differentiate and scaling up when you want to integrate. The words differentiate and integrate should not be now taken literally. Prof. Ji-Huan He likes to say that scaling is everything. In E-infinity at least, this is as near to the truth as anything could be. Let us discuss one example demonstrating the application of this idea and generating El Naschie’s hierarchy of Heterotic strings. You recall in classical mechanics when you want the equation of equilibrium you write a Lagrangian, then the vanishing of the first variation of this Lagrangian gives you the equation of equilibrium or motion. In very simple cases when the Lagrangian is a function rather than a functional, variation is replaced by simple differentiation. In E-infinity things are far simpler than that to the extent that some who expect to lift heavy weights are shocked and left in a state of disbelief because of the simplicity which they did not expect. For certain manifolds involving E-infinity the curvature may be given by very simple expressions. Squaring the curvature you find a normalized energy. If you can identify a certain parameter as a loading and you can calculate the distance which this loading can potentially move, then you have an equilibrium equation. Alternatively if somehow you convince yourself that you have what is equivalent to a Lagrangian, then repeated scaling gives you the answer for equilibrium equation or equation of motion. Suppose I convince you that half of the inverse of the electromagnetic fine structure constant may be regarded physically and correctly as the numerical value of the Lagrangian. In this case repeated scaling using the golden mean will give you the solution of corresponding equilibrium equation or equation of motion. Again this sounds far more complex than when you really do it. Let us do it.
Half alpha bar is equal to 137.082039325 divided by two equal to 68.541020. Multiply this now repeatedly with the golden mean 0.618033989.
The result is then the following hierarchy 42.360680, 26.180340, 16.180340, 10, 6.180340, 3.819660 and so on. This is the well known Heterotic string hierarchy in the transfinite form. The first value is the inverse coupling constant for non-super symmetric unification of all fundamental forces. The second is the coupling constant for super symmetry or the number of bosonic strings. The 16.180340 relates to the additional dimensions added. The 10 are the super string dimensions. The 6.180340 are compactified dimensions and the final number is 4 minus k where 4 are the dimensions of spacetime and k is equal 0.18034 which appears in all other numbers. The explanation of all that is given in detail somewhere else but obtaining the result is less than elementary as far as computation is concerned. Let me show you first how we obtain the original equation. El Naschie showed long ago that the average curvature of Cantorian spacetime at the core is equal to 26.18033989. Squaring this you get the energy in normalized form which is 685.410197. If you remember, this is the dimension of Ray Munroe’s E12. Not quite but very near. It is also very close to the sum of the dimension of all the 17 two and three Stein spaces. Not quite but very near. Introducing a loading lambda index I, then we can define a potential distance equal to the square root of the sum of all theoretical values of the coupling involved in reconstructing the inverse electromagnetic fine structure constant. This is 60, 30, 8 + 1 = 9, and the quantum gravity coupling 1. Adding altogether comes to exactly 100. The square root is therefore 10. Lambda I is therefore equal to 685.410197 divided by 10. This gives us exactly the value we started with namely 68.5410197. When you scale it you get lambda 1 equal to 42.360680, lambda 2 equal to 26.180340 and so on. This is extremely simple isn’t it? It is simple but difficult to understand. It is only difficult to understand because of our habits of thinking which we do not want to get rid of easily. Remember we must free ourselves from all ties as Prof. M. B. once said. I am sure you have hundreds of questions. I can assure you if you are not shocked and if you have no questions, then could not possibly have understood anything.
Best regards,
E-infinity communication No. 9
Wyle scaling and deriving the spectral dimension 4.02 of Loll, Ambjorn and Jurkiewicz using E-infinity
There is no reason why we should not continue with our discussion and give further examples of the use of golden Wyle scaling in E-infinity. We have to take the opportunity first to draw the attention of the readers to the literature where details are given. The most important three papers which can be consulted on this subject are the following: From classical gauge theory back to Weyl scaling via E-infinity spacetime by M. S. El Naschie, Chaos, Solitons & Fractals (CS&F), 38, 2008, p. 980, A Feynman path integral-like method for deriving the four dimensionality of spacetime from first principles by L. Marek-Crnjac, CS&F, 41, 2009, p. 2471 and Density manifolds, geometric measures and high-energy physics in transfinite dimensions by S.I. Nada, CS&F, 42, 2009, p. 1539.
To explain the main idea in the most excessively simplistic terms you could say the following. There is a fundamental difference between changing direction in space and stretching a line in space. In the first case nothing really physical happened while in the second case something almost physical happened. That was the crux of Einstein’s objection against Wyle’s gauge theory. Lacking a natural scale, in a fractal setting a stretching can be set on the same footing as changing an angle. You read the rest please in the relevant literature. By the way many people know this trick and that is why fractal spacetime is becoming fashionable and will become even more fashionable as time goes by, so you can play it again Sam.
Let me now return to examples of Wyle scaling and we can do two things for the price of one. We give another example and derive the spectral dimension given for instance in the paper of Ambjorn, Jurkiewicz and Loll in Scientific American or the improved version they published in a book Edited by Daniele Oriti entitled Approaches to Quantum Gravity, published by Cambridge, 2009 a few years after the first derivation by El Naschie using Bose Einstein statistics. The excellent paper by the three authors who are world renowned for simplictic triangulation, in other words, tiling the space with simplexes which again means Regge calculus or finite element of John Argyris who was one of Mohamed El Naschie’s teachers in Stuttgart, Germany and Imperial College, London is entitled Quantum Gravity: the art of building spacetime on page 341-359. The important formula is on page 352.
Let’s begin at the beginning. In Heterotic string theory anomaly cancellation requires either O(32) or E8 E8. In this theory there are left and right moving sectors. The left is purely bosonic. We start with 26 dimensional strings where 16 have been compactified on a lattice, for instance E8 E8 lattice or spin 32 divided by Z2 lattice. The right movers on the other hand are super symmetric. Thus the right movers are explicitly spacetime symmetric. Recall that spacetime symmetry of super space is found until this moment by trial and error. Now in the right moving sector at the lowest level we have 8 plus 8 equals 16 states. In the left moving sector we have three distinct objects leading to 8 states plus 16 states plus 480 states. This makes altogether 504 states. To obtain the entire spectrum we follow Fock space rules of quantum field theory and multiply left movers with right movers. That way we obtain from 504 times 16 the well known 8064 states. This is a very well known result. You can find this result in popular literature like Scientific American or text books on string theory like Kaku’s book. Knowing that there are truly childish people hanging around on internet blogs with an enormous chips on their shoulders, we would like to give you the literature to check everything yourself so that these children do not write that we are telling you fibs. OK, here they are. Scientific American, Superstrings by M.B. Green, p. 183-203, in the caption of Figure 11.14. Reprinted in a book called Particles and Forces, Editor Richard A. Carrigan, published by W. Freeman, New York (1990). M. Kaku’s book Introduction to Superstrings and M-theory, published by Springer, 1999, see page 384. Now enter Mohamed El Naschie. He realized that the same result is easily obtained in an entirely different manner and in some respects a far simpler geometrical way by assigning the right amount of instantons to each site of Klein’s modular curve as a holographic boundary of string theory. Without going into details the instanton number in this case is 24. The number of triangles as you know from earlier communications is 336. Multiplying the two numbers you get the total number of instantons, namely 8064. Mohamed was under the false impression that this must be a well known method. He published it and mentioned the whole thing just as a marginal thing. He published it a few times without paying too much attention to any novelty. It was then after heated discussion that Nobel laureate Gerard ‘tHooft convinced him that it is an entirely new theory. I forgot to mention some of the extensions and the insight which El Naschie gave to this point also without realizing that it is an entirely new theory. First look at 504. This was realized by Mohamed as the summing of an exceptional Lie group. Here is a sum about which some people who are ignorant about the exceptional Lie group hierarchy rejected off hand and of course wrongly so. Add the dimensions of E8, E7, E6 and E5 together then you will have 248 plus 133 plus 78 plus 45 equals exactly 504. What most people did not know is the fact that E5 is nothing else but SO(10) of grand unification which is competing with SU(5). SO(10) is really an E5 based on its Dykin diagram. This fact was known to El Naschie as well as many other careful people which do not include of course John Baez who had too much to do on his shows on his diverse blogs for entertaining people. It was of course a mistake to stop at E5. In mathematics it would mean you are not using a complete set. To use a complete set you have to take all the exceptional groups. The sum of that as everyone knows in the meantime was found by El Naschie to be 4 times 137 equals 548. Similar reasoning would show that classical Heterotic strings are only approximation and the real numbers of massless states is not 8064 but 8872 when we ignore everything after the dot. Now El Naschie did not calculate it in this way. He reasoned differently using the holographic boundary. Taking compactification into account you do not have 336 triangles but 338.885438 weighted number of triangles. In addition he derived the exact instanton density and found that it is not 24 but exactly 26.18033989. Multiplying both exact numbers you find 8872.135951. Wonderful. This is now our numerical Lagrangian or potential or whatever you want to call it. Sixteen times differentiation is an E-infinity equal to sixteen times scaling with the golden mean. Multiplying our Lagrangian with the golden mean to the power of sixteen, you get 4.01999999 on a pocket calculator. This is Loll, Ambjorn and co’s result. For all practical considerations it is 4.02. Please do this calculation yourself.
It is interesting to ask why the pioneer of the holographic boundary did not find this result first. I do not know but we cannot all work on everything simultaneously. A reasonable explanation may be the following. The expert on the holographic boundary did not care about the 336 because they know once they compactify we get infinity and physicists do not like infinity and do not work with it. Mohamed El Naschie on the other hand took a gamma distribution weighted infinitely compactified Klein modular curve which added approximately 3 to the 336 to get approximately 339 which is a finite result and meaningful. He was also able to deal with a fuzzy K3 manifold and find the instanton number to change from the classical 24 to 26.18033. This is incidentally equal to the dimension of transfinite Heterotic strings as well as the corresponding Euler constant as well as the curvature which we discussed in an earlier communication. 26.18033 turned out to be an extremely important number. People interested in number theory knew that much earlier but did not know about any relevance in physics. Nobel laureate David Gross must be credited with super natural intuition to have invented Heterotic string theory. Witten and his friends were probably the first to introduce K3 to physics. So you can see string theory is not in any way as useless as some of its opponents want us to believe. Nothing is useless except underestimating people. Nothing is as harmful as belittling the achievements of other people. Nothing is as revolting as the yellow jokes of envious souls who ceaselessly inundate us with their superfluous comments, polluting the blogosphere with their inferiority complexes. You can find explicit calculations of the spectral dimension in several papers recently published in Chaos, Solitons & Fractals. For instance From Menger-Urysohn to Hausdorff dimensions in high energy physics by G. Iovane, 42, 2009, p. 2338 and On the Menger-Urysohn theory of Cantorian manifolds and transfinite dimensions in physics by Guo-Cheng Wu and Ji-Huan He, 42, 2009, p. 781. Also A Feynman path integral-like method for deriving the four dimensionality of spacetime from first principles by L. Marek-Crnjac, 41, 2008, p. 2471 and Density manifolds, geometric measures and high-energy physics in transfinite dimensions by S.I. Nada, 42, 2009, p. 1539.
I think by now anyone who has attentively read the last nine communications must be able to derive on his own the basic and fundamental dimensions of fractal spacetime, namely the topological dimension 4, the Hausdorff dimension 4 plus the golden mean to the power of 3, the scaling dimension 4 minus k equal to the golden mean square to the power of 10 and finally the spectral dimension which is 4.02. The work of Loll and her crew is remarkable in that they got the exact result without having the complete theory. They were right to think that only a computer can get this result right. I say only a computer or E-infinity theory and nothing in between. They may have wrongly led people to think that 4.02 is a Hausdorff dimension or a topological dimension for quantum spacetime. It is not. It is an exact dimension for the spectral dimension of quantum spacetime. In a sense this is an even more important dimension because it measures how spacetime unfolds. It looks at it as a dynamical process, not as a statical process. It looks at it like ink spreading in water. They should be commended for their handiwork. What a pity that they were not aware of all what we have done in the last 15 years using E-infinity theory and fractal spacetime and what a pity that people can sometimes be so individualistic that they cannot see the achievements of others. Cooperation is always much better than meaningless struggle against others. At the end of the day, at least in E-infinity, we have benefitted from all the sound ideas of well developed careful theories like string theory, loop quantum mechanics, holographic principles, Penrose tiling, noncommutative geometry and so on and so forth. I sincerely hope you agree with us, at least on this.
E-infinity communication 10
Continuation of previous discussion and comments
The best thing to start this section is to repeat the potent comments which were lodged on this very same blog by Anonymous. We agree with all that is said in this comment but would like to amplify it. Let us start by repeating here the comments of the respectable Anonymous person and then we will continue.
Start of Anonymous Comment: The comment regarding 137.082039325 is by far the worse. It shows total misunderstanding not of E-infinity only, but of science. It shows that the person in question is mixing between mingling experimental data with mathematics on the one side and the clear distinction between model building, mathematical that is and a fully fledged theory. If I understand ‘tHooft correctly he always laments that leading scientists who develop excellent mathematical models more often than not market it using false labels and sell it as a theory. Surely these are modern times of commercial internet and wheeler dealers, even on the level of prestigious prizes in physics. E-infinity is a theory. It is genuinely a fundamental theory. 137.082…. is from the world of theory. If you read at a minimum El Naschie’s famous review article, A review of E-infinity, then you know the experimental value is obtained by projection from 137.082… to find the almost exact experimental value of 137.03….. El Naschie was at pain to explain the difference and the projection formula is given explicitly. It was given explicitly in two dozen papers by El Naschie, his students and collaborators of which I am aware. You must understand the difference. A theory like string theory in ten dimensions or M-theory in eleven dimensions or F-theory in twelve dimensions or E-infinity in infinite dimensions cannot take any measurement in these higher dimensions. Measurements, whether we like it or not are taken in our 3 + 1 familiar spacetime. It is very doubtful that any man however superior, like Ed Witten, Penrose or Einstein himself could construct a laboratory in five or more dimensions to satisfy the theory through measurement. Of course we then translate everything from theory to experiment. That is why we have things like mathematics, imagination and intuition. To be sure, the ideal E-infinity value has not been postulated. It was deduced. How? Just read the papers. A mother, however kind, can do nothing more than feed the baby. Regardless of how careful and kind she is, and regardless how young and fragile the baby is, her job ends with giving the food on the spoon into his mouth. The baby must digest the food. I swear to God I am not being patronizing, big headed or arrogant. This is not at all my nature. I am a humble almost mediocre mathematician. I was just thrown out of equilibrium by the comment. I was disheartened that after so much, there is such a deep fundamental understanding of basic principles of scientific research. Let me explain to you how I understand El Naschie’s heuristic derivation of 137.082… He gave other mathematical derivations as well as physical derivations. However the present heuristic one is what I find best. It comes out of a heuristic derivation based on the renormalization group equation of quantum field theory. Take the experimental values of the electroweak and round them to a neat integer. It will not take much to find out that the inverse alphas are 60, 30, 9 and unity. In addition you replace the group theoretical fudging factor by the golden mean scaling. Thus the familiar Clebsch is replaced by 1.618033989. Pretty close are everything to the familiar values. Our reconstruction of the inverse electromagnetic fine structure constant becomes in this case the following. 60 multiplied with the inverse golden mean plus 30 plus 9 plus 1 where 1 is the coupling constant of the Planck masses to the Planck Aether. El Naschie says if you do the calculation correction you will find 137.082… comes out like magic. This shows that the correct theoretical values are 60, 30, 9 and 1. The first three were verified experimentally with the accuracy of measurement of our present day facilities at CERN and Fermi Lab. Some got the Nobel Prize for it. The unity requires Planck energy to verify experimentally which is probably for ever outside of our reach, experimentally speaking. However the present calculation and much of El Naschie’s reasoning are compelling proof that this is correct. The whole procedure is referred to in detail in El Naschie’s paper with the telling title Transfinite harmonization by taking the dissonance out of the quantum field symphony. Published in Chaos, Solitons & Fractals, 36, 2008, p. 781. Particle physics may be likened to a magnificent symphony….. We conclude by restating our conviction that nonlinear dynamics, chaos and fractals will revolutionize the way we think about our high energy physics problems. End of comment.
Now to dwell on what exactly is physical and what exactly is mathematical is not at all the important point. It may be important but it is not important for our purposes here. The vital point is that Einstein’s objection against Weyl is torpedoed and sank like the Bismarck in the North Sea once spacetime is a fractal. The fractal spacetime knows nothing called a priori scale. That is the crux of Bidharn’s intelligent observation forming his conjecture. That is why all theories using sophisticated scaling including what is relatively crude or brutal force computer methods using mesh or lattices along the lines of K. Wilson have all been successful in obtaining results reasonably close to the experimental verification. Again there are no experimental values at all for anything related to quantum physics. The whole idea is that every measurement depends on the energy with which you probe spacetime which means your observation, including the dimensionality of your space, are a function of your resolution. I am afraid we have to give up completely our old classical intuition and adopt a new intuition suitable for quantum physics but what am I saying here? This is really well known and the point is the following. People who work with quantum mechanics, including the Copenhagen version, got so familiar with their measurements that they intuitively do the right thing sometimes without knowing. So let me repeat again, the inverse electromagnetic fine structure constant is 137 or 127 or 42 or 26. It all depends on the energy at which you are probing things. The 26 for instance is where all fundamental forces, including electromagnetism, are exactly of the same strength and this is by the way a super symmetric theory. If you are talking about the Planck scale then we know of course that all the difference between everything disappears and you could say that the electromagnetic fine structure constant became unity. But then you never call it in this case the electromagnetic fine structure constant. You call it in this case the Planck mass coupling. So please be careful here. These are pitfalls but they are well known pitfalls.
Let me give you another two derivations of the inverse electromagnetic find structure constant at higher energy. Let us take the energy connected to Ed Witten’s eleven dimensional 5-Brane theory. The number of states, as is well known, is given by 528. This is more than our E8 E8 496. In this case it is easily shown that particle physics will be represented by our familiar 339 of the holographic boundary. However the graviton sector of Einstein is now a graviton sector of Kaluza Klein. I am not very careful with my terminology here. All what I really mean is that my dimensionality is given by D=5 and not D=4 of Einstein. Therefore I do not have 20 independent components in the Riemann tensor but rather 50. Just to remind you, the number of the independent components is given by n square multiplied with open bracket n square minus one close bracket divided by twelve. For n=4 you get 20 and for n-5 you get 50. The correct value can be shown to be 52 for technical reasons and not just 50 because we will have to use an exceptional Lie group called F4 to take care of the gravity section in this case. El Naschie’s fundamental equation now becomes 528 equals electromagnetism plus 339 plus 52. Trivial arithmetic then shows that electromagnetism must be 137 as it should be.
A final exercise for the reader is to derive 137 but this time from the much larger value of the sum over all 8 exceptional Lie groups family Ei. You remember that this is equal to 548. The analysis is very similar but now the gravity section will be represented again differently. It is not simply obtainable from the Riemann tensor nor is it an F4 group. It is represented by the roots of E6 or alternatively or equivalently by the kissing number of spheres in 6 dimensions. The value in this case is 72. Repeating the same elementary calculation of El Naschie’s fundamental equation you will find that the electromagnetic fine structure constant comes out as 137. There are a few subtle points here and subtle questions which I do not want to burden you with here. However I cannot resist the temptation of showing you a remarkable analysis in 11 dimensions where all the fundamental forces including electromagnetism reach the same magnitude of 339. It is extremely elementary to perform the calculation but the picture behind it is far from being trivial. You start by putting a quantum path in 11 dimensions. A quantum path has a Hausdorff dimension of 2. To put it in 11 dimensions you follow Connes or El Naschie’s bijection formula and raise 2 to the power of 11 minus 1. This is equal to 2 to the power of 10 which is equal to 1024. The number of states corresponding to that is 3 times 339. When you add to that the 7 compactified spacetime dimensions, it comes to exactly 1024 and you have from the 11 dimensions only 4 left. These are our spacetime dimensions which we are all familiar with. Next time we would like to discuss the Menger-Urysohn dimensional system and show you the wonder of the vacuum and how E-infinity has a handle on the vacuum which the great super string theory does not have. Until the next time, thank you for your patience.
E-infinity communication 11
Further elucidation and towards the general theory of transfinite dimension
We have a lot of work to do and we hope we can do it today but I doubt that we can seriously start explaining Menger Urysohn theory of transfinite dimension in this communication. Judging by past experience negative dimensions cause physicists a great deal of difficulty but let us start without much ado.
First we have to tie up loose ends from the last communication. In particular what is so special about 11 dimensions? Where did the 11 dimensions crop up for the first time? The first time it was super gravity. This is a theory in where there are nothing but gravitons and their super symmetric partners, gravitinos. Incidentally both have never been observed. Never the less this is a full blooded general theory. For a long time it was considered to be the only game in town, pretty much like super strings nowadays. The next 11 dimensional theory may be the M-theory of Edward Witten. This theory unifies the various competing super string theories but the exact form of this theory is not known. The theory is formulated in 11 dimensions. The 11 dimensions are normally justified in the following way: We need 7 dimensions for SU(3), SU(2) and U(1) of the standard model. In addition we have 4 spacetime dimensions and this gives us 11 dimensions. This is Ed Witten’s standard explanation stripped to the bones. Also from Ed Witten we have a 5-Brane model which requires 11 dimensions. Now the 5-Brane model has 528 states. We encountered this number before. Now it is time to tell you something about maximally symmetric spaces. This is mathematics connected to killing fields. Let us take n to be the dimension of a space, then if the space is maximally symmetric then the number of isometries will be equal to n multiplied with n plus 1 divided by 2. Taking n to be 32 you would have 528 isometries which is equivalent to the number of states in Witten’s model. If you write the model explicitly to obtain 528 you will notice that you have string-like objects as well as 2 and 5-Branes. The combinatorial formula is as follows: You have 11 over 1 plus 11 over 2 and 11 over 5. When you work out the combinatorial correctly you find that this is 11 plus 55 plus 462 making exactly 528. You see there are no zero objects, that is to say there are no points particles in the model. Also the 3 dimensional and the 4 dimensional objects are missing. You see number theory is trying to tell you something which is physical. If you get the hint, as Mohamed El Naschie did, then you will find that 11 over 0 plus 11 over 3 and 11 over 4 gives you exactly 496. Surely you remember this? This is the dimension or isometries of E8 E8. Now a second exercise into the simple addition of two numbers, 528 plus 496 gives 1024. This is exactly what we got when we put a quantum path in 11 dimensions and found the total dimensionality by the bijection formula of E-infinity which tells you that N0 is 2 to the power of the dimension minus 1. This is 2 to the power of 10 equals 1024. I hope at least some of you like it. Number theory is a powerful subject which physicists should take very seriously. A summary of elementary number theory in high energy physics was given by El Naschie and I recommend this paper strongly. It appears in Chaos, Solitons & Fractals, 27, 2006, p. 297. There are a few theorems given in this paper and let me mention here theorem No. 5 on page 310. This theorem shows that the theoretical value of the inverse find structure constant is given by certain expressions which lead to 137.082039325. So as long ago as 2006, which means four years ago, this value was given by a mathematical theorem and there are dozens of other derivations leading to the same. Again particle physics is a symphony. Everything has to work together harmoniously. This harmony principle is what leads to this value. Remember the comments of Sir Roger Penrose about his tiling. He said the tiling has a memory and when you violate the rules, the tiling remembers and takes it revenge when you suddenly find you cannot go on. Words which El Naschie always remembers to say in his lectures goes back to Niels Bohr…. Only in the multitude lies clarity. Believing that this is really the case, let me go on amplifying the preceding discussion a little bit before we turn to another subject. You remember how we got 528 from summing over exceptional Lie groups. It was E8, E7, E6 and E5. This led to the important section formed by 504 in string theory. Now let us add E4. Most people who do not know E5 will not know E4. For sure American mathematician John Baez, who calls himself a one man internet army, did not have enough time left to familiarize himself with E5 or E4. E5 turned out as we saw before to be SO(10) of grand unification. E4 on the other hand is the SU(5) proposal for grand unification. The dimension is 24. Adding the 24 to 504 we get Witten’s 528. In fact if you go on adding E3, E2 and E1 you arrive to El Naschie’s sum, namely 548. I will not discuss this any further here but ask a final question. Why stop with D equals 11 of M-theory? An Iranian born scientist who lives in America went on to invent a further theory with 12 dimensions for spacetime. Let me show you some marvelous things about this from a number theoretical point of view and I will leave it to you to think of the rest. Take SL(2, 12). The dimension is given by the well known formula and leads to 1716 isometries. On the other hand the Riemann curvature tensor leads for the same dimension to exactly the same number as independent components, namely 1716. Please try it yourself. This coincidence, which is not coincidence, is behind many of the wonderful things about F-theory. For instance we could do something similar to what we did much earlier. Take the fundamental 5 forces. Give each 339. The total would be 5 times 339. Think yourself in a bosonic 26 dimensional space of strings. You have to compactify 22. Add this 22 to the 5 times 339 and you find 1716 plus 4 dimensions left from the 26 to be our ordinary spacetime. I promise now to move to transfinite theory of dimensions.
The father of the inductive theory is an extraordinary Jewish Russian scientist, Paul Urysohn. This wonderful man died tragically while swimming in the Biscay at a very tender age. Hilbert knew the value of the young man from Russia. In a famous anecdote David Hilbert asked his assistant Richard Courrant what happened to the paper of Urysohn which was supposed to be published in Hilbert’s journal. Courrant said he sent it for peer review. Hilbert immediately snapped, I told you to publish the paper. I did not say you should send it for dam refereeing. We clearly live in different times where the thin veil of so called peer review is used and misused on a daily basis in high places. I do not think I want to go into that but I just wanted to tell you this because David Hilbert knew there is nobody except him who could appreciate and understand the work of this young Russian mathematical genius. Now I cannot give you the mathematical derivation of Urysohn’s work, not even in the improved and simplified version by the great Jewish Austrian mathematician Karl Menger who was forced like so many other great people to emigrate to the United States because of the ignorant unscientific Nazis who took over Germany and then marched into Austria before putting the entire of Europe on fire. What I will do here is an extremely, extremely simplistic way to come to the nitty gritty which theoretical physicists like ourselves can use to tame quantum mechanics.
Let us start with 3 dimensions, say a cube. What is the dimension of the boundary of a cube? Clearly this is an area which is two dimensional. That means it is 3 minus 1 equals 2. Apply the same thing to a 2 dimensional square. The border is a line which is one dimension so it is 2 minus 1 equals 1. Take a line. How can we separate the line or what is the border of the line? The border is a point which has a dimension equal to the line minus 1 equals zero. True mathematical genius or any kind of abnormally inclined human being never stop where almost everybody else stops and they ask questions which are unexpected although very simple and natural. Suppose we want to separate the point into two parts. What would the border of a point be? Formally and naively we just continue our implicit formula which we used so far, namely the border of n dimensional object is always n minus one. Applying this to the point we have 0 minus 1 equals minus 1. Mathematicians call the point with zero dimension the zero set and what we just calculated and found to be minus 1 is called the empty set. That is where everything in mathematics and set theory starts. I hasten to say that in physics the zero set is misunderstood on many occasions and as for the empty set, it is more than misunderstood at the majority of physical occasions. In his theory El Naschie did not even stop at this point. He continued asking the question, namely what is the border of an empty set? Formally this is minus 1 minus 1 equals to minus 2. This is a set which is even emptier than the empty set. Using certain relations between topological and Hausdorff dimensions, El Naschie continued this game to minus infinity or total nothingness. He was able to show that the Hausdorff dimension corresponding to this nothingness which is minus infinity is a simple 0. In addition it is a trivial result of his bijection formula that the zero set has a Hausdorff dimension equal to the golden mean while the empty set has a Hausdorff dimension equal to the square of the golden mean. The same result automatically follows from Connes’ theory of noncommutative geometry applied to the Penrose universe as explained in the book of Connes which we mentioned in an earlier communication. We will stop here and promise to give the details in the next communication. Best wishes,
E-infinity communication 12
The Menger-Urysohn transfinite theory of dimension as used in E-infinity theory
Before continuing our discussion we must give you some literature on the subject as well as give the various people who contributed to E-infinity their due by mentioning their achievements. The easiest way to start studying the transfinite theory of dimension is to look at the work of Nada, Crnjac, Iovane, He and Zhong. Prof. Shokry Nada is a professor of topology who got his Ph.D. from Southampton, UK and although he is originally Egyptian he is since many years part of the full time staff of the University of Qatar, Dept. of Mathematics. I strongly recommend contacting him personally but only on scientific questions. He has no patience for gossip or triviality and will not answer to things like those using the internet for entertainment on an unacceptable level. I give here without any particular order seven papers from these various authors that will help in studying the Menger-Urysohn theory tailored to E-infinity and physics. All papers are from Chaos, Solitons & Fractals. On the mathematical theory of transfinite dimensions and its application in physics, 42, 2009, p.530. The mathematical theory of finite and infinite dimensional topological spaces and its relevance to quantum gravity, 42, 2009, pp. 1974. Partially ordered sets, transfinite topology and the dimension of Cantorian-fractal spacetime, 42, 2009, pp. 1796. On the Menger-Urysohn theory of Cantorian manifolds and transfinite dimensions in physics, 42, 2009, p. 781. Density manifolds, geometric measures and high-energy physics in transfinite dimensions, 42, 2009, p. 1539. From Menger-Urysohn to Hausdorff dimensions in high energy physics, 42, 2009, p. 2338. From the numerics of dynamics to the dynamics of numerics and vice versa in high energy particle physics, 42, 2009, p. 1780. It is however recommendable to go back to the original contribution of Menger and Urysohn. The papers of Urysohn are most in French and German. Menger published mainly in German and English. There is an important classical book by Hurwitz, a Polish mathematician writing in German. You will find all of that referred to in the papers of Nada, Iovane, He and Crnjac. Regarding the golden mean in physics, we forgot to mention a few papers which were initially considered outlandish but in the meantime, that is no longer the case. They are important papers by Leonard J. Malinowski, Electronic golden structure of the periodic chart, CS&F, 42, 2009, p. 1396. The other papers you will find on Elsevier’s Science Direct.
To understand E-infinity fast we recommend that you take no short cuts. You must read at least one review paper by Mohamed El Naschie or Marek-Crnjac from the beginning to the end. At a minimum you should read El Naschie’s paper in David Finkelstein’s journal, Int. Journal of Theoretical Physics, Vol. 37, No. 12, 1998 from the beginning to the end. It is fair to say that after flirting with nonlinear sciences due to his background in the theory of stability and bifurcation as well as Rene Thom’s catastrophe theory, El Naschie started to seriously enter into nonlinear science and chaos due to the encouragement of his friend, the chaos pioneer, Otto Rössler. Much of what he knows about chaos was taught to him by his Israeli friend, the well known scientist Itamar Procaccia from the Wiseman Inst. as well as the legendary figure of chaos, Mitchell Feigenbaum. El Naschie was highly impressed by the overall personality of Mitchell although he did not share his passion for red wine and smoking. El Naschie is almost a vegetarian, unlike Mitchell who lives from red meat.
Maybe it is time here to correct some old mistakes and omissions. El Naschie in retrospect was always at pain to acknowledge that Indian meteorologist Marie Selvam may have been the first to notice the E-infinity theoretical value of the inverse electromagnetic constant. He mentioned that on many occasions before but he asked me to mention that again on this occasion. In addition we must acknowledge that Carlos Castro did some important work on E-infinity and it is with sadness that we note that he moved towards blogs and internet publications, leaving serious discussion with his old friends on science. However these few words are said because we cannot rewrite history as we like. The truth remains always the truth, no matter how painful it is. A person who is outside our group but works with incredible dedication on fractal time is a German American Susie Vrobel. We should mention her work like Fractal Time and the Gift of Natural Constraints, Tempos in Sci. & Nature, Structures, Relations and Complexity, Vol. 879, June 1999. Another person outside our group who works with dedication on fractals in cosmology is the South African Jonathan J. Dickau and we recommend his paper Fractal cosmology, CS&F, 41, 2009, 2102. Now we should return to our main subject.
The vital so called bijection formula of E-infinity theory basically says the following. If you want to know the Hausdorff dimension of a Cantorian set in n dimensions then all that you need is the inverse of the zero set raised to n minus 1. As we said earlier and we will prove it again, the Hausdorff dimension of the zero set is the golden mean. So if we want to know what the Hausdorff dimension is in two dimensions, we take the inverse of the golden mean and raise it to two minus one. That means we would have 1 divided by the golden mean which because of the nice property of Feigenbaum’s golden mean renormalization is exactly equal 1 plus the golden mean. For three dimensions you can easily work out the result to be 2 plus the golden mean. For four dimensions you obtain the famous formula which is 1 over the golden mean to the power of 4 minus 1. This is 1 over the golden mean to the power of 3 which is our famous number 4 plus the golden mean to the power of 3 equals 4.23606799…. What is interesting now is to look into the zero case and the empty set case. Let us see what the dimension for n equals 1 is. This would be the inverse of the golden mean to the power of 1 minus 1. This would be equal to the inverse of the golden mean to the power of 0. This means it is unity. In E-infinity this is called the normality condition. The one dimension is a special case. Here the Menger-Urysohn extension of the topological dimension and the Hausdorff dimension coincide and are equal to unity. Let us go one step further and ask what the Hausdorff dimension is in dimension 0. I mean now the Menger-Urysohn dimension 0. Then we have 1 divided by the golden mean to the power of 0 minus 1 equals minus 1. That means it is the golden mean to the power of 1 which is equal to the golden mean. Thus we have proven the assumption. Now we go to the second most vital step and ask what the Hausdorff dimension is when the Menger-Urysohn topological dimension is equal to minus 1. Remember this is the empty set as defined classically. The result is we have 1 over the golden mean to the power of minus 1 minus 1 which means to the power of minus 2. This means we have the golden mean proper to the power of plus 2. Now we have resolved indirectly the famous two-slit experiment. The most conservative explanation is due to the work of Mohamed El Naschie together with his late teacher, fatherly friend and mentor, Prof. Dr. Dr. habil Werner Martienssen who sadly died a few weeks ago. Martienssen and El Naschie decided to give the wave the Menger-Urysohn dimension of the empty set. This is a Hausdorff dimension equal to the golden mean to the power of two. The particles on the other hand get a Menger-Urysohn zero and a Hausdorff dimension equal to the golden mean. All these are of course interpreted as probabilities but I am running ahead of my theory. I will repeat all of that later on. El Naschie asked a trivial question but in doing so, he solved a major problem. He said why stop at the empty set? Why not ask if there are emptier sets? Being a naive engineer he said why should I integrate from minus 1 to plus infinity. From many engineering problems of struts on an elastic foundation extending relatively from minus infinity to plus infinity, some time referred to as full or half space in theory of elasticity, he is used to integrate from minus infinity to plus infinity. So he found emptier and emptier sets with Hausdorff dimensions equal to golden mean to the power of 3, then golden mean to the power of 4, then to the power of 5 and so on. At minus infinity you will have the golden mean to the power of infinity. Since the golden mean is smaller than one, to be raised to infinity you have an absolute 0. That way El Naschie discovered as it would turn out later, a unidirectional system giving a hint at the unidirectionality of time starting at a singularity. When he reached this result he wrote a paper in some Pergamon mathematical journal and dedicated it to his teacher and mentor, Ilya Prigogine who was rather excited about it and said so in correspondence. Related results were also communicated to K.F. von Weizsäcker who was very excited about it and wrote as much. Weizsäcker advised El Naschie to take the work of David Finkelstein seriously and told him that his own work was quite near to that of Finkelstein. My recounting of all these anecdotes may not be very accurate because I know them all second hand from someone who was told by someone that El Naschie told him. However the big paper dedicated to Ilya Prigogine is there for everyone to read and so is the complimentary letter of Weizsäcker to El Naschie. I am saying all these relatively unimportant facts because there is a whole industry now on the internet whose main reason d’être is to shed doubt and spread doubt and rumors and lies about E-infinity and its founders. El Naschie’s theory was not invented in 2008 when all this noise started with an article published in Scientific American. El Naschie has been working on his theory since the late 80’s. Only Garnet Ord and maybe Laurent Nottale were first chronologically speaking.
I think we have covered substantial ground on Menger-Urysohn theory and the extension of the classical empty set to the absolute empty set by El Naschie. The relevant papers which are cited in the pure mathematical literature quite a bit will be given to you in the next communication. It is important to understand now that we have subdued infinity. In Cantorian spacetime and fractal spacetime there is no ultraviolet or infrared catastrophe. Everything is regularized automatically because we are working in a naturally renormalized geometry. Everything has at least two major dimensions to describe it. Therefore the old fashioned uniqueness of dimension theorem does not apply and do not restrict us anymore. That is why Cantor’s theory is a paradise from which we should not be evicted as David Hilbert asserted. E-infinity is based on Cantor’s paradise. There is no room here for cheap jokes except from those uncorrectable philistines whose jokes just pollute every site in the blogosphere. I am sorry for using these harsh words but E-infinity or not, we are luckily all human. Until next time, all the best.
E-infinity communication 13
Transfinite dimension applied to the two-slit experiment with quantum particles
Before we start again we ought to give some literature about the extended Menger-Urysohn system. The first important paper here is El Naschie’s 1994 paper On Certain ‘Empty’ Cantor sets and their Dimensions, CS&F, Vol. 4, No. w, 1994, p. 293. Three related papers which may be useful are Is quantum space a random Cantor set with a golden mean dimension at the core?, CS&F Vol. 4, No. 2, p. 177, 1994. Statistical geometry of a Cantor Discretum and Semiconductors, Computers Math. Applic. , Vol. 29, No. 12, p. 103, 1995 and Time symmetry breaking, duality and Cantorian space-time, CS&F, Vol. 7, No. 4, p. 499, 1996. Finally a very important general paper is by two Romanian mathematicians Mircea Crasmareanu and Cristina-Elena Hretcanu called Golden differential geometry, CS&F, 38, 2008, p. 1229. This particular paper incorporates the golden mean in a fundamental way to produce golden differential geometry which is extremely valuable in understanding E-infinity theory.
Now there are a few realizations which one should draw from El Naschie’s E-infinity. First an infinite dimensional system is necessarily chaotic. This is really a theorem. We first knew about it from the experimental work of Ji-Huan He using a computer and drawing infinite dimensional cubes. He did not really make it infinite. You do not have infinity on a computer. He went as far as a 26 dimensional cube. Then we realized that the great mathematical engineer and mathematician David Ruelle presented and used this theory in some early work. I am not sure but I think Ruelle also gave a proof. You will probably know the name of David Ruelle if you work in nonlinear dynamics for his contribution including Ruelle-Taken’s scenario for turbulence. There is an anecdote to tell here. Some hated that a civil engineer should be such a good mathematician which David Ruelle is. He could not get his paper on turbulence published even though he was an Editor in Chief of a journal. At the end he got fed up, submitted the paper to himself and accepted and published the paper anyway. The scientific community therefore owes him two things, an apology and thanks. Of course it is a procedure which should not be generalized but there are until today those who are consumed by jealousy and who keep attacking Ruelle for publishing his own paper in his own journal. In his famous book he recounts how a certain person used to go from one library to another library to tear out his paper from the journal copy in that library. Unbelievable what jealousy and hatred does to people, even scientists. Mind you this was even before blogs and internet vandals masquerading as scientists. Any case enough of that.
An important other point if the role played by the zero set and the empty set in physics. Physicists seem not to have incorporated this in a systematic way and have not developed the intuition to work mathematically with nothing instead of something. Thank God for B. Mandelbrot, M. Barnsley and Otto Peitgen. They gave us some intuition about fractals. Thank God for material scientists who were the first to jump on fractals and use them. In high energy physics on the other hand things were and are still relatively very slow indeed. The reason is they do not know how to use mathematics properly for fractals and connected to physics. They only use the marvelous device of Hausdorff dimension. Alas the Hausdorff dimension is not a topological invariant. The marvelous thing about the new theory, as shown in the work of Alain Connes and Roger Penrose is that both, the topological Menger-Urysohn dimension and the Hausdorff dimension are connected in one formula. In the theory of Connes this is done implicitly. In El Naschie’s work this is explicitly as evident from his bijection formula discussed earlier on. For this reason we will have to discuss at length the work of Tim Palmer who came to similar conclusions but in a qualitative way and could not make quantative computation because he does not assign the right dimension to the empty set. You could say Palmer discovered fractals for quantum mechanics but only qualitatively. His biggest triumph and achievement in quantum mechanics was his ingeniously apt formulation – quantum mechanics is blind to fractals. However by not admitting Menger-Urysohn minus 1 dimension to the classical empty set which represents the vacuum at its first level, he could not resolve the problem of quantum mechanics except qualitatively. Palmer’s paper is on the ArXiv in a revised form since 2009. The paper was published in yet another revised form in the Proceedings of the Royal Society and we strongly suggest that you read it. Some profound work preceding the work of Palmer is due to the group around the very competent mathematician and nonlinear dynamics connoisseur Prof. George Nicholas. Three from the same family published a very influential paper on the two slit experiment in Chaos, Solitons & Fractals many years ago. Again this paper was motivated by the work of El Naschie on the same subject using a simple analogy to the three point chaos game. This chaos game as well as the four point chaos game is well explained in the relevant literature referred to in the papers of G. Nicholas and M. El Naschie all published in Chaos, Solitons & Fractals. What has passed unnoticed was several papers written around the same time or slightly after in which Mohamed El Naschie uses the theory of Menger-Urysohn to explicitly show that the two slit experiment with quantum particles and the wave collapse could be explained completely rationally, you could say almost classically, using the Menger-Urysohn theory of dimensions. This theory can be worked out in infinite dimensions only when spacetime has an infinite dimensional topology. This is the case with the Hilbert cube or El Naschie’s four dimensional cube inside another four dimensional cube and so on ad infinity. The idea which started with the work of El Naschie was refined considerably in its mathematical formulation first in Italy by Prof. Gerardo Iovane from the University of Salerno, Dept. of Mathematics and subsequently in several papers by the remarkable Russian mathematician A.M. Mukhamedov, from Kazan State Technical University, Russia. Maybe some will recall that hyperbolic geometry was invented in the University of Kazan. In this connection we would like to draw attention to a recent paper by Prof. Mukhamedov entitled Towards a deterministic quantum chaos, published in Problems of Nonlinear Analysis in Eng. Systems, No. 2(32), Vol. 15, 2009. There are many papers about the two-slit experiment published in CS&F including those of Prof. S. Nada who used density manifolds, Prof Marek-Crnjac who used quotient manifolds. Both ideas go back to work by El Naschie and for proper understanding you need knowledge of geometrical measure theory and/or string theory. Some very simple straight forward papers were published earlier on by El Naschie like The two-slit experiment as the foundation of E-infinity of high energy physics, CS&F, 25, 2005, p. 509. Let me give you an excessively simplified version but having the same flavor.
The probability of finding a Cantor point in a one dimensional Cantor set is a topological probability equal to the golden mean. The probability of finding no Cantor point is the complimentary probability, namely 1 minus the golden mean. This is the golden mean squared as a pocket calculator will assure you. Based on this elementary observation you can reason that the probability for a Cantor point to be at a point number 1 or a point number 2 must be the sum of two probabilities and it turns out to be the sum but where the golden mean square gets a negative sign. So the sum is really the difference. The difference between golden mean and golden mean squared is equal to the golden mean to the power of 3. On the other hand the multiplication theorem of the theory of probability teaches us that being at point 1 and point 2 simultaneously is the multiplication of the golden mean of the golden mean squared. This is again equal to the golden mean to the power 3. This simple analysis shows us that based on probability theory we cannot distinguish between a point at 1 or 2 or a point at 1 and 2 simultaneously. This is the indisguishability condition. That is why you cannot say what is a particle and what is a wave unless you prepare the experiment accordingly. In quantum mechanics, as in Cantorian spacetime, a point can be in two different places at the same time. When you take measurements by imposing yourself on the empty set, the empty set is no longer empty and the wave collapses. It is a simple as that. Simple as it may be, this is a hefty dose for this communication and after you have had a look at the literature, I will go through all that with you once again in far more detail. However you must work with me. I can repeat all this to you a hundred times but unless you do it yourself, and although it is very simple, it will slip through your fingers and you will shake your head in bewilderment wondering what it is all about. It is very simple. However it seems nothing is as difficult as simple, unfamiliar things. On the other hand, familiarity breeds contempt and you become careless. We promise you to be neither. Best regards
E-infinity Communication No. 14
Defining the probability in Cantorian spacetime, negative dimensions and the transfinite theory of Menger-Urysohn
Defining the probability in a Cantorian setting initially seems to be a hopeless undertaking. Some may be worried that such a task could lead to the same end station of George Cantor’s mental institute or was it only a nerve clinic. After all Anthony Eden ended up in something similar after his Suez Canal adventure. Let me explain. Suppose we have n distinct objects. Suppose we have an experiment where choosing a single object of the n is truly random and cannot distinguish between the different objects. Combinatorics says that the probability of haphazardly picking a particular object is equal to 1 over n. Suppose the ensemble element n is infinitely large, the probability is thus zero and combinatoric probability fails miserably to help us in a Cantorian setting because we have not only infinitely man Cantor points, but actually uncountably many. This uncountable infinity brought some people to despair. Cantor’s diagonal methods were labeled nonsense by nobody less than Kroniker. He stood in the way of Cantor’s promotion to professorship in a respectable German University like Berlin or Munich. Poor George was left in a small provincial German university. Finally he was mocked and teased by his colleagues to the extent that he several times suffered from a breakdown of nerves. It may also have been the intense contemplation of the Cantor set and its meaning. When you have taken out all the middle thirds out except for the n points iteratively and continued this process ad infinitum, then you have no length left. The set is measure zero. How on earth could you have not only something left which has the cardinality of the continuum, but you have something with a respectable and considering the situation quite sizeable finite dimension. OK it is a Hausdorff dimension but dimension never the less equal to ln2 divided by ln3 which is 0.63…. Not bad for something which is not there. Friedrich Hegel in his dialectic coined an ugly Greek/Latin half breed word coincidentia oppositorium. He must have had the Cantor set in his mind. Remember a line is a dimension 1 and is a continuum. Compare this with the Cantor set which is not really there because it is measure zero. It has a dimension 0.63 and thus a majority holder in the company and as if this is not enough, it has the cardinality of the continuum itself. In more earthly words, a Cantor set has as many points as a continuous line. No wonder Shakespeare said ‘there are things in heaven and earth Horatio’. No wonder that until this day many atheoretical physicists would rather not know anything about Cantor sets. This all would convince you that Cantor sets have nothing to do with reality and consequently it should have no place in physics. I call now to the witness stand Prof. Friedrich Pfeiffer from the University of Munich. This professor is probably one of the world’s most prominent professors of mechanical engineering. He was not only the Head of the Department of a Centre of Excellence, namely the University of Munich, Germany but also he was the Editor in Chief of the German Journal Ingenieur Archv. If you published a paper in this journal then you reached the promise land in Germany. The professor’s specialty is chaotic research of mechanical systems applicable directly in the industry. Before joining university again, the famous professor worked for an even more famous car producer in Bavaria, BMW. Some of his contributions were to take unnecessary noise from the motor and the clutch and let a BMW car be as silent as a Rolls Royce. No wonder BMW bought Rolls Royce. In a manner of speaking the famous professor was taking the Cantor set causing chaotic noise out of the BMW car. Cantor set, esoteric or not for physicists, are as real as hell for engineers and many other professions. It is strange that of all people high energy scientists and quantum physics theoreticians who deal with things which nobody has ever seen or experienced firsthand should consider Cantor sets esoteric and resist its integration into quantum physics as a basis of new micro spacetime geometry. There are of course many exceptions which we mentioned earlier, David Finkelstein, Heinrich Saller, Parisi, Ord, Nottale and many others. However the typical theoretical physicist neither appreciates nor most of the time knows anything about Cantor sets. Now let us return to our subject properly. When the combinatorial method fails this is no surprise. We have the geometrical method. You know in the darts game you also have infinitely many points but then you define probability geometrically by the quotient of different areas. Great. However calamity struck. A Cantor set has no measure. So we have no length. We are dealing with an esoteric phenomena measured zero. The geometrical method and geometrical probability goes out of the window. Of course there are sophisticated methods based on measured theory. Mark Kac described probability theory as a measure theory with a soul. We would like to keep this soul and remain in probability theory as much as we can. When all things fail, you rub Aladdin’s wonder lamp and ask the genie to bring you in some topology. A line, thin as it may be has a dimension 1. The Cantor set also has a dimension and from a well known theorem by Mauldin and Williams a random Cantor set of the triadic type also has a dimension when you assume uniform probability which is the simplest assumption you could possibly make and this Hausdorff dimension is equal to the golden mean. Now we can define a topological probability so to speak. Such quotient would be made of the dimension of the Cantor set divided by the dimension of the line. This is the golden mean divided by 1 which is equal to the golden mean. At long last we have at least something with which we can do some computation. If the simplicity of the solution confuses you and if the question you pose to yourself confuses you even further, do not despair. The great Poincare himself managed to confuse himself in an exam for mathematics and failed. He took the examination once more, succeeded and then invented topology. When he became so famous he failed once more. This time he did not realize that he was wrong. He failed to recognize the work of Cantor. He considered Cantor’s geometry a gallery of monsters. He agreed to a certain extent with Kronecker that Cantor’s ideas are maladies which inflicted mathematics and he was sure that mathematics would soon recover from it. You would rightly say nowadays that you could take all this nonsense from anybody, including Kronecker the arch enemy of the non-finite but for God’s sake, not Poincare. However we cannot rewrite history. Poincare did not recognize his three body problem. The geometry of non-integrability is the geometry of chaos and the limit set of chaos and the backbone of any chaotic system is as James Yorke taught us, a Cantor set. God invented the integer. Everything else is the work of man. This is of course total nonsense, typical for Kronecker on this subject. We have now so many infinities and hierarchies of cardinalities beyond Kronecker’s and even Cantor’s imagination. They are sufficient to make Kronecker turn in his grave, infinitely many times. I hope you got the right taste for the thing to come next to resolve the two-slit experiment which is the basis for the Cantorian proposal for quantum spacetime. And yes, a point, whatever we mean with this word can exist at two different locations at the same time in the infinite dimensional topology of Cantorian spacetime and similar spaces. One of the nicest people one could ever meet anywhere at any time is an English/Canadian physicist whose name is Garnet Ord. Ord is the man who coined the word fractal spacetime. He corresponded and discussed many things with Richard Feynman. Einstein is great but Feynman is something else altogether. Ord intuitively knew about the power of fractals and Cantor sets but if you see Ord and El Naschie discussing things regarding Kronecker and Cantor you would think these two extremely close friends are in two totally opposed sides as far as finite and infinite are concerned. I will keep many anecdotes about El Naschie and Ord for the next communication but I advise you for the time being to familiarize yourself with the greatest mathematical genius of all time, Georg Cantor by reading the wonderful book of Joseph Warren Dauben entitled Georg Cantor, His Mathematics and Philosophy of the Infinite, Princeton University Press, 1979. All the best. (23.3.10).
E-infinity Communication No. 15
Set language and probability language dictionary of E-infinity as a two-slit experiment with quantum particles
A philosophically inclined cleric in England invented diagrams which are quite useful to use to move from one language to another in E-infinity theory. I mean set theoretical language and the language of probability theory or events. This was important for El Naschie in developing E-infinity and may be helpful for some in understanding this part which is crucial. The classical language of events or probability language speaks of probability space, events, impossible events, not d or the opposite of d, either/or or both, both, mutually exclusive and if then. In set theoretical language and in the same order you could say universal set, subset, empty set, compliment of d, the union operation, intersection operation, operation of intersection for a totally disjoined set and a set being a subset of another set. The set notations are different from one author to another but are well known. El Naschie studied mathematics in Germany under Kaluza. The anecdote connected to examining a rebellious though peaceful member of the extra-parliamentary opposition is laid down in his reminiscence of his student years written on the occasion of his 60th birthday, A tale of two Kleins unified in strings and E-infinity theory, CS&F, 26, 2005, p. 247. I hope referring to these things is not interpreted draconically as cult which is way over the top to say the least. Everyone has his own style of writing. Some like these anecdotes as a welcome distraction from the boredom of too much formality. It is immodest to recommend one’s own taste but it is hypocritical to recommend somebody else’s taste. If we must we would rather be something, we would rather be immodest than hypocritical, so if our kind friends would bear with us and consider that we are doing all that for fun and free of charge to the benefit of everybody else, then at least be so tolerant to leave everyone express himself in the way he likes. If you do not like something, do not read it. We are not offensive to anyone and mentioning the outstanding achievements of people like Richard Feynman, Nottale, Ord, Rössler and Tim Palmer should not be provocative to anyone. We would like that the young people have the right examples to follow at least if they are serious about science.
Now let us suppose you are on the unit interval of which a Cantor set was made. The dimension of the interval before digging so many holes in it is unity. This is the same for the Hausdorff dimension as well as the topological Menger-Urysohn. If you are a member of the Cantor set then the dimension attached to you would be zero for Menger-Urysohn and the golden mean for Hausdorff. If you are not then you are definitely in the empty set. The corresponding dimension would then be minus 1 and the golden mean to the power of square. These simple facts follow from Connes’ formulation of Penrose universe. This is just another formalism of the bijection formula with a slightly different mental picture. You can have whatever mental picture you want as long as this helps you to have this mystical feeling of understanding. Now you have two basic operations from the set theoretical point of view, union and intersection from which you get two dimensions for two elementary manifolds. The quotient manifold is given by a dimension which is the quotient of the dimension of the sub manifolds. That way you find the elementary fact that the dimension of a manifold which can sustain the preceding set theoretical conditions is nothing else but the infinite dimensional Hilbert cube which is not identical but very similar in many aspects to the space studied by Ji-Huan He. It is the same space which you obtain from putting a four dimensional cube into another four dimensional cube and so on ad infinitum. Again I am probably going to fast now but everything you know from string theory and high energy physics can be obtained as a deformation of this infinite dimensional Hilbert cube which has a Hausdorff dimension four plus the golden mean to the power of three, a topological expectation value of exactly 4 and a formal dimension of infinity. It is infinity in a hierarchal sense. It is infinity because you are taking infinitely many concentric four dimensional cubes to reach this result. Do not forget, Ruelle’s theorem. A classical system is necessarily chaotic when the dimensionality is infinity, even when it is hierarchal. In a sense you are holding infinity in the palm of your hand as in the famous poem of William Blake. For a geometrical visualization and easy access to the connection to string theory, I strongly recommend that you carefully study the excellent paper entitled Twenty-six dimensional polytope and high energy spacetime physics by Ji-Huan He, Lan Xu, Li-Na Zhang and Xu-Hong Wu, CS&F, 33, 1, 2007, p.5-13. In addition we cannot stress enough the importance of reading the work by the exceptionally gifted young Italian professor of applied mathematics, Gerardo Iovane. One of Gerardo’s computer graphics representations of fuzzy K3 Kähler manifold of E-infinity was entitled ‘E-infinity Cantorian Universe: A fractal manifestation of love’. Many of Gerardo’s papers can be obtained free of charge either directly from Elsevier’s science direct who do not always charge for every paper when you come to it through Google Scholar or can be found on certain pirate blogs claiming to belong to E-infinity members which is not always true.
Mohamed El Naschie is a walking encyclopedia when it comes to certain anecdotes of famous people with whom he had the privilege of talking. Again I hope this remark is not taken as cult. He said quoting Sir Prof. Herman Bondi that a fool can ask more questions than a wise man can answer’. However he hastened to say following Weizsaker who was again quoting Heisenberg ‘one has to learn, sometimes the hard way, that asking the right question is normally half of the answer’. We are frequently perplexed because the questions we are asking are imprecise or meaningless or undecidable. Undecidability is not the work of the devil. In fact undecidability in the sense of Gödel implies chaos and chaos implies fractals and fractals imply E-infinity theory. Very frequently when one does not understand something, one should not try too hard. There are very frequently mental blockages caused by our very selves. It is best to go and sleep and not to try too hard. During sleep the subconscious work and it is frequently according to surrealistic artists far more intelligent than consciousness. That is at least the theory of the sleep walkers advocated by Koestler. Accordingly I am now going to sleep. It is one o’clock after midnight local time. (24.3.10)
E-infinity communication No. 16
Summing over paths, dimensions, exceptional Lie groups and knots in E-infinity and tidying some loose ends from previous communications. Part I
I am taking over from my colleague and will start by apologizing for various gaps in the presentation. The number of states which are of interest are 496 for superstrings, 528 for the 5-Brane model, 548 for summing over 8 exceptional Ei Lie groups, 685 for summing over 17 two and three Stein spaces and the same sum for 12 exceptional Lie groups some of which are not from the Ei family. Finally and most importantly 8872 for 219 three dimensional crystalographic group. The knowledgeable reader will remember that the 17 two dimensional crystalographic group, that is to say the 17 Alhambra Andalusian tiling corresponds to 230 three dimensional group. This is an error and a common one. The 17 correspond to 230 minus 11 making 219. These are the 3D crystalographic group which truly corresponds to the 17 two dimensional crystalographic group. This is all explained in details in the literature and the papers mentioned in the previous communication. The connection between Heterotic string theory and the 219 crystalographic group is truly remarkable. As far as I know it is Mohamed El Naschie who drew attention to this fundamental fact for the first time. Before I finish remind me to tell you about two missed opportunities. The first is connected to Nobel laureate Gerard ‘tHooft and the second is connected to Nobel laureate Steve Weinberg. It is not only an anecdote I will quote paragraph and verse hoping this will at least make you trust E-infinity a little because unless a certain amount of trust is assumed at the beginning, you cannot made headway easily.
Let me explain the last two lines in come details. When you explain Einstein’s special theory of relativity, what do you say? You simply say that you no longer think of our space as being 3 plus 1 space and time but as a fused four dimensional spacetime. You would also say that the simultaneousity is not possible because every point has four different coordinates in this spacetime. Finally you add that the speed of light cannot be exceeded. When you want to be more sophisticated you say that these conditions are not independent of each other as is obvious from the Lawrence transformation which preceded Einstein’s work. To give the impression of historical sophistication one would probably add that Poincare knew all of that long before Einstein and that in his work he spelt out indirectly that E is equal to m multiplied with C squared, where C is the speed of light. The general relativity is much easier to explain because you only say that in the very, very large spacetime is curved. This geometrical curvature is what we perceive as gravity. If you want to be sophisticated you just add that since the speed of light is the maximum speed in spacetime, then gravity’s effect cannot travel between two gravitating masses instantly and must travel at a maximum with the speed of light. We do not notice the curvature because it is noticeable only on very large scales. When you say all that everybody is happy. Why are things different in E-infinity? Why, despite so many explanations do people want to understand in a simple way what this E-infinity is? Here is a simple way. While classical spacetime is smooth and Euclidean at intermediate distance and curved at very large distance it is not smooth and not continuous at very short distance. That is all folks. To model the large scale geometry we use Riemannian geometry. The curvature tensor is the driving force behind Einstein’s equation. Similarly to model quantum spacetime at these very short distances we use fractal geometry in its simplest form. The simplest form of a fractal is a Cantor set. We take infinite numbers of Cantor sets to do the job. How on earth could anyone draw a conclusion from that that Cantorian spacetime is infinite. No it can be finite and have infinite dimensions. Even from elementary school mathematics we know that we can sum an infinite series and find a finite answer. That is the whole point behind fractals. When you hold a piece of fractal in your hand, you are de facto holding infinity in your hand. E-infinity theory is very similar to the no boundary proposal of Hawking. It is all extremely simple. Sometimes I am reminded with the heavy weight lifters who are used to carry this 300 kilos and go to lift something which is very light although it does not look it and in doing so, he hurts himself because he was not expecting it to be so light. It is similar to Sonny Liston when he directed a blow to Muhammad Ali’s head but Ali shuffled sideways and Sonny tore the muscle in his right arm. People are expecting E-infinity to be difficult. That is why they find it difficult. Just relax and have faith. It is far simpler than you think. I will not answer any question unless the concerned person gives me his word of honor that he has read at least one single review paper written by an expert on E-infinity theory. In the next communication I will go into slightly more detail and talk about the missed opportunities of ‘tHooft and Weinberg, two people for whom we have the highest possible regard a scientist can have for another scientist as far as science is concerned.
E-Infinity Communication no. 17
Some lost opportunities
What we have to say in this communication might seem at the first instance to be a diversion. Believe me it is not and I just ask you to bear with me a little. I will give two examples where E-Infinity could have been of a great help but it was not utilized and not even considered. We are all no matter how liberal and open minded we think of ourselves subject to prejudice which is deep rooted. Try as much as we want, we cannot escape from two things: The noise of our upbringing and the noise in our brain. Nobody is saved from this “condition humane, to use a term of John Paul Sartre. The two examples we will give are related to the work of two towering figures of the 20th century theoretical physics who are still with us and are still working and contributing vigorously to the literature. The two are Nobel Laureates Steven Weinberg and Nobel Laureate Gerard ‘tHooft. Let me start with a much simpler example of Weinberg. I don’t think there is a single person who has anything to do with theoretical physics who wouldn’t know the great man Steven Weinberg who has written the Definitive Treatise on Quantum Field Theory in three volumes published by Cambridge Press. In addition Steven is a great intellectual personality and his influence goes far beyond physics. For instance in his book facing up he presented in a courageous and logical way the point view of Zionism. This is of course a ticklish issue particularly nowadays and with regards to the Middle East and the rise of the Muslim religion in political form. But Weinberg’s Zionism has an undeniable human and logical face. He is right to warn from the rise of any religious discrimination. He is also right to warn from repetition of the holocaust in any form or guise. Some years ago I was told that the great man was invited to a Conference dedicated to transfinite physics. Weinberg did not hear this expression before. He declined the invitation politely being a responsible and courteous person. What a pity that he wasn’t there. Gaining such mega brain to transfinite physics would have completed the revolution which started with Richard Feynman and continued with the work of Garnett Ord, Laurent Nottale , Mohamed El Naschie, Goldfain and dozens of other scientists including Sidharth, Svotzel, Otto Rossler and yes Renate Loll, Jan Ambjorn and of course Fay Dawker in England who followed a slightly different line initiated by David Finkelstein in America and Heinrich Saller in Germany. Let me be now specific. Consider volume 3 of Steven Weinberg’s book on quantum field theory. Volume 3 is dedicated to super symmetry and the book was published by Cambridge in the year 2000. On page 188 to 192 of the book, Weinberg considers super symmetric unification of the strong and electro weak. He calculates the inv unification coupling constant and finds it by virtue of equation (28.219) to be 1 divided by 17.5. In other words the inverse coupling constant of unification is simply 17.5. Now to us working in E-Infinity we recognize immediately this result as wrong. We do not need to make many calculations to realize that somewhere a misfortune and probably trivial computational error was introduced to Weinberg’s analysis. Let me explain why: First, if you are dealing with super symmetric unification, then you are implying gravitational force whether it is explicit or not. Consequently, Weinberg’s analysis is not simply a grand unification but complete unification of all fundamental forces namely electro magnetism, weak force, strong force and gravity. Now we know for sure that the exact value approximated to the nearest integer of the inverse coupling in such case must be 26. The 17 and half is too far for 26 to be even remotely correct. For instance one of the past students of Weinberg and a leading string theoretician and a colleague of Weinberg in the meantime is Joseph Polchinski. In volume 2 of his book String Theory, he calculates the same value and comes to the result that the inverse unification coupling constant must be something near to 25. In fact on page 347 of Polchinksi’s said book, two values are given for the unification by virtue of equation (18.312). The first value is the unification energy of about 10 to the power of 16 Gev. The second value is the inverse unification of 25. Of course Polchinski says it is for grand unification. However and as we reasoned earlier since super symmetry is involved, it is for complete unification including gravity. Now how did we notice so quickly. The reason is that we have at our disposal, a set of very simple principles and even simpler sets of equations which leads us directly to the correct result. In detail we should say the following: We know the exact theoretical value of the four fundamental couplings of the electro weak theory which we need to reconstruct the exact theoretical value of the inverse electromagnetic fine structure constant namely 137.082039325. The value needed for that is Alpha bar 1 equal to 60, alpha bar 2 equal 30 and alpha bar 3 equal 9. In addition, we have the Planck coupling 1. Using the well known reconstruction of E-Infinity we obtain 137.082039325. As anticipated by E-Infinity theory the exact theoretical value 60, 30, and 9 are extremely close to the nearest integer approximation to the experimental value found in the literature. It is a very elementary business to find by averaging a value for the unification inverse constant. You simply take the geometrical mean of the 3 said values. In other words you take the third root of the multiplication of 60, 30 and 9 + 10. You could of course take 9 instead of 10. If you take the 9 you get 25.3 as an approximation. If you take the 10 you get 26.2 as an approximation. An E-Infinity exact analysis will give you of course the exact transfinite value namely 26.18033989. This is nothing but the inverse golden mean to the power of 2 and taken 10 times. It is interesting to note that the value of approximately 17 refers only to partial unification. This partial unification is easily obtained by averaging. You just take the square root of 30 multiplied by 10 and get 17.3. This is unification of strong force with electro weak alone without considering electromagnetism. You see how easily we can do calculations because the golden mean binary system makes cumbersome computations elementary. But this is also not all. We have conceptual simplicity. We know the building blocks of spacetime. They are the random Cantor set of the golden mean Hausdorff dimension. The great Dutch scientist Nobel Laureate Gerard ‘tHooft made the search for the building blocks of Nature and the title of his beautiful popular book: In search of the ultimate building blocks published by Cambridge University Press 1997. That brings us to the next example of a missed opportunity. In his book on page 2 in figure 1, Gerardus ‘tHooft plays with a wonderful idea namely making smaller and smaller kites from a sheet of paper. Miraculously and as if an invisible hand is moving ‘tHooft’s hand, he designed in figure 1 a logarithmic spiral connected to the golden mean without saying so. For practical considerations, which are totally justified, ‘tHooft stopped before making the ultimate theoretical conclusion that he is reaching an element of a wild topology with a Cantor set ramification which harbors the golden mean. Then on page 174 ‘tHooft finds at long last a unification theory which he could praise since he does not like string theory. This theory not surprisingly is Loop Quantum Mechanics of Lee Smolin and Rovelli. ‘tHooft writes “In this theory the only thing relevant is the number and kind of knots linking the loops………..Accidentally Knot Theory is one of the most difficult branches of modern mathematics”. In E-Infinity we beg to differ profoundly. It is a prejudice to think knot theory is difficult. Knot theory is simple. You can do experimental with the rope and nothing more. Most of the knots are in 3D and they become unknots in 4. There is of course more complex knots in 4 which become unknot in 5. Mohamed El Nastier used Knot Theory skillfully to point a connection between knots, Feigenbaum scenario and ramification at a Cantor set. In one of his very readable papers titled: Fuzzy muti-instanton knots in the fabric of space-time and Dirac’s vacuum fluctuation, El Naschie discussed ‘tHooft’s work in detail and points out how he can obtain ‘tHooft instanton as a volume of a symmetry group. This is a trivial consequence of E-Infinity theory. You see the special orthogonal group SO3 has a volume equal to 8 Pie square. Mohamed El Naschie gave the exact value in transfinite form as well as many other interesting points. You see this way, the instanton becomes more physical as a knot which has volume and becomes nearer to a particle like state or a collection of 16 particle like states. The schism between string theory calculation of the 8064 and the holographic calculation of the same disappear. In a certain way, it was ‘tHooft who pointed out to Mohamed El Naschie that he has a new theory. It was however a lost opportunity to combine forces and pull in the same direction rather than sit on the fence and have to bear what Shakespeare called: the slings and arrows of outrageous fortune. However at E-Infinity we are ready to follow Shakespeare as well. We will take arms against a sea of trouble and by opposing we will end them. And it is nobler in the heart to suffer.
E-Infinity
E-Infinity 18
Part 2 of Communication No. 17
Not only Confucius is advising restraint when faced with the awesome power of irrational hatred or the poisonous device of twisting facts. Aristotle finds no way to face the artificial wit of those who have nothing in their heads apart of comic strips and slapsticks except to fortify yourself in continuing serious discussion. That is what we will do here.
There is nothing called weird topology. Of course you can call it what you want but there is no such expression in use. The correct expression is wild topology. The only person who could be very upset about that in a professional way that is must be John Baez. In his n-Category café he made a meal out of wild topology only to find that he is of course wrong. That is what happened to you when you spend too much in cracking jokes, writing silly articles with ha-ha-ha instead of reading seriously to expand your horizon. So many people calling themselves anonymous spend unreasonable amount of time on worthless blogs achieving nothing except maybe getting rid of their internal frustrations with themselves. John Baez of Riverside University proclaimed loudly that there is no 8 exceptional Lie group. Of course he was wrong and his victim was right. We have E8 with 248 generators. Then we have E7 with 133 and then we have E6 with 78. That is not where it stops. Because most people know F4 with 52 and G2 with 14. But these last two are not E line. The correct E5 is somewhat surprisingly SO(10) with 45 generators. Then we have E4 and this is a counterpart of SO(10) namely, SU5 with 24 generators and you can go on that way until you exhaust the E Line. The sum was found by El Naschie to be 548 to the nearest integer. Next blunder of John Baez was regarding 2 and 3 Stein spaces. He never heard of them. What a blog master does not know about does not exist by definition. It is replaced systematically by silly jokes and ha-ha-ha. That is what Charlie Chaplin would call modern times, or theoretical physics a la blogs with café au lait. We could go on indefinitely like that but this will violate the rules laid down by Prof. Mohamed El Naschie about refraining from personal remarks and jokes that have nothing to do with science. Wild topology is a very important part of general topology. It is connected also to knot theory. The Russian literature is abounding with such examples. In particular, a Great Russian mathematician living in France made this connection and that is where Mohamed El Naschie learned his stuff about the connection between knot theory and cantor sets. He added a trick to that which he learned from Nicholas Hoff. Do not ask me who is Nicholas Hoff? However I know from Mohamed El Naschie that he is one of his teachers. He used to be the Head of Aeronautical and Astro-nautical Department in USA. When faced with big nonlinear problems, Hoff chopped the thing into two parts. He ignored the nonlinear terms and solved the linear part. That is mathematically acceptable. But then he did something unconventional based on intuition. He chops the linear part and somehow regards only the nonlinear end state. That is a bit unusual to say the least. It is unusual for pure mathematicians although I am not one. Hoff did not carry the ballast and regarded only in state. When he reached the top with a ladder, he kicked the ladder and onlookers wondered how he reached the top. Mohamed El Naschie did not follow the intricate knot doubling of an entanglement. He took only the end state. He took the limit set. This limit set is the Cantor set. Something similar was done though not quantitatively by Tim Palmer. Before them something similar was done by Michael Berry. People working in nonlinear dynamics have an engineering sense. This is a world apart from the algebraic computational approach of a great man like Nobel Laureate Gerard ‘tHooft. There is no doubt of the admiration of Mohamed El Naschie to Gerard ‘tHooft but the latter would be the first to acknowledge that a Nobel Prize is not a passé partout. It is a great pity that this great man hasn’t fully taken in the fact that a random cantor set has a golden mean Hausdorff dimension and that a random cantor set is the end state and by duality it is the very very beginning. As such you are started with a golden mean binary system. You have now a chance to solve things with unheard of simplicity similar to what John Wheeler has always proclaimed. At such level it is absolutely misguided to take the theories of Newton or even Einstein as a guiding light. It is totally wrong at this level to differentiate between physics and maths. It is completely naïve to think that there is really a distinction between reality as we think we know it and ir-reality whatever we mean by this word in our so called real world where our labs exist and measurements are taken. There are those religious fundamentalists. We regard them as fatal and misguided apart from being non scientific. But there are equally inclined fundamentalists who think that everything is only measurable in the laboratory. These things must be expressed in lengthy complicated equations. We call them scientific fundamentalists. They are just as misguided. Dawker continuous onslaught using political means on religion and anything resembling it is but one symptom of this narrow minded fundamentalism. One question these fundamentalists never ask themselves: why are they so fanatic about denying anything they don’t see including God, whatever this is? A fast more rational way is to analyze the brains of these people as well as the brains of the oppositely inclined people to find what all the fuss is about. A look into the mirror, a chat with a beautiful woman, or reminiscing about past time may help fundamentalists to know the real reason for why they are insisting on whatever they are insisting on particularly when we could not know the answer. E-Infinity starts where George Cantor started. It will be surprising that those who believe in chairs and tables as chairs and tables and we jolly good will build some could swallow E-Infinity immediately. But I think I may be wrong here. If you really understand fractals, and if you really did not just learn it by heart from a text book because others have accepted it, then you will understand E-Infinity. The wonder of E-Infinity is the wonder of fractals and the empty set. Gerard ‘tHooft once said: “I understand what negative dimension is, it is a Grassmanian variable”. That is fine but it is not fundamental. The fundamental thing is to say a negative dimension is the dimension of the empty set. In fact the empty sets. All empty sets are fractal like. The totally empty set is the incarnation of nothingness. For traditional physicists with or without the highest decorations, this is a difficult part to swallow. Tim Palmer understood part of the dilemma. He put it in the most eloquent way we know of. Far more eloquent than anything which Ord, Nottale or Mohamed have written. It is eloquent because it is short, sweet and simple. He said quantum mechanics is blind to fractals. This is the main problem. That is why wild topology is important. At this level philosophy is not decoration. At this level philosophy is part and parcel of the physical shebang.
What I wanted to clarify in this communication was initially the question of unification, Weyl scaling and the rest. Inevitably diversion took place. We have to stop and start again in another communication. I will call it if I am the one to write it: an equation searching for a Lagrangian. Or was it six characters searching for an author. If Barandello is correct then they have to be six equations searching again for a Lagrangian. For the time being, I bid you goodbye for I have to take the plane or was it the train to San Fernando. A last minute note, you need some literature for wild topology. I cannot recommend strongly enough the classical book of John Hocking and Gail Young “Topology.” It is republished in Dover, copyright 1961. Another Russian book on set theory is by N.J. Wilenkin. It is called Set theory for Entertainment. Finally there is a marvelous book by Christian Beck, an English Professor of Physics at Queen Mary College called: Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields. It is published in World Scientific, copyright 2002. Beck reproduced everything that Mohamed El Naschie has found but using a computer. The work is recommended to anyone who is deluded to think that El Naschie’s work is numerology. Beck calculated the possibility that this is numerology and found that the possibility is 10 to the power of minus 38.
Goodbye for now but not for long.
Communication No. 19
Miscellaneous comments, some corrections and continuation of Part 3 of Communication No. 17
Let me start with a minor correction. We said that Christian Beck in his famous book linking non linear dynamics to high energy physics: “Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields” has calculated the probability of an element of numerology and found it to be absurdly small. We quoted a number. The correct number is even far absurdly small than our memory had it at the time. The number quoted on page 247 by Beck is ten to the power of minus 60. Let me repeat. This is 1 divided by ten and another 59 zeros beside it. You have to be incorrigible fanatic to talk of any numerology in the work of Beck who worked directly with a computer as well as the work of El Naschie who knowingly or unknowingly was working with the golden mean binary system which can match any computer.
We all know that super string depends crucially on a number well known from number theory namely 496. Is this numerology? Not even the greatest enemy of string theory could make such claim. We know that loop quantum mechanic would not work at all unless we multiply everything with a mysterious numerical factor, the Imerze-barbiro number. I probably spelt the name wrongly but never mind is loop quantum mechanic numerology? I don’t think our Nobel Laureate Gerard ‘tHooft would like that at all, he believes reasonably well in loop quantum mechanic because it does not have extra dimensions. Why? Because extra dimensions cannot be seen. Is this prejudiced? I don’t think he believes it is prejudiced. Forgive me for being confused at this point. Did anybody see an instanton? Did anybody see the unification monopole of Polyakov and ‘tHooft? Don’t get me wrong, I believe they will be found or something similar will be found. However it is clear that beauty is in the eye of the beholder. Funding is a problem. Some say “Cherchez la femme.” In our community it is more the case that we should “cherchez la monnaie”. Continuing in the same vein, what about this quadratic equation which Mohamed El Naschie has given? The equation was expressed in terms of alpha bar. It reads alpha bar square minus 140 alpha bar plus 400 equal zero. It has of course two solutions: the first is alpha bar is 137 plus k0. This is 137.082039325. Note it is the theoretical value of E-Infinity. It is not the experimental value. The second value is 3 minus k0. This is a scaling of one of the dimensions of spacetime. Now take the mass of the proton. According to E-Infinity it has to be alpha bar square divided by 20. That way you get the real experimental value of the proton. Take the experimental value of alpha bar and do the same. You will not get the correct value for the mass of the proton. You will get a good approximation but not the correct value. To understand that, you have to read El Naschie’s paper on the Mass Spectrum. However before this you have to depart from the naïve belief of reality physics and maths. You cannot take Newton nor Einstein as a paradigm in a naïve way. I may come to this point later on when we have the time. For the moment I would like to give a second example. In fact I would like to give many examples to refute the naïve belief of dividing physics from maths in a simplistic way. I said previously that the real negative dimension is the empty set. The best example I know is El Naschie SU(n) hierarchy. I will not give now details but I give it to you as given in the papers of El Naschie. It is minus 1, zero, 3, 8, 63 and then finally 3968. This is not ‘tHooft’s beloved Yang-Mill theory. It is the super Yang-Mill theory. If in any doubt, please see Kaku’s book:”Introduction to superstring and M theory, page 385. El Naschie’s hierarchy just stated is connected to n equal zero 1238 and 63 in the same order. I will explain that later on in detail. I just want to say that those who think this is numerology know nothing about numerology or physics and Beck calculation is a numerical confirmation for that. In fact the work of Renate Loll and Jon Ambjorn lies squarely on the side of El Naschie and Beck. They use with great skill the number crunching efficiency of a computer. El Naschie on the other hand does the same using a number system made extra for that. The Great Russian academician Alexey Stakhov said in his recent book which was acclaimed on the level of a Nobel Prize that missing the use of the golden mean binary system was a disaster and nothing less for the development of mathematical physics. I have made many claims here and I did not give you details. I owe you a few. I will do this as soon as we can piece things together for you. I also promise to show you that the theory presented for quarks confinement using E-Infinity is absolutely correct. We had our doubts initially but now there is no doubt whatsoever. That also will be addressed very soon.
Communication No. 20
The Golden Mean in High Energy Physics before, during and after E-Infinity
We will have to leave it to the philosopher and historian of science to determine the complex history of the golden mean in high energy physics. As far as we are concerned, we feel that Mohamed El Naschie must be accredited with integrating the golden mean in high energy physics in a systematic way and on a grand scale. He did not do that intentionally. It just happened. The golden mean more or less manifested in the computation as fundamental for any minimal consistent and accurate quantum field theory formulation outside the rules of classical quantum field theory. Without any attempt to be historically correct we must draw attention to very important papers where the golden mean manifested itself. I must say that the authors which I am about to mention were initially not traditional mainstream. They are not renegades. They are somewhere in between. They are meantime part of the establishment but it was not always like that. The first is an exceptional
Russian mathematician who worked initially in turbulence, A. Polyakov. The second is a superb solid state physicist, mathematician and hobby engineer, Subir Sachdev. If my memory serves me right, although this is slightly on the gossip side, I think Sachdev’s American wife is the daughter or the grandchild of Dwight Eisenhower, the great President of USA and the hero of D-Day in the Second World War. The paper of Polyakov is entitled: Feigenbaum universality in string theory, published in Journal of Theoretical Physics (JETP), vol. 77, No 6/March 2003, pp. 260-365. Polyakov found the period doubling of Feigenbaum in quantum field theory. Please read Mohamed El Naschie’s paper on the connection between the hyperbolic region of period doubling and the Hausdorff dimension of fractal spacetime. A critical value in the hyperbolic region is his famous 4.23606799. When you talk Feigenbaum, you talk golden mean renormalization groups. In fact it was
Mitchell Feigenbaum, Otto Rossler, Julio Casati, Boris Cherekov and Itmar Proccaccia who initiated Mohamed El Naschie’s interest in nonlinear dynamics, KAM theorem, period doubling and thus the golden mean threshold. Mohamed El Naschie merely extended that to high energy physics. The second paper by Sachdev was published in Physics Letters B 309, 285(1993), Polylogarithm identities in a conformal field theory in three dimensions. You can find it free of charge published in arXiv: hep.th/93605131, 25 May 1993. An extremely instructive and neat summary of the application of the golden mean is a nice paper by the very versatile, Slovenian mathematician L. Marek-Crnjac. The paper is titled: The golden mean in the topology of four-manifolds, in conformal field theory, in the mathematical probability theory and in Cantorian space-time, published in Chaos, Solitons & Fractals 28(2006) 1113-1118. A wonderful paper by Professor Christian Beck from Queen Mary
University, London and Muhammad Maher from the same department is: Chaotic quantization and the mass spectrum of fermions, published in Chaos, Solitons & Fractals, 37 (2008) 9-15. This paper was refereed and recommended for publication in Chaos, Solitons & Fractals by Professor, Dr. Dr. Werner Martienssen from the University of Frankfurt. In this paper you can see the influence of nonlinear dynamic and cantor sets in modern physics and determining the mass spectrum of elementary particles in a similar but not identical way to E-Infinity. It was not always golden mean from the beginning. El Naschie used initially deterministic fractals. He started initially by using the classical triadic cantor set with the Hausdorff dimension ln2 divided by ln3. You do not get golden mean for deterministic cantor sets. Paradoxically it is randomness which introduced golden mean harmony. You can see that from a paper published in Vistas in Astronomy. The author is
Mohamed El Naschie. The title of the paper is: Quantum Mechanics, Cantorian Space-time and the Heisenberg Uncertainty Principle. This paper dated 1993, vol. 37, pp. 249-252, did not include the golden mean yet. Mohamed El Naschie rediscovered the average Hausdorff dimension of a quantum path. This is equal to 2. It is the individual Hausdorff dimension of a quantum path which is equal to the golden mean. The interplay between 2 and the golden mean produced the approximate value for the Hausdorff dimension of the core of quantum cantorian spacetime which is approximately equal to the exact value. To be specific, it is 2 divided by ln of the inverse golden mean which gives us an approximation to the exact value 4.23606799. There is a nice paper summarizing the application of the golden mean by El Naschie titled: The Golden Mean in Quantum Geometry, Knot Theory and Related Topics, Chaos, Solitons & Fractals, Vol. 10 No. 8 page 1303-1307 (1999). Another
paper which seems to have strong influence on groups working in the Parameter Institute in Canada is El Naschie’s Quantum Groups and Hamiltonian Sets on a Nuclear Spacetime Cantorian Manifold, published in Chaos, Solitons and Fractals, vol. 10 no 7, pp. 1251-1256 (1999). The golden mean as such and its connection to E8 became fundamental in the work of Mohamed El Naschie after one of his students, Dr. Ahmed Mahrus from Newcastle, Department of Physics, UK, drew the attention of Mohamed El Naschie to the golden mean binary system. Academician and Nobel Prize nominee Alexei Stakhov expanded this system in a recent magnificent work published by World Scientific entitled: The Mathematics of Harmony. Mohamed El Naschie started his adventure with period doubling and renormalization relatively early. He was at the time Director of Projects and one of the main editors of a prestigious Middle Eastern Journal. Later on he published a paper on the subject
titled: Order, Chaos and Generalized Bifurcation. The paper is published in the Journal of Engineering Sciences, King Saud University, vol. 14, no.2 (1988), pp 437-444. I did hear some good news for those who do not have easy access to expensive scientific Journals. I am told that a charity organization did put all the scientific papers of Mohamed El Naschie in a free access blog. I do not know where or when this was done, but those who will search will find it. I hope this will facilitate serious study of E-Infinity. Of course those who prefer other activities will not be deterred from following their natural inclinations. We hope however that the majority will follow their scientific inclination. We hope also this little contribution is helpful and we will be shortly returning with more.
E-Infinity
Communication No. 21
Why does the golden mean pop up everywhere in mathematical physics?
A true scientist aiming at scientific truth could not be afflicted by a worse malady other than prejudice. Some notable scientists who have done occasionally excellent work elude themselves in confusing prejudice with scientific skepticism. I am far too skeptical to believe anything easily, you will hear them say. Give him anything new and he will answer immediately: I do not need to read it. I waste my time on big names only. I know before I read that this is no good any case. Too many great people have tried before and failed. Who the hell could be this guy from Rumania to teach me a Caltech man about the quantum or anything at this level? When it comes to the golden mean things could be dozen of times worse. Ignorance is invariably covered up pure arrogance and never ever forget the witty jokes, the hallmark of a hole head. In what follows we would like to give well known elementary evidence that it is absolutely natural for the golden mean to be
the foundation of quantum mechanics and high energy physics and much much more. Many of these evidences have been discussed at length by Mohamed El Naschie, Marek-Crnjac and their students. For convenience summary, we recommend a paper titled: A short history of fractal-Cantorian space-time by L. Marek-Crnjac, published in Chaos, Solitons & Fractals 41 (2009) 2697-2705.
1. The golden mean is the solution of a simple quadratic equation with appropriate sign. A quadratic equation is the simplest non linear equation we know of. Only a linear equation is simpler. Einstein said if everything would be linear nothing would affect nothing. Therefore, for things to affect things we need at least a quadratic equation to describe physics of a minimal complexity. The simplest vibrational set fulfilling Einstein’s requirement is a 2 degree of freedom linear oscillator. The characteristic equation for such set when normalizing all constants is a quadratic equation with a golden mean solution. This is discussed at length in a paper by Mohamed El Naschie titled: On a class of general theories for high energy physics, published in Chaos, Solitons & Fractals. 14 (2002) 649-668. The Slovenian mathematician Marek-Crnjac suggested that fusing infinitely many but hierarchical sets of this type leads to E-Infinity theory. The connection
to the basic concepts of string theory is evident. It might be interesting for some to note that structural engineers habitually replace complex structures by systems of springs and masses. In some sense this is a finite element realization of a structure. In the same sense and taking a bird’s eye view this is the connection to Regg Calculus. No wonder that Mohamed El Naschie used this method when you remember he is a Structure Engineer and a past student of John Argyres, the inventor of finite elements who used it skillfully in the modern fuselage structure of airplanes at the dawn of the aerospace age.
2. The golden mean is the most irrational number. Its continued fraction expansion involves only unity. Consequently it is the backbone of the KAM theorem. There is no stability in a Hamiltonian system without a rational number. Since the golden mean is the most irrational, it is the threshold of the most stable periodic orbit in a dynamic system. The marriage between KAM and quantum mechanics resulted in Rene Thom’s VAK conjecture which has been re-generalized by El Naschie and used in high energy physics.
3. A random cantor set possesses a Hausdorff dimension equal to the golden mean. Of course there are many cantor sets which are random and have a Hausdorff dimension close to or different from the golden mean. However the most simple and generic random cantor set has the golden mean as a Hausdorff dimension. A wild topology always ramifies at infinity into a set of wild point, equivalent to a random cantor set. Generically this is equal to the golden mean. If you regard the final state and forget about the mechanism leading to it, then you have golden cantor sets geometry at ultra high energy corresponding to the wild topology. The basic mathematical work in this direction was done by an American topologist Alexander. The most famous examples are Alexander Horned spheres and Antoine Collier. Mohamed El Naschie merely carried these ideas to high energy physics and was probably inspired by S. Kaufmann in the U.S.A.
4. The fundamental theory of 4-manifold depends crucially on the Fibonacci and thus the golden mean. This has been considered at length in the corresponding mathematical literature. Many references to this work are given in El Naschie’s papers and elsewhere, for instance Crnjac’s work.
We must stress that the excellent work of T. N. Palmer depends crucially on an understanding of number theory. In fact classical quantum mechanics depends on number theory. You must always remember the trivial fact that without complex numbers, there is no classical quantum mechanics. We think enough for today as my hands are getting heavier and we hope to be back as soon as possible.
Communication No. 22
The inverse problem of quantum field theory in E-Infinity theory and symplectic tiling
We mentioned few very good reasons why the golden mean should pop up at so many different places and so unexpectedly in quantum high energy physics. We reasoned that quadratic equation with golden mean roots is the simplest non trivial algebraic equation that there is. We mentioned the maximal irrationality of the golden mean and the role it plays in KAM theorem and the stability of dynamic systems. We alluded to wild topology and its connection to the simplest form of random cantor set which by a well known theorem due to American mathematician Mauldin and his student William will have a golden mean as a Hausdorff dimension. There are far more reasons than what we mentioned. The reasons are sometimes very subtle and none is so subtle and so important than the relation to Penrose Tiling. I remember vividly attending a lecture by Professor El Naschie in the Einstein Institute for Gravitation in the Max Planck Institute near Berlin, Germany. A very
imminent and famous German astrophysicist was present when El Naschie was talking about the Penrose Tiling and E-Infinity theory. The imminent German scientist became very agitated and said: “This is all a simple tiling, how could you scale things denying the existence of a natural scale and how could use that for high energy physics”? El Naschie was equally agitated but remained calm. Of course it was El Naschie’s mistake. He thought everyone has seen the wonderful example for non commutative geometry presented in the book of Alain Conne. El Naschie explained to the imminent astrophysicist that of course he should have said Penrose fractal tiling. El Naschie meant that every tile in Penrose Tiling could be tiled again using Penrose tiling and so on ad infinitum. The great German astrophysicist and he was definitely a great astrophysicist was not familiar with fractals. He belonged to a generation which worked decades before Andre Linde, the
Russian astrophysicist, moved to America and introduced fractals to the big bang. Talking to Mohamed El Naschie later on, the irony became even bigger. El Naschie learned much about fractal geometry in astrophysics and quantum mechanics from a remark in a paper published in the 60’s by the very same German astrophysicist. The remark concerned the famous paper of Carl Menger which he dedicated to Einstein at his birthday. Many of us and I am not an exception refer to papers without reading them attentively. Let us return to Penrose tiling. Without the golden mean, there is no Penrose tiling but why is it like that? El Naschie gave a naive example which I find very instructive. Being a structural civil engineer, his example comes yet again from engineering. Suppose you are building a wall. You are using bricks. Watch a master mason performing his job. He fits the bricks together. These are the integers in number theory. Now and then he takes a smaller
or a larger brick so that things fit a little bit better. These are the rationals. However no matter how clever our master bricklayer is, he will never get a smooth monolithic wall without resorting to mortar. The mortar between the bricks helps to produce a smooth monolithically connected wall. These are the irrationals of number theory. If you take the golden mean 0.6180339 you notice that it is almost equal to half i.e. 0.5 + 0.1180339. It is a rational plus an irrational tail which is easily expressed again in terms of the golden mean. In this case it is simply 1 + k all divided by 10. The k is the famous golden mean to the power of 3 multiplied by 1 minus the golden mean to the power of 3. We met this number frequently when we discussed transfinite corrections. It is this technique of writing things in the most convenient form using the simplest and most efficient binary system that there is which allows E-Infinity to produce exact results with
simplicity which is difficult to comprehend when we use the old mentality of algebraic manipulation brute force, patching and approximating at different stages until things become approximately correct but truly ugly and cumbersome to handle.
I am too young to have been together with Heisenberg, Paul Dirac and Neils Bohr at the Tate Gallery in London when they visited it. Mohamed El Naschie told me however the following story which he again heard it from Heisenberg directly. The story is too beautiful not to be true even if it is true and just ingeniously invented by El Naschie to impress us. Paul Dirac loved perfection. When he presented a draft paper to Neils Bohr and the latter made his usual correction, Dirac becomes extremely sad. One day the three mean were in London. Paul suggested taking the opportunity to visit the Tate Gallery. I knew the old location of this gallery because Mohamed El Naschie took me there when I visited him in London. They were standing all in front of a large picture either by Turner or Claude Monet. I cannot remember. Dirac stood silent for a while then he moved forward and pointed to a spot at the bottom of the painting and said in his calm voice:”This point
is wrong”. I do not think I can say anymore. That must be one of the best signs of Paul Dirac’s conviction that it must be beautiful or it is not true or correct. If you look to the classical form of the renormalized equation of unification of fundamental action in high energy physics, you realize that it is extremely involved, clumsy and cannot be described as being beautiful. However it is approximately true. To use the same language of Dirac, there is a point there which is wrong. Not so with the same equation which E-Infinity produces. The equation of E-Infinity is perfection per excellence in this case. We will discuss it in detail. But I wanted to introduce the idea first. I know from El Naschie that he traveled to a country in Northern Europe known for its beautiful tulips, in order to introduce his equation to a man whose opinion he values above every other mortal. To make a long story short, the great man told Mohamed El Naschie:”Yes,
it is amazingly simple but there are too many things before that. Your equation comes from where? It just comes out of the blue”. I cannot reproduce the sense of disappointment which Mohamed El Naschie felt at this point. I could almost hear what is going in his mind. Of course there are many things before that. I thought you know that I know that. In fact how a great man like you could think that I could not know that. The problem is Mohamed El Naschie knew what the great man knows but the great man did not know what Mohamed El Naschie knows mainly fractals, chaos and complexity theory. You could write any simple equation if you can. If you are clever you can always guess the answer and idealize it. Next you ask yourself: What do I need to have? Normally a Lagrangian in order to get this result. This is the well known inverse problem. El Naschie worked on many problems in the calculus of variations. He knew from his classical engineering work
the power of the inverse problem as a method. The inverse problem is always more difficult. For instance, relative to multiplication division is more difficult. Similarly we have many opposed mathematical procedures where the inverse is always more difficult. It is not impossible to find whatever you want when so many clever people have worked so hard to give you a theory which is almost but not completely perfect. Classical quantum field theory is such theory. It is almost but not completely perfect. Those who have studied it carefully will be led to a simple solution of the inverse problem as will be explained later. To be able to complete the job, you have to free yourself of all prejudice and have no hiccups about fractals, transfiniteness and golden means. The result of this effort will be evident from the following equation of E-Infinity theory. In general the equation reads as following: the inverse unification coupling is equal to the
following sum: alpha bar 3 plus alpha bar 4 plus the natural logarithm of the ratio of the unification mass or energy divided by a reference mass or energy. This logarithm is multiplied by the factor which has to do with super symmetry. In the classical form this equation is not much different but it is clumsy and filled with numerical factors which a beginner could not make heads or tails of. It takes a long time to familiarize oneself with it. The E-Infinity theory version is by contrast perfect. Let me mention first that the discovery of the Logarithmic scaling decay was a major step in high energy physics. I forgot who discovered it first and I forgot a lot of things about its history which I read once upon a time. I do not remember where I read it but I did. We all would be very grateful for readers of these comments and I mean readers who are only interested in science could draw our attention to this logarithmic law and give us its background
and history. That would be very nice indeed and would save us a lot of work to dig in old papers and textbooks. E-Infinity took this logarithmic law and changed it de facto to a golden mean scaling of E-Infinity hierarchy.
In the next communication we will show you these things step by step. We hope also that you will notice that the inverse problem becomes tractable and lead to a simple solution of quarks confinement using E-Infinity modification of the original classical solution. Hopefully without wanting in any sense to sound pompous and give our enemies fuel to burn more wood, let me say that if we in E-Infinity could see so far then it was because we stood on the shoulders of giants. These giants are: Gerard ‘tHooft, David Gross, David Politzer and Frank Wilczek, to mention only a few of the key players in this field.
Communication No 23
Unification and Confinement
Although the negative coupling constant of strong interaction related to confinement was discovered long ago by the young Gerard ‘tHooft in an unpublished work and was rediscovered several times later we hold different views about confinement. In E-Infinity theory we take the view proposed by El Naschie that an exact solution to confinement must be related to the structure of space time as well as the nature of the Planck scale. Similar to quantum gravity E-Infinity believe that the key to understanding confinement thoroughly is the Planck scale. We are of course aware that some regard experimental phenomena involving jets as an indirect observation of free quarks of a sort. However we find the picture a little bit involved using experimental and theoretical reasoning in a much interwoven way and therefore a clear cut mathematical derivation is not there. We are almost there. With almost I mean 95% there. However theoretical physicists are
perfectionists. They could not leave a single spot anywhere without worrying about. This attitude is classical and ‘tHooft himself is a classical example for this way of thinking. Having said that, we must stress that El Naschie is by no means the only one who thinks that confinement should be linked to Planck energy scale physics. A British physicist working in Hungary, Lawrence B. Crowell, holds very similar views but we will not go into there in this communication. Let us go back to our exact renormalization unification equation.
Let us probe things at the Planck scale. This is 10 to the power of 19 GeV. As an energy reference scale, we take this time not the Z0 scale but the electron scale. To be precise we take a Cuba pair. The mass is twice that of an electron which is roughly 10 to the power of – 3 GeV. The natural logarithmic term in this case is ln of the ratio of the above mentioned scale. In other words it is ln 10 to the power of 22. School mathematics teaches us that this is 22 multiplied by ln 10. Anyone looking at this number and is also familiar with E-Infinity transfinite corrections will guess immediately in a split of a second that 22 should be 22 + k and ln 10 should be 2 + 2k. Here k is our famous number 0.18033989. More accurately it is the golden mean to the power of 3 multiplied by 1 minus the golden mean to the power of 3. Our logarithmic term is therefore exactly equal to 52 plus 2k. Assuming super symmetry we have to divide by 2 and obtain 26 + k.
This is exactly 26.18033989. Our equation is now as follows. On the left hand side we have the inverse unification coupling. On the right hand side we have the inverse strong coupling alpha bar 3 added to alpha bar 4 added to the value of the logarithmic term divided by 2. On the other hand, we know very well that super symmetric unification takes place at the inverse coupling 26 + k. This term on the left hand side cancels the logarithmic term divided by 2 on the right hand side. Therefore we conclude that alpha bar 3 plus alpha bar 4 must be equal to 0. Now we know for sure that alpha bar 4 is equal unity. It follows therefore that the inverse strong coupling alpha bar 3 must be equal to minus 1. This is exactly the point which people searching for confinement missed again and again until young ‘tHooft found it and spoke about it in a discussion which took place in a conference in Marseilles, France many years ago. Our derivation is however exact.
El Naschie gave some eccentric elucidation using analogies from visitors in outer space. You can read this in his papers which are now available free of charge in a blog financed by a charity institute. Physicists love trivial eccentric analogies to illustrate the main points of what they are proposing. Einstein loved to ride over a photon and fly with a speed of light. Others have even more outrageous ways of illustrating their point. To make a fuss about these things shows only how shallow the person in question is. I am really putting it here very politely. After all we are assuming that all readers of this blog are scientists. The odd 2 or 3 do not really count. We will return to confinement and harmonization in the next communication.
Communication No. 24 (Part 1)
The Fundamental Equation of Unification of E-Infinity Theory as a harmonization of the corresponding renormalization group equation of classical high energy physics.
After considering many fundamental problems in non linear dynamics and its possible connection to quantum physics, a paper entitled: Complex Dynamic in a 4D Peano-Hilbert Space appeared in Il Nuovo Cimento in 1992. This was a famous journal where Einstein published some of his work but after a short period of stagnation the Journal was re-launched and renamed: The European Journal of Physics. The author of the paper is Mohamed El Naschie and he wrote it during his Cambridge time. It can be found in Vol. 107 B, N. 5 – Maggio 1992 pp.583-594. In this paper Mohamed El Naschie made an explicit connection though in general terms between quantum mechanics and fractals. Sub section 8, page 593 is titled: Quantum mechanics, Dirac’s vacuum and foam space-time. He was not accurate at that time by thinking that Wheeler’s foam space time is a proper fractal. Nevertheless the point was made. In fact he mentioned explicitly that lifting the Menger foam
to 4 dimensions leads to something very similar to our space time. He was clearly not yet aware of the fundamental role played by randomness and was still working with a deterministic cantor set. In a paper published at about the same time in the Journal of Franklin Institute, Mohamed El Naschie expanded the idea, discussed in more detail the Menger sponge and expressed his hope that fractal cantorian spacetime can easily resolve all the paradoxes associated with quantum mechanics. Mohamed El Naschie continued his effort until he determined the exact expectation value of the topological and Hausdorff dimension of quantum space time. He perfected the theory mathematically after discovering the relevance of Jones’ Invariant and quantum groups to his work. About 7 years after his the mentioned Nuovo Cimento paper, the mathematical basis and the connection to hyperbolic geometry and KAM theorem became crystal clear. Just as two examples for his work
in this period, I may quote the following two papers: 1. Jones’ Invariant, Cantorian Geometry and Quantum Space-time. 2. Quantum Groups and Hamiltonian Sets on a Nuclear Spacetime Cantorian Manifold. Both papers are published in the 1999 volume of Chaos, Solitons & Fractals. In other words latest by 1999 the following fundamental facts were proven by Mohamed El Naschie and elaborated upon extensively by J. Huan-He, Marek-Crnjac and Erwin Goldfain. First the building blocks of quantum spacetime are random cantor sets with a golden mean Hausdorff dimension. Second, the expectation value of the topological dimension of the space time core is exactly 4. On the other hand, the corresponding exact Hausdorff dimension is exactly 4 plus the golden mean to the power of 3. That is to say, it is 4.23606799.... In 2008 the paper by Jan Ambjorn, Jerry Jurkiewicz and Renate Loll appeared in Scientific American. The paper uses computer simulation extensively. Well
you could say it is not exactly computer simulation but computer computation of a Regg calculus approximation of a quantum gravity model. You could also view it as an improvement of Regg calculus using Ambjorn’s triangulation. Professor Jan Ambjorn from the Neils Bohr Institute in Copenhagen is a world authority on triangulation technique of this kind. Leaving details aside and to make a long story short, the main thrust and results of the paper are the following: Quantum space time is best modeled using cantor sets as building blocks. Second, assuming Prigogine’s arrow of time on the quantum level, things work out perfectly. Third and most importantly, a space time dimension of 4.02 was worked out from first principles for the first time. To see how much emphasis the authors put on this fact, let us quote verbatim what they wrote on page 29 of their July 2008 Scientific American Paper. They wrote “Imagine our elation when the number of
dimensions came out as 4 (more precisely as 4.02 plus minus 0.1.)It was the first time anyone had ever derived the observed number of dimensions from first principles. “ We in E-Infinity theory group can imagine their elation but we ask you at the same time to imagine our alienation when no reference whatsoever was made in this paper or any of similar papers published at the same time in Physics Review Letter and the Journal of the Institute of Physics to our work. The first derivation of the dimensionality of quantum space time from first principle was made as you can check yourselves at least 9 years earlier. In fact the value found by Ambjorn et al namely 4.02 was found to be a spectral dimension and obtainable using other methods. One of these methods is the Bose Einstein statistics of El Naschie was found approximately 16 years earlier. When you see the structure of the paper in Scientific American you see that the logic expressed in the
fractal figures on page 28 and 29 including the Menger Sponge and the Serpinski Gasket follow the same logic of El Naschie’s paper in Il Nuovo Cimento and the Journal of the Franklin Institute. Rediscovering things again and again was not uncommon in the past. We mean no disrespect to anybody when we point out these facts. What is important is how we react or others reacted to these facts. Something positive came out of this any case. Cantor sets are in quantum mechanics to stay. Random cantor sets with golden mean Hausdorff dimension are meantime an experimental fact. No amount of propaganda could possibly change these facts. Should we have unintentionally compromised anyone then we as a group apologize collectively as long as the issue of priority is restored. For science priority is unimportant. For scientists it is important. The Yang-‘tHooft know that better than anyone else. He wrote a great deal about something similar which happened to
him in connection with the strong interaction. This is the subject which we will discuss shortly. Before doing that however let us recall something extremely important in various respects which involves Nobel Laureate Gerard ‘tHooft. In an excellent book on quantum gravity, edited by Daniel Oriti published in Cambridge this year but dated 2009, Gerard ‘tHooft makes the following answers to questions on page 155. The question was by L. Crane about non integer Hausdorff dimension in quantum gravity. ‘tHooft answered as follows: “We thought of such possibility. As far as the real world is concerned.............................................Weltman once thought there might be real physics in non integer dimensions but he never got anywhere with that............................................ I do know what negative dimensions mean................................it is anti-commuting coordinate.” Those in the know must be exhilarated to see how
far ahead of anybody E-Infinity group was. With all the due respect and it is a genuine respect we have for Gerard ‘tHooft, his statement could not remain unchallenged. Mohamed El Naschie derives the exact value of the Hausdorff dimension of quantum spacetime namely 4.23606799 from ‘tHooft’s dimensional regularization and he gets real physics out of it. We use the word real physics with large doses of salt. Talking about real physics in a simple way in the realm of quantum gravity can be the most misleading thing which one can do. We do not intend to dwell on the illusive nature of reality. However those who are philosophically inclined to use words like real physics should know about dozens of incidents where inclination to reality made reality disappear. The paper in question was entitled: On ‘tHooft dimensional regularization in E-Infinity space, Chaos, Solitons & Fractals 12 (2001) pp 851-858. This paper was preceded by another paper
which was entitled: “ ‘Thooft dimensional regularization implies Cantorian Space time.” The paper was presented at a conference which the great man ‘tHooft himself attended. We sincerely hope that this closes the subject which Oh! So many lesser mortals try to keep artificially alive again and again on certain blogs. When you multiply 4.23606799 by ten you get 42.3606.........and this is the non super symmetric inverse grand unification coupling. Let us discuss how this value as well as the super symmetric value namely 26.18..........could be obtained from the fundamental equation of unification. This is what we will do in part 2 of this communication.
Communication No 25 Part 2
In Part 1 of this Communication we discussed the fact that the basic Hausdorff dimension of E-Infinity follows from ‘tHooft’s dimensional regularization and we hinted at several other things, for instance negative dimension as well as the connection between E-Infinity and certain types of Regg triangulations. We will return to negative dimension, fractal and the so-called physics as distinct from mathematics. Here however we will concentrate on the unification equation.
In E-Infinity the ideal value of the electro weak, electro magnetic, and strong coupling are given by inverse forms and they are remarkably integer values. We use them to reconstruct the inverse electro magnetic constant. We have done this several times before in these communications. Let me remind you. We have alpha bar 1 equal exactly 60. Alpha bar 2 is half this value namely 30. Alpha bar 3, we divide in 2 alphas. It is equal to 8 + 1 = 9. All these values are remarkably close to the experimentally found value. Please note these values belong to an idealized world. It is the world of Plato. It is the world of ideals. It is the world of the supreme entity which organizes this world. It is the world of the theoretical and mathematical physicists who are constructing general theories emulating the work of the Supreme Being and hoping at the end to find some indication that these theories are right. The indication comes out from the messy real world. It
comes out from the messy laboratories. It comes out from billion of datum measured with different accuracies subject to human scatter, fallibility and noise. In addition we have alpha bar 4 which is equal unity. This value denotes the coupling between the Planck masses to the Planck ether, something which we most probably will not discover directly. The square of the Clibsch factor is also given in its ideal transfinite form namely the inverse of the golden mean which is pretty close to the classical value namely 5 divided by 3. Try it out. 5 divided by 3 is equal to 1.666.........The inverse golden mean is on the other hand 1.618033......Now we can write our first renormalization equation result for which we obtain the exact theoretical E-Infinity inverse electro magnetic fine structure coupling constant. The equation reads: alpha zero bar equal 60 multiplied by 1.618033.....plus 30 plus 9 plus 1. This adds together to exactly 137.082039325........
Next we would like to derive the non-super symmetric unification inverse coupling. We follow almost the same procedure. This so called coupling constant which is of course not a constant but a function of the resolution which means a function of energy is equal to alpha bar 3 plus alpha 4. That means equal to 9 plus 1 = 10. Then we have the logarithmic term. The exact theory does not have a logarithmic term. The exact theory replaces logarithmic term with golden mean scaling. However let us work first with an approximation using the logarithmic term because it is educational and helps the beginner to see the light. Grand unification takes place roughly at the ‘tHooft-Polyakov monopole. This is a mass of about 10 to the power of 16 GeV, when expressed in energy unit. Of course we cannot take a natural logarithm of a figure with dimension. We have to resort to a trick to make it dimensionless. The trick is not arbitrary. We more or less take a
reference energy which is the energy at which we take measurement. In this case we take the electro weak. More precisely let us take, as in the classical theory which you can find in any textbook, the energy scale of Z0. This is 91 GeV. When you take the natural logarithm of the ratio of both energies and we implore you to do so with your computers or pocket calculators, you will find it 32.3305...... We can assure you that the exact value is 32.3606...... and that this value is nothing but a scaling of the exact inverse electro magnetic fine structure constant of E-Infinity. In other words the logarithmic term is in reality nothing more and nothing less than 137.082039325........ multiplied by the golden mean to the power of 3. Adding our earlier 10 to this value you will find that the inverse unification constant becomes 42.3606...... This is exactly as we anticipated ten copies of the Hausdorff dimension of E-Infinity spacetime. What we have
done here needs more elaboration. We will do that later on in detail. For now it is sufficient to know that 42.3606.... is the non-super symmetric inverse unification coupling constant. To obtain the super symmetric value, we contemplate in the following manner. The minimal super symmetry requires that every particle should have a super symmetric partner, a so-called sparticle. Since we are working with inverse value, the doubling enters as an inverse value. In our equation this is expressed via a factor 1 divided by Roh. In this case Roh is 2. Our factor is thus a half. This half has to be multiplied by the natural logarithmic term. This term we just calculated to be 32.606.... Half of that is exactly 16.18033.....Adding to this value our 10, the final result is 26.18033.......This is a delightful result. With minimal effort and incredible accuracy we obtain the two fundamental coupling constants of unification of all fundamental forces. The result
agrees with most of the approximate results existing in the literature using various methods. These results vary in the first case between 40 and 45 and in the second case between 24 and 27. We will continue our calculation in the next Communication but we would like to close the discussion here by returning to ‘tHooft’s elucidation of negative dimensions. On such occasion it is quite in order to make some remarks on the nature of reality.
You remember Gerard ‘tHooft said in the discussion reproduced on page 152 of the book of Oriti which we cited earlier on: “A negative dimension could be understood as an anti commuting coordinate”. In other words, we think ‘tHooft means a Grassmanian value, something which is routinely used in super symmetric theory. We note parenthesically that ‘tHooft is normally skeptical about super symmetry and does not believe that it really exists in Nature. Then ‘tHooft continues by saying that we could think of an anti commuting differently and understand it as a negative dimension which replaces integration by differentiation. He said explicitly “differentiation is the inverse of integration.” A pure mathematician would find this language absurd. However from our E-Infinity viewpoint, we agree with the imaginative language of ‘tHooft and refute the pure mathematician allegation of absurdity. Of course we know that 1 divided by integration is
not equal differentiation. Such a sentence has no meaning. Nevertheless a theoretical physicist, we know, what ‘tHooft is trying to tell us. We find it a great pity that the truly great Nobel Laureate Gerard ‘tHooft is not familiar with the mathematical theory of dimension and the work on E-Infinity and particularly El Naschie’s theory about negative dimension. You see ‘tHooft could reformulate himself by saying that he knows that a negative dimension namely a minus 1 dimension is the dimension of the empty set. The beauty of the whole thing is that we could use a more tangible language and say that something which has the dimension of minus 1 could be modeled by a fractal. As fractals become thinner and thinner, the dimension becomes more and more negative and when the fractal totally disappears, its dimension becomes infinitely negative. Many could say that fractals are just geometric figures and not physics. In such case I just pass. If
someone feels that Grassmanian variables are more physical than fractals, then I have to give up and I have nothing anymore to say. The point is the following: To encompass the whole world we have to integrate from zero to infinity. Such world could be classical. However a quantum world has a ground state. A ground state could only be understood when we go behind the Zero. From this point of view to encompass the entire world, we have to integrate from minus Infinity to plus Infinity. That is what E-Infinity solved. In the same time this is one of the major problems with string theory. Nevertheless it was string theory which taught us how to deal with this problem in principle. It was on the other hand, the mathematical theory of Menger-Urhyson which gave us the tools to solve it. Our solution is what we call E-Infinity theory.
Communication No 26
From Dirac ‘s Sea to the Density Manifold of E-Infinity
Mathematics in some philosophical view is nothing but a highly efficient notation. Novel mathematics depends crucially on modern notation. Some would go further and could say that far more important than notation is the number system which we employ. Imagine what could have happened if the Europeans would not have had the good sense to accept Fibonacci’s advise and adopt the Arabic number system instead of the Roman number system. In previous communications we discussed extensively the role of the golden mean binary in E-Infinity. In particular we drew attention to what academician Alexey Stakhov named harmony mathematics or the mathematics of harmony to use a less Slavic and a more English expression.
In the present short communication we aim at expanding on few things. First we show how to drive the dimension of a manifold combining the virtues of non-commutative geometry and topological quantum field theory. Second we outline the way from Dirac’s theory of holes to Hilbert cube density manifolds and E-Infinity using simple random Cantor sets construction.
Let us start from Connes’ non-commutative dimension named Index of N in M where N is a sub manifold of M and M is explained in non-commutative geometry. It turns out that this index is equal to 1 divided by L plus 1 divided by 1 – L. Here L is a trace of E as explained in the excellent book of Connes referred to in previous communications. This is our first dimension. Now let us go to the theory of sub factors. The sub factor dimension corresponding to the previous dimension is given for instance in the excellent book of Saunders as 1 divided by 2d – 1. Remembering that quantum topological field theory is derivable from sub factors, we could imagine intuitively without much ado that equating both dimensions would result in a theory of manifolds combining the two concepts. Thus equating the inverse of L multiplied by 1 – L and the inverse of 2d– 1, we find an equation which is easily solved by setting L equal to d or d equal L. Expressed in term of d, we get the following quadratic equation ...............
d square plus d minus 1 = 0. The two-solution is d equal to the golden mean and d equal to the negative inverse square of the golden mean. Inserting the positive value into our index dimension we get the classical value 4 + the golden mean to the power of 3. We see that the most important topological invariant of a manifold combining non-commutativity with topological quantum field theory is the Hausdorff dimension of the core of E-Infinity theory.
Now let us move to Dirac’s theory of holes. Consider the construction of a classically random Cantor set. We take a unit interval. It starts at 0 and ends at 1. For later reference remember that the inverse of the beginning is Infinity. While the inverse of the end remains 1. These peculiar inverses of the boundary play a crucial role in the mathematics of the theory. Now the random Cantor set as is by now well-known has a dimension equal to the golden mean and a measure 0. But what is the dimension of the infinitely many holes left by construction in the Cantor set initial unit Interval? Since the Menger-Uryhson of the initial interval is obviously 1, then the dimension of all these holes is 1 – the golden mean which is equal to the golden mean to the power of 2. On the other hand it is by now well-known from our previous communications that the golden mean to the power of 2 is the dimension of the classical empty set. Now let us look in a purely formalistic way to the dimension which is formed by the union of the Cantor sets and the holes. It seems intuitively obvious and mathematically it is of course correct that the union of both dimensions would restore the situation to a unit interval. This is the trivial result of saying that the golden mean plus the golden mean square is equal to unity. When you are on this line of union you are either on a point or on an empty space. And the probability for that is the highest possible namely Unity. We talk of probability because the Hausdorff dimension could be interpreted in this case as a topological dimension as explained in previous communications. Now we ask another question namely what is the dimension of point belonging simultaneously to empty space and Cantor set? Such a requirement is classically impossible from a classical physical point of view. However formally it is the multiplication of both dimensions namely the intersection of sets. We have thus the dimension golden mean multiplied by the golden mean square which is the golden mean to the power of 3. The next interpretation is crucial. The golden mean to the power of 3 refers naturally to a much emptier set than the classical empty set. This set spans a sub space described by quotient space. This quotient space is given by the quotient formed from the union of the two sets divided by the intersection of the two sets. In other words, the dimension of the quotient space is the division of the golden mean plus the golden mean square divided by the golden mean multiplied by the golden mean square. Working out the simple arithmetic, we find yet again that the dimension is equal to the inverse of the golden mean to the power of three. This is essentially our famous 4 plus the golden mean to the power of three. In a nutshell, our Hilbert cube is a model for the core of E-Infinity theory and implies that it is a quotient manifold or a density manifold akin to what is known in measure theory as Gaussian manifold. The preceding explanation is a short version of a lecture given to post-graduate students by Professor Mohamed El Naschie in the Faculty of Science, University of Cairo five years ago. We thank him for permitting us to reproduce here though in a very concise version.
Note added: In his lecture El Naschie emphasized the reciprocity between what is known as quantum dimension in the theory of quantum group and E-Infinity. For instance the quantum group dimension is given by the inverse of a dimension d to the power of 3 minus d to the power of 3 all divided by the inverse of d minus d. Setting d equal to the golden mean we find that the quantum group dimension is exactly equal 4. This is the exact topological dimension of space time in E-Infinity as well as in reality. From that alone we see that we have a strong indication that the golden mean calculus as used in E-Infinity is a natural quantum calculus. Thus without saying it explicitly, E-Infinity employs tacitly quantum calculus disguised as a golden mean calculus.
E-infinity Communication No. 27
The Lee Symmetry Group Bonanza in high energy physics – A remarkable hierarchy.
Shadowing is a powerful method in nonlinear dynamics. Two decades ago some of us attended a lecture on this technique by the remarkable scientist, James York who incidentally gave chaos its name. An engaging melange of theory and demonstration ended with applause for arriving at Mars in a precise orbit using chaos. In general one could always or let us say most of the time shadows a transfinite scheme using an integer approximation. Let us take the plunge and give a remarkable example for what we mean without much ado. There is of course the danger for the uninitiated to confuse our reasoning with numerology. However those who have persisted until now in reading our communications are necessarily sufficiently sophisticated not to jump in such trivially wrong conclusions. We consider only SU(n). The standard model is based on SU(1)SU(2)U(1). This is not one but a combination of three groups. In general the dimension of SU(n) is equal to n square minus 1. Starting by n = 0 one finds the dimension of the zero set. However, wait a minute! If we are familiar with the exact mathematical theory of dimensions of Menger-Uryhson then we should start at least from the classical empty set. The classical empty set is of the dimension minus 1. Taking SU(-1) we find the dimension zero. Remarkably when we take SU(0) we find the dimension minus 1. In a sense we have a limited cycle confining us to what in physics corresponds to the beginning of the fractal world. Let us leave this and move to the next stage which is n = 1, then n = 2. Thereafter we follow the hierarchy which is self-explanatory. In other words, we take the dimension of SU(2) which is 3 then calculate the dimension of SU(3) which is 8 and then the dimension of SU(8) which is 63 and so on. This and so on refers to one step further only namely SU(63). For this SU(63) the dimension is exactly 3968. Look at this figure carefully. 3968 is 8 times 496. Remember 496 is the dimension of E8 E8. Consequently the dimension of SU(63) is nothing but the number of states of super Yang-Mill theory. You can ascertain yourself of this fact by consulting the relevant text books particularly the excellent book of M. Kaku. To understand the relevance of the result look at the spin reduction. The number of states should be divided in this case by 2 to the power of 6 which is 64. Our reduction equation would be in this case the dimension of SU(63) divided by 64 equal to 62. Doubling of this number leads directly to the familiar standard model representation for instance 124 as well as El Naschie-Lisi representation namely Dim E8 = 248. There has been several attempts by El Naschie and Lisi as well as Ray Munroe to give interpretation to this number as number of particles. As far as we are concerned one of the best interpretations is the following: The 124 could be split into 114 + 10. The 114 is twice 57, where the 57 is the dimension of the modular space M(19,3) which has a dimension 57 as is well known. The 10 on the other hand could be interpreted as 1 graviton with up and down equal to 2 plus 8 dimensional Higgs field. Some suggested the 2 graviton states plus 5 Higgs plus 3 space dimensions. We do not intend to discuss this in detail here. The main point was to show why Lee groups can come very near to exact solution. In general we may have two equivalent alternatives in high energy physics. Either we keep adding new exotic particles or we keep a good eye on the fine structure of space time. The theory of technicolor which was developed by various remarkable Israeli scientists is an example of the first strategy. On the other hand E-Infinity theory of Goldfain, Nottale, El Naschie, Ord, Crnjac, He and their associates as well as Alain Connes’ theory of non-commutative geometry is an example for the second strategy connected to a meticulous study of the fractal fine structure of space time.
We thank Mohamed El Naschie for giving permission to reproduce his Lee symmetry group hierarchy leading to super Yang-Mill particle-like states which were presented for the first time three years ago at various occasions.
E-Infinity
E-infinity Communication No. 28
Gödel’s theorem, Banach-Tarski theorem in E-infinity quantum mechanics
There is no doubt that skepticism is a valid attitude of many conservative scientists and particularly in physics. Never the less skepticism is one thing and name calling is another. Mudslinging is neither a legitimate nor an effective response in any event. Unfortunately some main stream scientists particularly mathematical physicists could easily cross the line between open minded skepticism and inflexible certainty which could easily be confused with ideological commitment. They close legitimate debate by insinuating moral deficiency in those expressing dissenting views. This sounds as if we are talking about politics in one of last century's banana republics; however we also have in mind conventional quantum field theory. Ironically conventional quantum field theory faced the same skepticism of the invalid type a few decades ago in a way not dissimilar to what some proponents of E-infinity are facing today. Reading some of the pseudo arguments used against E-infinity you immediately feel that the main aim is not so much to refute our arguments but rather to discredit our motives. Branding your opponent as a crack pot or pseudo scientist is a convenient way of ducking real questions and possibly novel answers. Of course those clever character assassins based their falsehoods on minimal truths to confuse the unsuspecting reader. For instance the laughable claim that Obama was the illegitimate child of Malcolm-X is so absurd that is had no affect what so ever. On the other hand any falsehood can acquire currency if it is published in a reputable journal such as say for instance, Nature particularly when it is supported by people who have a vested interest in spreading the falsehood.
In E-infinity theory issuing a flat out denial that the theory is wrong is not an option. In this case you turn to manufacturing doubts. A factory of spreading doubts is put in place and it works wonders when financial interests and funding are in question. At the end you have the depressing tale of a systematic rejection of a consistent mathematical theory based on prejudice and hearsay attributed to well established scientists who never really expressed their opinion directly and who names are used by sock puppets to spread the rumors.
Take fractal spacetime as advocated by Ord, Nottale, El Naschie, Goldfain, Ji-Huan He and their followers. Superficial criticism tries to make the unsuspecting reader believe that this idea of a fractal spacetime is total quackery. No attempt will ever be made to show that the idea is based partially on the work of leading scientists such as Richard Feynman and G. Parisi. Those who doubt these words should consult the excellent paper by G. Parisi called Hausdorff dimensions and gauge theories, published in Physics Letters, Vol. 81B, number 3,4, February 1979, p. 357. When El Naschie and L. Crnjac determined the mass spectrum of the standard model with unheard of precision the work was scornfully criticized as numerology. However when Frank Wilczek in a paper entitled Mass by Numbers published in Nature, Vol. 456 27 Nov. 2008 presents a very similar view of the mass spectrum, no one dared to question the profundity of the truly profound work not because they understood it in a different way but because the author carries the Nobel laureates hat. We have many more examples which are worth studying in details. For instance in a paper by F.J. Dyson entitled Divergence of Perturbation Theory in Quantum Electrodynamics, published in Physical Review, Vol. 85, No. 4, February 15, 1952, p. 631 the famous mathematical physicist does not shrug from indicating a connection between the number of terms in an expansion namely 137 and the number of particles created spontaneously as well as hinting towards a deeper connection to the inverse electromagnetic fine structure constant. Similar results based on P-Adic expansion as well as exceptional Lie symmetry groups which was obtained by El Naschie a couple of years ago is on the other hand rejected off hand as crack pot for a very deep reason, namely that El Naschie is an engineer and does not have a very inspiring first name. If the reader does not believe it he should scrutinize a certain dissenting voice which is disproportionately reported on the World Wide Web. And yet another example: Goldfain and later on El Naschie found a connection between high energy physics and Feigenbaum’s universalities. This work was disregarded or made fun of. However in December 2002 D. Polyakov and I. Kogan reported similar results in an Archiv paper entitled Ghost-matter mixing and Feigenbaum universality in string theory, arXiv:hep-th/0212137 v3 23 Apr 2003. This paper was well received because Polyakov enjoys in the meantime the good will of the establishment in the West. It is interesting to note that Polyakov came to his result before ‘tHooft regarding the famous monopole because of his knowledge of fluid turbulence. The great Russian born physicist was proud to note his expertise in turbulence. On the other hand the profound paper by Alfred M. Mukhamedov communicated to Chaos, Solitons & Fractals by M.S. El Naschie which was entitled Towards a gauge theory of turbulence, 29, 2006, p. 253 was almost totally disregarded. If you ask why, the sad answer is because it was published in a journal regarded by the establishment as a non main stream journal. This is enough for many to ban publication of this kind. What can we say? Yet another example: A paper published by Subir Sachdev called Polylogarithm identities in a conformal field theory in three dimensions, Physics Letters B, 309, 285, 1993, arXiv: hep-th/9305131 25 May 1993. In this paper the golden mean features as a fundamental quantity of conformal field theory in three dimensions. The paper was of course well received because the author is a prominent Yale University professor. On the other hand our generalization of the theory to a fully fledged golden quantum field theory is regarded by the same single critic as crack-pottery, a word with which this doubtful critic is in love. Again, what can we say? This word says about the man who is using it more than the man who is accused. We would like to mention three papers by El Naschie involving Banach-Tarski theorem and Gödel’s theorem and its relation to E-infinity quantum mechanics. These papers are Banach-Tarski theorem and Cantorian Micro space-time, Chaos, Solitons & Fractals, Vol. 5, No. 8, p. 1503, 1995, Statistical mechanics of multi-dimensional Cantor sets, Gödel theorem and Quantum spacetime, The Franklin Inst., Vol. 330, No. 1, p. 199, 1993, Gödel, Cantor and modern nonlinear dynamics, First Int. Symposium on Gödel’s Theorems, Edited by Z.W. Wolkowski, World Scientific, May 1991. The papers were relatively well received, for instance by the legendary Martin Gardener. However the establishment remained silent, if not hostile. Interestingly Prof. Lawrence B. Crowell from Alpha Inst. of Advanced Study in Hungary expressed similar views in his book published in 2005 by World Scientific entitled Quantum Fluctuation of Spacetime. In the next communication we will discuss these things in more depth as well as the role of Fuzzy platonic spaces as a model for quantum physics which was published in the journal Mathematical Methods, Physical Methods & Simulation Sci. & Techn, Vol. 1, No. 1, June 2007, p. 91.
In the meantime we hope you will read carefully not only what is written explicitly in this communication but also what is written between the lines.
E-infinity Communication No. 29
Gödel’s theorem, Banach-Tarski theorem in E-infinity quantum mechanics
There is no doubt that skepticism is a valid attitude of many conservative scientists and particularly in physics. Never the less skepticism is one thing and name calling is another. Mudslinging is neither a legitimate nor an effective response in any event. Unfortunately some main stream scientists particularly mathematical physicists could easily cross the line between open minded skepticism and inflexible certainty which could easily be confused with ideological commitment. They close legitimate debate by insinuating moral deficiency in those expressing dissenting views. This sounds as if we are talking about politics in one of last century's banana republics; however we also have in mind conventional quantum field theory. Ironically conventional quantum field theory faced the same skepticism of the invalid type a few decades ago in a way not dissimilar to what some proponents of E-infinity are facing today. Reading some of the pseudo arguments used against E-infinity you immediately feel that the main aim is not so much to refute our arguments but rather to discredit our motives. Branding your opponent as a crack pot or pseudo scientist is a convenient way of ducking real questions and possibly novel answers. Of course those clever character assassins based their falsehoods on minimal truths to confuse the unsuspecting reader. For instance the laughable claim that Obama was the illegitimate child of Malcolm-X is so absurd that is had no affect what so ever. On the other hand any falsehood can acquire currency if it is published in a reputable journal such as say for instance, Nature particularly when it is supported by people who have a vested interest in spreading the falsehood.
In E-infinity theory issuing a flat out denial that the theory is wrong is not an option. In this case you turn to manufacturing doubts. A factory of spreading doubts is put in place and it works wonders when financial interests and funding are in question. At the end you have the depressing tale of a systematic rejection of a consistent mathematical theory based on prejudice and hearsay attributed to well established scientists who never really expressed their opinion directly and who names are used by sock puppets to spread the rumors.
Take fractal spacetime as advocated by Ord, Nottale, El Naschie, Goldfain, Ji-Huan He and their followers. Superficial criticism tries to make the unsuspecting reader believe that this idea of a fractal spacetime is total quackery. No attempt will ever be made to show that the idea is based partially on the work of leading scientists such as Richard Feynman and G. Parisi. Those who doubt these words should consult the excellent paper by G. Parisi called Hausdorff dimensions and gauge theories, published in Physics Letters, Vol. 81B, number 3,4, February 1979, p. 357. When El Naschie and L. Crnjac determined the mass spectrum of the standard model with unheard of precision the work was scornfully criticized as numerology. However when Frank Wilczek in a paper entitled Mass by Numbers published in Nature, Vol. 456 27 Nov. 2008 presents a very similar view of the mass spectrum, no one dared to question the profundity of the truly profound work not because they understood it in a different way but because the author carries the Nobel laureates hat. We have many more examples which are worth studying in details. For instance in a paper by F.J. Dyson entitled Divergence of Perturbation Theory in Quantum Electrodynamics, published in Physical Review, Vol. 85, No. 4, February 15, 1952, p. 631 the famous mathematical physicist does not shrug from indicating a connection between the number of terms in an expansion namely 137 and the number of particles created spontaneously as well as hinting towards a deeper connection to the inverse electromagnetic fine structure constant. Similar results based on P-Adic expansion as well as exceptional Lie symmetry groups which was obtained by El Naschie a couple of years ago is on the other hand rejected off hand as crack pot for a very deep reason, namely that El Naschie is an engineer and does not have a very inspiring first name. If the reader does not believe it he should scrutinize a certain dissenting voice which is disproportionately reported on the World Wide Web. And yet another example: Goldfain and later on El Naschie found a connection between high energy physics and Feigenbaum’s universalities. This work was disregarded or made fun of. However in December 2002 D. Polyakov and I. Kogan reported similar results in an Archiv paper entitled Ghost-matter mixing and Feigenbaum universality in string theory, arXiv:hep-th/0212137 v3 23 Apr 2003. This paper was well received because Polyakov enjoys in the meantime the good will of the establishment in the West. It is interesting to note that Polyakov came to his result before ‘tHooft regarding the famous monopole because of his knowledge of fluid turbulence. The great Russian born physicist was proud to note his expertise in turbulence. On the other hand the profound paper by Alfred M. Mukhamedov communicated to Chaos, Solitons & Fractals by M.S. El Naschie which was entitled Towards a gauge theory of turbulence, 29, 2006, p. 253 was almost totally disregarded. If you ask why, the sad answer is because it was published in a journal regarded by the establishment as a non main stream journal. This is enough for many to ban publication of this kind. What can we say? Yet another example: A paper published by Subir Sachdev called Polylogarithm identities in a conformal field theory in three dimensions, Physics Letters B, 309, 285, 1993, arXiv: hep-th/9305131 25 May 1993. In this paper the golden mean features as a fundamental quantity of conformal field theory in three dimensions. The paper was of course well received because the author is a prominent Yale University professor. On the other hand our generalization of the theory to a fully fledged golden quantum field theory is regarded by the same single critic as crack-pottery, a word with which this doubtful critic is in love. Again, what can we say? This word says about the man who is using it more than the man who is accused. We would like to mention three papers by El Naschie involving Banach-Tarski theorem and Gödel’s theorem and its relation to E-infinity quantum mechanics. These papers are Banach-Tarski theorem and Cantorian Micro space-time, Chaos, Solitons & Fractals, Vol. 5, No. 8, p. 1503, 1995, Statistical mechanics of multi-dimensional Cantor sets, Gödel theorem and Quantum spacetime, The Franklin Inst., Vol. 330, No. 1, p. 199, 1993, Gödel, Cantor and modern nonlinear dynamics, First Int. Symposium on Gödel’s Theorems, Edited by Z.W. Wolkowski, World Scientific, May 1991. The papers were relatively well received, for instance by the legendary Martin Gardener. However the establishment remained silent, if not hostile. Interestingly Prof. Lawrence B. Crowell from Alpha Inst. of Advanced Study in Hungary expressed similar views in his book published in 2005 by World Scientific entitled Quantum Fluctuation of Spacetime. In the next communication we will discuss these things in more depth as well as the role of Fuzzy platonic spaces as a model for quantum physics which was published in the journal Mathematical Methods, Physical Methods & Simulation Sci. & Techn, Vol. 1, No. 1, June 2007, p. 91.
In the meantime we hope you will read carefully not only what is written explicitly in this communication but also what is written between the lines.
E-infinity communication No. 30
From Gödel and Banach-Tarski theorem to randomness and Undecidability in physics
Prof. Karl Svozil from the Dept. of Physics, University of Vienna is one of the early pioneers of fractal spacetime. Svozil worked closely with Otto Rössler as well as the leading experimental physicist Anton Zeilinger. In addition he made many contributions of Chaos, Solitons & Fractals. Randomness and Undecidability in Physics is the title of his modus operandi as a book writer which was published by World Scientific in 1993. In this book Svozil drew attention probably for the first time to the connection between Banach-Tarski paradox and nonlinear dynamics. El Naschie acknowledged that this led him to his work on the Banach-Tarski paradox in cosmology. An account of El Naschie’s work on this subject is given on pages 179-190 of a book entitled The Pea and The Sun by Dr. Leonard M. Wapner. The book was published by A.K. Peters Ltd, Massachusetts, USA in 2005.
In excessively colloquial terms Gödel’s theorem implied that you cannot lift yourself by pulling yourself by the toes. The Banach-Tarski theorem on the other hand represents the ultimate free lunch that is getting something from nothing. We promised in our last communication, No. 27, to discuss these two theorems in connection with high energy physics in some detail. However we feel that we should postpone such a discussion to a later communication and concentrate for the moment on something else which is deeply connected with randomness and Undecidability.
When you cannot know for sure anything you normally invoke probability as the method of last resort. Probability in physics means statistical mechanics. In this communication we would like to present this subject in connection with quantum field theory and E-infinity theory from a slightly unconventional point of view. It is said that good scientists always search for analogies between theories while the best of them search for analogies between analogies. Encouraged by such a philosophical attitude we would like to present some illuminating analogies which are of relevance to what we intend to demonstrate. Let us first start with the analogy between quantum field theory and statistical mechanics. Imagine a table where on the left hand side we have quantum field theoretical concepts and on the right hand side we have statistical mechanical concepts corresponding to the former. Let us start with the most obvious and most simple correspondence and analogy. In our view this is Feynman’s path integral. In statistical mechanics the corresponding notion is the partitioning function. The second simplest concept in quantum field theory is the change in the vacuum. The corresponding statistical mechanical concept is phase transition as well as bifurcation. It follows then that quantum fluctuation of quantum field theory corresponds to statistical fluctuation in certain expectation values. Next we know that the Lagrangian corresponds to the H function of statistical mechanics. Slightly less obvious is that gauge theory of quantum field theory corresponds to intersection and union of hyper areas in n dimensional spaces which is similar to what we do in E-infinity theory. The quantum h corresponds to the inverse of beta which is equal to k Boltzmann multiplied with T. The final analogy is probably the least understandable at first sight to those not familiar with statistical mechanics. The mass m of quantum field theory corresponds to the inverse of the length of correlation. To gain an intuitive feeling for the meaning of this analogy we have to remember the relation between the correlation length and the mean free path known from atomic physics. This mean free path measures a probability of collision. On the other hand we know that the smaller the distance we are probing, the larger the energy involved because of Heisenberg’s uncertainty principle. Since mass is directly related by relativity to energy, this mass must be inversely proportional to the correlation length and therefore m is equal to one divided by L. In E-infinity theory this explains the relation between the mass and the inverse of the electromagnetic fine structure constant and the geometry and topology of spacetime as will become evident later on.
Next let us attempt to write down a similar table for the analogy between quantum field theory and E-infinity theory. The most obvious thing here is again the Feynman path integral. This path integral is effectively summing over all paths which a quantum particle could take. In E-infinity we sum over all weighted dimensions. The corresponding expression is thus the expectation value of the Hausdorff dimension which is equal to four plus the golden mean to the power of three as is well known. Next quantum fluctuation of quantum field theory will correspond to deterministic chaos in E-infinity theory. More precisely it corresponds to an average symmetry as explained elsewhere in more detail. Gauge theory of quantum field theory is the intersection and union of Cantor sets in E-infinity formulation. The Lagrangian is replaced in E-infinity theory with a scaled energy where differentiation and integration is replaced by golden mean scaling up and down as explained in previous communications. The mass of quantum field theory appears in E-infinity theory as an average equal to the dimensions of certain symmetry groups all related to the inverse electromagnetic fine structure constant which is regarded in E-infinity theory as more fundamental than any other so called constants of nature. The change in vacuum in quantum field theory corresponds to the universal fluctuation of E-infinity which is equal to the golden mean to the power of three. Finally our classical quantum h corresponds to various powers and combinations of the golden mean as explained in detail in E-infinity theory. We admit that the preceding account is quite condensed but we are hoping that the reader has read at least some of our earlier communications and is therefore in a position to fill in the gaps of such a condensed summary.
E-infinity communication No. 31
The quantum gas model of E-infinity – the work of L. Marek-Crnjac
Over the years many of the E-infinity researchers introduced by way of analogy physical pictures for the mathematical structure of E-infinity Cantorian fractal spacetime. For instance El Naschie drew analogy to polymers. Somewhat related is the fractal sphere packing of Marek-Crnjac. However this could not be said to be a physical but rather geometrical topological interpretation of E-infinity. To be sure a physical interpretation for E-infinity by way of analogy is somewhat scarce.
On the other hand and rather recently a Slovenian mathematician L. Marek-Crnjac noticed a most instructive and useful analogy between the mathematics of ideal quantum gases and that of E-infinity. The model presented by Crnjac is as beautiful and charming as the picturesque mountain landscape near to Maribor where this Slovenian scientist lives and works. The idea is published in a formal paper but we may outline the gist of it.
Building on work by Mohamed El Naschie related to conformal field theory, the golden mean and its relation to polylogarithmic identities, Crnjac was able to show that a vital quantity related to a chemical potential could be seen as being analogous to the Hausdorff dimension of the core of Cantorian spacetime. In this case the familiar Hausdorff dimension is obtained from a polypseudolog to be divided by the square of one minus d. Substituting the golden mean for d the polypseudolog gives 4.236067899 and that for a negative dimension equal minus one. Recalling the dimensional theory of Menger-Uhrysohn as discussed by El Naschie, Nada, Iovane and He as well as L. Crnjac we see that this value in this presentation explicitly admits its indebtedness to the empty set for which the Menger-Urysohn dimension is equal minus one.
There are many highly interesting aspects of this new discovery of Crnjac. For instance the Bose-Einstein and Fermi-Dirac statistics are unified in a completely novel manner. This unification gives considerable support to the idea of super symmetry. In other words we have an argument for the existence of unified theory for both fundamental statistics apart of super symmetry. The work of Crnjac is still at the very beginning and is tying many fields together.
However we decided to mention it because it is new and very credible evidence that the basic tenants of E-infinity theory are correct.
At the beginning one may be surprised at this new result of Crnjac. However deep reflection shows that this is completely normal and expected on the basis of conformal field theory. To explain this point we have to recall a few facts. The concept of quantum field theory is almost 75 years old but it is only in the last 25 years that quantum field theory was taken seriously and used in high energy physics. However there are problems, the main one being that quantum field theory still uses the Lagrangian approach. This is the mainstream approach. It is also an insightful approach. Alas it is the least precise approach to quantum field theory. Mainstream scientists normally omit this point and even criticize E-infinity theory for not making use of the conventional Lagrangian approach. They omit mentioning that the Lagrangian approach, although useful for perturbative analysis, masks the genuinely non-perturbative aspects. Here comes conformal field theory to save the day. Conformal field theory like E-infinity theory lends itself easily to non-perturbative treatment. This aspect stems from the fact that infinite dimensional conformal symmetry can be presented in two dimensions. This is very familiar to people working in E-infinity. You recall that Klein’s modular curve has 336 isometries. The compactified Klein modular curve of E-infinity on the other hand has infinitely many isometries equivalent to infinitely many progressively smaller and smaller triangles. These triangles ramify at infinity into a Cantor set yet the whole thing is in two dimensions. That is to say a two dimensional projection and what is more interesting, by taking a gamma distribution the infinitely many isometries can be weighed statistically to be equivalent to almost 339 isometries. In other words we have three isometries more than the classical 336 isometries of the classical Klein modular curve. These simple but deep mathematical facts help those working not only in E-infinity theory but also in string theory. The connection comes from the mathematical relation to Affine Lie algebras, Monster groups and the modular invariance related to modular manifolds. The relation to Kac-Moody algebra, 3-point functions and fusion algebra was hinted at by El Naschie in many of his earlier writings.
We are deeply indebted to Prof. L. Marek-Crnjac for allowing us to publish this extract and we would be grateful if anyone using this condensed summary ensures they refer to the original work of Prof. L. Marek-Crnjac.
E-infinity group.
We must think about all these things and develop them further and keep each other informed. Let us, following Leo Tolstoy, try to forgive our opponents and wish them peace of their soul so that they can leave us in peace to complete our work.
Best wishes to everyone,
E-Infinity Communication
E-infinity communication No. 2
Why the golden mean?
Dear All,
Encouraged by the balanced and civilized remark of Dr. Munroe we will attempt to update you on anything new in E-infinity which comes to our attention and answer any reasonable well posed question which anyone of you has as far as we can. We hope that arguments and tone will remain within what you would do in a scientific conference or a discussion in a learned journal. By all means you can make the odd witty remark or polite joke. Just consider that you are not anonymous and that you are responsible for what you are saying. Science is a worthwhile and respectable endeavor. Even non religious people do not commit scandalous actions in a house of worship whether it is a church, synagogue, mosque or a budhist temple. Science is a kind of temple for us scientists and even if you do not believe in that, please out of respect to other believers, leave this site for science and let us discuss science without resorting to personal hidden agendas, irrational hate or jealousy and inferiority complexes. If you would love to see a trial you will have a long one in the High Court of London. Here we talk only about science. Thank you for understanding and now I can move to the next item.
I would like to give here a plausibility explanation of why the golden mean popped up in quantum mechanics. To understand our point of view you must know a little bit of nonlinear dynamics. The most important thing which you have to know in that respect is the KAM theorem. For instance you could consult a book by Heinz Georg Schuster called Deterministic Chaos, published by VCH Verlag, 1989 but any other book on nonlinear dynamics would do. KAM is an acrynomn relating to the name Kolomogorov, Arnold and Moser, the three mathematicians who developed it. Loosely speaking it states that the last periodic orbit which can be destroyed by perturbation is the one which has the most irrational winding number. You can think of a winding number as the ratio of two frequencies as in resonance. The more irrational the winding number is, the more stable the orbit is. Since there is nothing more irrational than the golden mean because it is the least well approximated by a rational as you can see from continued fraction expansion it follows then that this orbit is the most stable. The stationary states which can be observed experimentally is therefore connected as close as possible to the golden mean. You can think of elementary particles as a stationary state of quantum mechanics. When you connect quantum mechanics to KAM then you are effectively making Rene Thom’s VAK hypothesis. Rene Thom conjectured that the Hamiltonian quantum mechanics has a form of attraction although it is non-dissipative and conservative. This strange attraction has a vague resemblance to strange attractors of dissipative systems. That is why it was called the vague attractor of Kologomorov by Rene Thom. El Naschie’s work and the subsequent experimental confirmation of the golden mean in quantum mechanics in the Helmholtz Centre is the proof that the VAK conjecture is correct. You see this is all mathematically perfect and correct but of course we are extending mathematics to physics. When ever you extend mathematics to physics you leave the secure ground of absolute logic and enter into the messy realm of reality. But that is why theoretical physics is for us far more interesting than pure mathematics. In theoretical physics you need not only maths but something more difficult to pin down given to man by God if I may say so and this thing is called intuition. That is the reason why the golden mean will keep coming out in every measurement of quantum mechanical phenomena. Thank you for your patience and we await your questions if you have any before we move to the next point.
E-infinity communication.
E-Infinity Communication No. 3
Further reasons why the golden mean?
Dear Ray
There may be a slight misunderstanding here. Irrationality of Phi is expressed by the fact that the fraction expansion involves only unity. In other words this is 1 divided by 1 plus 1 and 1 is divided again by 1 plus 1 and so on indefinitely. At infinity the result is the golden mean. Some call the inverse of the golden mean the golden mean. This is only semantics and totally irrelevant. We should also not loose sight of what we want to talk about. Whether the Fibonacci progression is more fundamental or the golden mean may be a question of interest to a number of theoreticians involved in a learned discourse. It is also irrelevant that John Baez had a vested interest to draw attention to the work involving the golden mean published by a Russian scientist in a far more restrictive area as compared to the fundamental generalization of E-Infinity. Important are only the facts that a fundamental theorem about stationary states relates elementary particles to the golden mean. The theorem is the VAK which as we said an extension of KAM to quantum mechanics. The experimental verification is a fact. Scientists engage in an honest historical analysis of science will show at some time who was first and who was not. Now let me go back to the fundamental question of why the golden mean?
A Slovenian scientist and mathematician following Mohamed El Naschie expand the idea of mechanical oscillators. Many papers have been published on this subject by L. Marek-Crnjac. Take a two degree of freedom oscillator. Two masses connected by two linear springs. Write the equation of motion. Set the value for the masses as well as the spring constants equal unity. The secular equation is then simply a quadratic equation. The Eigen values are golden mean related. The only positive real Eigen value is the golden mean. Imagine now that you have infinitely many such oscillators connected together. Consequently you can estimate the Eigen value using two well known theorems on Eigen values. These are the Southwell theorem and the Dunkerly theorem. They correspond to what we have studied in school about joining electrical resistance of Ome’s law. When they are successive you add the inverses and when they are parallel you add them. Eigen values are frequencies. Frequencies are energy and energy is mass. Extrapolating the whole thing to quantum mechanics as argued by El Naschie and Marek-Crnjac you have another plausibility explanation for why the golden mean will pop up in any accurate measurement in quantum mechanics phenomenon.
You can see all this theorem in any good book on Mechanical Vibration. There are of course many other ways to argue the appearance of the golden mean which I will discuss next as soon as you have made your comment.
Best regards,
E-Infinity communication
E-Infinity Communication No 4
Mathematical reason for the golden mean in quantum physics
Dear Ray
You are right. But like all of us you are right to a point. We and science exist because philosophy exists. We lose track of things for the same reason. Do not fall in love in general in eloquent formulations no matter how beautiful a sentence is. Reality is indeed fractal. But then you can lose track of reality when you gloss over. You do that when your observation is inaccurate. Your observation ergo reality is as good as the resolution of your instrument. The whole idea of random cantor set as a building block of spacetime is that it is the reality which we were not able to observe directly until now. But it is there. For the first time it manifested itself in quantum mechanics indirectly through the golden mean. Let me first give you some more demanding reasons why the golden mean is there.
We will have to make a jump. It is not systematic. It does not follow from what I said in previous communications. Please accept it for now on its face value. The most fundamental thing which we have for the whole shebang is Kline modular curve in the compactified version. I am not sure of this great man if this great man Roger Penrose knew that he rediscovered the same thing from a far more fundamental viewpoint as far as quantum mechanics is concerned. The entire world of quantum mechanics is encoded in Penrose universe. I know that theoretical physicists can be quite impatient with Penrose and secretly they curse him. I know that his twistor program has come to a halt. And he sometimes said something to the effect that it has failed. But that does not apply to his fractal tiling. Not at all. There is another mathematician who is interested in physics and who is truly an exceptional man. Not as general as Penrose but he runs in the same direction. The work of Alain Connes on non-commutative geometry is best illustrated by Penrose Universe. The work of both men derives its essence from Von Neumann Continuous Geometry. For the sake of this communication, I will refer to page, verse and chapter of Alain Connes Noncommutative Geometry published by Academic Press, copyright 1994. Please look at page 89. Examine figure II.3 entitled Penrose Tilings. Move to page 90. When you read the second paragraph your mind will lit. He says x is a quantum space and then he says that the entire thing is described as tiling. Then he gives an unheard of elementary equation which he calls dimensional function. The dimensional function is z plus the inverse golden mean multiplied by z. It was El Naschie who noticed the profound meaning of this equation and realized that applying some elementary matrix analysis to it, he obtained his bijection formula. You recall that the bijection formula relates the Hausdorff dimension for an n topological space to the backbone or Hausdorff dimension of a zero dimensional space. It is difficult to explain these things without mathematics on this blog. Let me give you a small example: a random cantor set has a golden mean as a Hausdorff dimension. The Menger Uhryson dimension which is nothing but an extended topological dimension for this set is zero. To find the dimension for n = 4, we take the inverse golden mean to the power of 4 – 1 equal three. This means the Hausdorff dimension is 4.23606799 or 4 plus the golden mean to the power of three. You met this famous formula before. But this time we are driving it from the work of Alain Connes. The same thing can be done using von Neumann’s Continuous Geometry which is the basis of Alain Connes work. However nothing can rival the simplicity of Penrose Fractal Tiling. This fractal tiling is the holographic boundary of the theory of everything. And now comes the unexpected expected result. It is impossible to have a Penrose Universe without golden mean proportionality. Penrose kit and dart inside kit and dart inside kit and dart and so on indefinitely could not be designed without the golden mean. In terms of the mathematics of Hamiltonian system, we say we could not have smooth tiling without gaps or overlapping unless we have golden mean proportionalities. Please read the elementary but fascinatingly beautiful Penrose Tilings. Penrose did not stop at that. He has driven a fantastic formula called the isomorphic length. The isomorphic lens was popularized by Martin Gardner. Believe it or not, when you multiply this length by 2 what do you get? You get exactly 4.23606799. In other words you get half of the Hausdorff dimension of the expectation value of El Naschie E-Infinity space. Remember this Hausdorff dimension is found in the most elementary fashion by putting a 4 dimensional cube inside another dimensional cube and so on indefinitely. To get the value you write a continued fraction. That is 4 plus one over four plus one over four and so on indefinitely in the familiar fashion and the final result after infinitely many iteration is 4 plus the golden mean to the power of 3. I will repeat again, this is double as much as we have with a Penrose isomorphic length. The isomorphic length is a wonder which is not a wonder. You stand anywhere in a Penrose universe holographic boundary. You look around yourself and see the world. You close your eyes and move a distance. At a distance not larger than the isomorphic length, you open your eyes again and you will think you have not moved at all. You have a recurrence of the whole universe around you giving you the illusion you have not moved. That is where infinity and finiteness become exchangeable. That is what Mohamed El Naschie observed and used. All very simple. This is the wonder of the hyperbolic geometry in compactified form. You have used related ideas in driving your E12 Exceptional Lie group. I have looked at your paper in Chaos, Solitons & Fractals. I have seen some of your figures. This is all thanks to the work of von Neumann’s noncommutative algebra and Alain Connes noncommutative geometry and Penrose incredible geometrical intuition and knowledge. Mohamed El Naschie, the engineer gave all these subjects the time they needed to digest and reproduce them in his own language in what we call today E-Infinity. I know this communication is far more complex than the previous one. There may be many gaps in my explanation. But I will come back to everything once again and my next communication will be far more comprehensive and far more elementary. But I would like to draw your attention to the last informal review which Mohamed El Naschie wrote on the subject. It is called: The theory of Cantorian spacetime and high energy physics (an informal review), published in Chaos, Solitons & Fractals 41(2009) 2635-2646. As for the golden mean and the symplictic character of quantum vacuum, I advise you to read a short and very simple and beautiful paper by Mohamed El Naschie relating the whole thing to the Banach-Tarski paradox and the no squeezing theorem. The paper is titled: New elementary particles as a possible product of a disintegrating symplictic vacuum published in Chaos, Solitons & Fractals 20(2004) 905-913.
I hope I was of some help and I advise to experiment yourself by doing some elementary calculations yourself. E-Infinity is hands on mathematics. We bring everything we know to bear on the problem. We start from absolutely abstract mathematics and go down without any problem to numerics as well as plausibility explanation. We need all the help we can get. Anything goes as long as it is logical and helpful to our understanding.
E-Infinity communication
E-infinity communication No. 5
Deriving the basic equations of E-infinity
Part I – Elementary introduction
Ignoring nonsense and concentrating on science I would like to explain how to derive the basic dimensional equations of E-infinity. I will start for simplicity with what is more familiar to people not acquainted with E-infinity as a starting point. Later on we will give the accurate, exact derivation and finally we will look into the correspondence to noncommutative geometry and other mathematical theory.
Dimension is the most important topological invariant we have. It is more fundamental than the Euler number or what have you. We start with topology to construct what Wheeler rather loosely calls pre-geometry. We imagine an infinite number of Cantor sets to be joined together statistically to form a pre-geometry and we use for this purpose a Gausian-like distribution which is called gamma distribution. There is some contradiction here because we are joining discrete sets but we use continuous distribution. Later on we will introduce the correct discrete gamma distribution. OK. What we are distributing now is not the sets but the Hausdorff dimension of the sets. We distribute them like Wheeler’s bucket of dust. You can check that in the literature. What is the expectation value of a Hausdorff dimension for the whole collection of this dust? Wheeler calls it Borel mix. I do not think Wheeler had exactly a Borel set in mind, but never mind. To find this average or expectation dimension you have to know the multiplier lambda and a shape factor r. The expectation value is given by r divided by the natural logarithm of lambda. A little contemplation will show that lambda must be the inverse golden mean because we are taking only random elementary Cantor sets in our Borel mix. Taking the simplest camel hump shape we see that r must be equal to two. You can find that in any text book on statistics and probability theory for instance the book of Pitman. Divide now two by ℓn the inverse golden mean. This gives you 4.156174. Please do this yourself using a pocket calculator. You will remember the exact value should have been 4.236067977. Well this is the price in accuracy which you have to pay for using the continuous gamma distribution. If you want to have the exact result then you have to get rid of the spurious nonlinear terms in the expansion of the natural logarithm. Mohamed El Naschie pointed out that the linear terms of the expansion correspond to the exact discrete distribution. If you do that and make no mistakes then the two divided by the logarithm of lambda changes to 1 plus the golden mean all divided by 1 minus the golden mean. When you do this you will find the exact solution which is 4.236067977.
Let me show how we get that from first principles exactly. You assume you have a random Cantor set with a golden mean Hausdorff dimension. You add to that all the Cantor sets in the world, that is to say infinitely many, each having a Hausdorff dimension equal to the golden mean to the power of n, where n goes from zero to infinity. This would give you a sum which when summed correctly then it is equal to 1 divided by 1 minus phi, namely the golden mean. To compare that with the initial Cantor set which you started with, you have to divide the whole sum again by phi. The result is the well known expectation value of the Hausdorff dimension of E-infinity. Suppose we do not know that phi is the golden mean. Leave it be just a formula for the expectation value of the Hausdorff dimension. This formula now reads 1 ϕ. Remember we had another formula from the gamma distribution which says (1 + ϕ) . Now comes the most important condition we have. The first dimension is the Hausdorff dimension, namely an expectation value. The second dimension is an expectation value of a topological dimension. To have a space worth the name space you should have no gaps and no overlapping. The requirement for that is that both preceding dimensions should be equal. Equating both formulas you find a quadratic equation for ϕ. Solving this equation you find that ϕ is indeed the golden mean, namely 0.618033989. This is space filling condition which Mohamed El Naschie introduced to derive this formula. Inserting the value for ϕ in any of the two formulas you always find that the dimension is equal to 4.236067977 as should be.
You can find the detailed derivation in many papers of El Naschie as well as in the work of Marek-Crnjac and others. We see that E-infinity pre-geometry is described by more than one dimension. We have a topological dimension on the average which is equal to the Hausdorff dimension on the average equal to 4 + the golden mean to the power of 3. In addition you have exactly four dimensional topologically speaking. However formally you have been adding infinitely many Cantor sets so that you really have infinitely many dimensions. To understand that the topological dimension is exactly four, I have to refer you to the bijection formula which is connecting El Naschie’s work with noncommutative geometry. In this particular case the formula says that the correction dimension is obtained from raising the inverse golden mean to the power of n minus one. To get the correct result, namely 4.236067977 you have to have n equals 4. When n is equal n then n minus one is equal 3 and hey presto, you see that the inverse golden mean to the power of 3 gives indeed the correct expected dimension, namely our familiar 4.236067977. I urge every reader to do it himself using the pocket calculator. Working yourself through all these little calculations you will start having a feel for the golden mean symphony which is governing quantum mechanics and high energy physics. In the next communication we will go deeper into all the subjects once more from a higher view point. This was just an elementary taste of what it is all about.
E-infinity communication No. 6
Derivation of the fundamental equations of E-infinity Part II
As Confuscious says, ignore perturbation and like Mohamed says, may peace be upon all of you. Let us continue with our scientific discussion.
We would like to tie up some loose ends and unintentional omission of some important aspects of the main equation as discussed so far in communication No. 5.
El Naschie draws attention to an important interpretation of the approximate formula of the expectation value of the gamma distribution of E-infinity. You recall that it was 2 divided by ln lambda. Taking lambda to be the inverse golden mean then ln lambda will have an important interpretation in terms of an entropy which mathematicians call topological entropy. The topological entropy for a certain map turns out ln 1.6180333 equal 0.481212. If you interpret the 2 as being the Hausdorff dimension of a quantum path following Abbot and Wise famous paper, then dividing 2 by this topological entropy is giving you the entropic content of a quantum path. In other words the approximate solution for the expectation value of the Hausdorff dimension, namely 4.156174 is effectively an entropic content. We see the important interplay between different notions which are not different at all when seen from a higher level. Those familiar with string theory and Brane theory may find an analogy for this in the fact that P-Brane and D-Brane which were thought to be different are in fact not different at all. The difference is in the eye of the beholder.
In this context we should apply the same interpretation but this time using a different lambda of a different map. The value of this lambda is itself our 2. Consequently we have 2 divided by ln2. This is a different entropic content using again a topological entropy definition. The value in this case is 2.885390. Try to remember this for later use because this value approximates the missing part needed to compactify chi 7. I mean the part you need in order to make 336 come nearer to 339 by compactification. Of course I am running ahead of my story but please remember this bit for later on.
Now let me give you the exact E-infinity derivation of our 4.23606799 dimensions which we can obtain without making any reference to anything except the pre-geometry we are constructing and the center of gravity theorem of probability theory. Let us take all integers from 0 to infinity. Let us regard this as infinite dimensions. Now we give each dimension a weight. This weight is equal to the golden mean to the power of n where n runs from 0 to infinity. What then is the center of gravity of the whole thing? This is a very simple problem in probability theory which mechanical engineers like to think of as searching for the resultant of many forces. It is now very easy to guess that the resultant which acts at the center of gravity will be given by the sum from 0 to infinity of n multiplied with the golden mean to the power of n and all multiplied with n again, where the last n is now the arm of the force while the force is n multiplied with the golden mean to the power of n. You have to divide this now by the sum of all forces which is sum from 0 to infinity of n multiplied with the golden mean to the power of n. In other words you have a sum of moments divided by the sum of forces. This gives you the expectation value of the average arm of the resultant. This arm is the expectation value of our dimension of the space made of these infinitely many weighted topological dimensions. These infinite series can be summed exactly as shown by El Naschie and it is an elementary matter to show that the final sum is simply 1 plus the golden mean all divided by minus the golden mean which all comes exactly to the well known and familiar value of 4 plus the golden mean to the power of 3. In sum pure mathematical communication Mohamed El Naschie was able to show that this is a well known result in the theory of the Coxeter groups. You can find this paper in Elsevier’s Science Direct.
In a forthcoming communication we promise to show you that all these results are derivable from the theory of subfactors as well as Alan Connes’ noncommutative geometry.
Now to close this 6th communication I would like to give you the exact derivation of the dimension of the holographic boundary of E8 E8 in the case of complete compactification. You recall that we started with 336. Then we keep adding triangles indefinitely in a hyperbolic way. To understand how to find a finite value for this compactification to infinity, you have to be familiar with the theory of Klein’s modular curve. You have to know that there is an important orbit with 42 points in the 336 curve. This 42 can be thought of as 4 and 0.2 lifted to 10 dimensions by taking 10 copies. That is to say it is 4.2 multiplied with 10. A little contemplation will make it evident that 4.2 is just an approximation to 4.23606799. Taking 10 copies of that you get 42.3606799. Now this is the exact coupling constant of unification in the absence of super symmetry. Thus in the ordinary case of non-compactified Klein’s modular curve, you get 336 from 42 multiplied with 8. In the exact transfinite case you get the correct result by multiplying 8 with 42.3606799. When you do that you get 338.885438. This is exactly 2.885438. You see now how close this is to what I told you earlier, namely 2 divided by ln2 equals 2.885390. Very close but not completely correct. The exact expression is well known from the theory of transfinite corrections as developed by Mohamed El Naschie following a similar theory dealing with operators due to Fritz John. The exact expression is 16k. Here k is equal to the golden mean to the power of 3 multiplied with 1 minus the golden mean to the power of 3. When you do that you find that it is 0.180340. Now you take 16 copies of it and you have your exact result 2.885438. Now let us derive the exact theoretical inverse electromagnetic constant using Mohamed El Naschie’s fundamental equation and the previous exact transfinite value. The number connected with Einstein’s gravity Reimannian tensor will remain 20. However the dimension of E8 is no longer exactly 248 but 247.983739. This is 248 minus k square divided by 2. Now our equation should read E8 E8 minus holographic boundary minus Einstein gravity equal inverse electromagnetic constant. Taking the appropriate dimension we have 495.967478 minus 338.885438 minus 20 equals 137.082039325. This is the exact theoretical value as promised. It is the integer value 137 plus a transfinite correction equal to k o where this new constant is equal to the golden mean power 5 multiplied with 1 minus the golden mean power 5 equals 0.082039325. You see the extent of our precision and how the constants of nature are obtained as a probability resulting from summing over infinitely many states but I think it is enough for now and promise to come back in our next communication in due course.
E-infinity communication No. 7
Derivation of the fundamental equations of E-infinity Part III
For the benefit of those who appreciate scientific discussion as opposed to defamatory allegations and those who would like to learn something about E-infinity as opposed to hearsay and parrot repetitions of misconceptions, we continue our discussion by giving literature which we omitted to mention in detail in Communication No. 6.
Pre-geometry and the main idea of Borel mix as used by Wheeler can be found on page 1205 of his monumental book Gravitation, by Charles W. Misner, Kip S. Thorne and John Archibald Wheeler, published by Freeman, New York 1973. You can find everything about gamma distribution in the book Probability by Jim Pitman, published by Springer, Berlin 1993. The expectation value of the gamma and other distributions are conveniently summarized in the table on page 476-477.
Now let me reminisce about Mohamed El Naschie and his lectures when I was just about to finish my Ph.D. He said something very remarkable about the Cantor sets. He said it is something which is there and yet not there. In his youth Mohamed was a friend and follower of Jean Paul Sartre. He was greatly influenced by Sartre’s major book Being and Nothingness (original title L’Etre et le neant: Essai d’ontologie phenomenologique). Mohamed read the book in the German translation, having German as quasi his mother tongue (Das Sein und das Nichts). He was even acquainted with it at the tender age of 12 in the Arabic translation by Abdul Rahman Badawi, the famous Egyptian philosopher at that time. In fact Mohamed’s father who was an army general was quite concerned and worried that his young son is reading these things of which he could not make out head or tail. Education is never in vain, even if it is philosophy. The Cantor set made quite an impression on Mohamed and he instantly connected it with what he had read in Sartre’s book. He even mentioned the work of Sartre as well as Martin Heidegger in one of his most profound very early papers on the subject. This particular paper was cited by many pure mathematicians thinking that Mohamed is a pure mathematician. In his review of E-infinity theory Mohamed likened the Hausdorff dimension of the Cantor set with the spirit of a body which has long decayed. He likened it with this famous ghost story by Oscar Wilde. This particular review which is highly cited and although it is only five years old and despite all the negative propaganda artificially created in certain quarters, A Review of E-infinity Theory is found on Google Scholar to have been cited 328 times.
I recall something similar which was said by Leibnitz about imaginary numbers. Leibnitz said there are amphibian between being and not being. This is a remarkable formulation which covers almost exactly Mohamed’s notion of a Cantor set. It is an amphibian between being and nothingness to use the terminology which he borrowed from Jean Paul Sartre. There is no shame in borrowing things from famous as well as lesser mortals. What is shameful is to attempt to hijack other people’s ideas and deliberately calling them your own.
Now I would like to go back to the derivation of the fundamental equations. I must warn from the outset however from a major mistake which some could easily commit because they judge in haste. There is no element in Mohamed El Naschie’s work which is numerology. We agree fully with what Dr. Ray Munroe said on certain blogs that numerology is a pattern of numbers for which we have not yet discovered the real underlying reality, physics or theory. However nothing is further from the truth when it comes to E-infinity theory. If you still suffer from this delusion then you have not yet understood the theory. That would be a real pity because the theory is not difficult at all when you free yourself from all prejudice.
Now we give yet another derivation of the expectation value of the Hausdorff dimension. You have to start with the well known dimension of noncommutative geometry. The dimension is given as index or dim of N and M. It is written however as [M:N]. This is given by 1 divided by L plus 1 divided by 1 minus L. Here L is trace of E. From the theory of subfactors we have a very similar formula but this time giving [M:N] by 1 divided 2d-1. Now set L equal d and you will find a quadratic equation for d or L showing that it is equal to the golden mean or minus the inverse golden mean. For the positive value we get our 4 plus the golden mean to the power of 3. I leave it to you to have fun with Alain Connes and the great mathematician of the theory of subfactors. I leave it to you now to work out for yourself the four dimensional fusion algebra which has 1, 1, golden mean and golden mean again as a solution. It sounds like a puzzle but the next communication will make it clearer. Let me end in the same way that some of the rather pleasant commenters on the Munroe blog end their comments by saying, have fun and I add try to ignore perturbation.
E-infinity
E-infinity communication No. 8
The E-infinity counterpart of calculus – and Wyle scaling
It is my intention to start by making specific reference to the literature concerning the last communication which was omitted. The important noncommutative dimension which we used was given in detail in Connes marvelous book Noncommutative Geometry, Academic Press, 1994. The formula is given on page 506. He uses slightly different notation but everything is explained lucidly. You can understand subfactors from the wonderful book by V. Jones and V.S. Sunder called Introduction to Subfactors by Cambridge University Press, 1997. You may recall that Jones is the man who found the remarkable relation between knot theory and statistical mechanics. Much of what Mohamed El Naschie did could be reinterpreted in terms of Jones’ work. The relevant formula may be found on page 31 of this book. Several other important formulas may be found on page 143 and you can move from there to study how quantum field theory can be derived from subfactors. Believe me it all sounds far more complex than it is. Oh and there is an important paper by Mohamed El Naschie which has much of this stuff published in the Int. Journal of Theoretical Physics, Vol. 37, No. 12, 1998 entitled Superstrings, Knots and Noncommutative Geometry in E-infinity Space. The Editor in Chief of this journal at the time was no one less than the legendary David Finkelstein. David you will recall is one of the main people responsible for introducing set theory in quantum mechanics. Following Heisenberg and Finkelstein, Prof. Heinrich Saller excelled in developing a highly mathematical theory starting from a combination of symmetry groups and set theory. We warn however that this is very demanding mathematically for the average physicist. However if you would at least have a glance at the work of Saller, you will see high energy physics and quantum mechanics from a profoundly important and different view point which is not that dissimilar from that of what El Naschie is doing using what is comparatively humble mathematical tools.
Some of our members think I should move now to explaining the main tools of computation which we have in E-infinity theory. This would be Wyle scale. Other members think that we should no introduce the transfinite theory of dimensions which is the mathematical founding of E-infinity theory. A sizeable minority think we should outline the connection to wild topology and a few other disciplines used frequently in E-infinity theory. We will have to do all of that in the few coming communications. However I am inclined to think that it is time to introduce at least one simple example of the scaling analysis.
If you look to the beginning of the work of Nottale you realize that this eminent and great French scientist agonized about differentiation. He was slightly on the loose side when he equated fractals with non-differentiability. That brought him into conflict with fractal experts like the great Israeli scientist Itamar Proccacia. However the dispute is merely a misunderstanding when temperament takes over reasoning. Nottale like Ord and later on El Naschie realized the need for calculus machinery but he also realized that differentiation and similarly integration covers up all the interesting phenomena and leads us in the wrong direction. Ord in particular realized that taking the limit is the source of all contradictions in quantum mechanics. That is how Nottale decided for a compromise, albeit an ingenious one. Nottale does not give up continuity. For him a Cantor set is excluded a priori as the basis of the calculation. Therefore he used non-standard analysis in his calculations. That is how he came to most of his excellent results. Garnet Ord somehow managed to avoid taking the limit of his difference equations. In a manner of speech and in a way which Ord does not particularly like, you could say that he invented his own quantum calculus. Of course Garnet sees it completely different nowadays and he has grown more sophisticated about these things but for the purposes of this communication, it is sufficient to think of it in this way.
In noncommutative geometry Alain Connes was faced with similar problems like the pioneers of E-infinity. Being one of the greatest pure and applied mathematicians in the history of mathematics he had of course a sophisticated solution. Without going into the detail, let me give you a very short dictionary of the noncommutative solution. Alain Connes introduced a quantized calculus for which the following short dictionary applies. First infinitesimal is replaced by compact operators. Second integral is replaced by a Dixmier trace which is incidentally also used by El Naschie. The table for classical quantum correspondence in noncommutative geometry is given on page 20 of Alain Connes book which we mentioned earlier.
El Naschie as well as Ji-Huan He and Marek-Crnjac and occasionally Goldfain use something else. They return to Wyle scaling. Fractals has a marvelous character expressed in Bidenharn conjecture. The conjecture says the obvious that there are no a priori scales, man or God given, in a fractal spacetime. Therefore the counter argument of Einstein against Wyle’s idea does not apply in a fractal spacetime. The history of the whole controversy and its solution in E-infinity theory is duly explained in several papers by Mohamed El Naschie as well as a review by L. Marek-Crnjac. I will give you the exact reference later on. The final result is disarmingly simple. You are more or less scaling down when you want to differentiate and scaling up when you want to integrate. The words differentiate and integrate should not be now taken literally. Prof. Ji-Huan He likes to say that scaling is everything. In E-infinity at least, this is as near to the truth as anything could be. Let us discuss one example demonstrating the application of this idea and generating El Naschie’s hierarchy of Heterotic strings. You recall in classical mechanics when you want the equation of equilibrium you write a Lagrangian, then the vanishing of the first variation of this Lagrangian gives you the equation of equilibrium or motion. In very simple cases when the Lagrangian is a function rather than a functional, variation is replaced by simple differentiation. In E-infinity things are far simpler than that to the extent that some who expect to lift heavy weights are shocked and left in a state of disbelief because of the simplicity which they did not expect. For certain manifolds involving E-infinity the curvature may be given by very simple expressions. Squaring the curvature you find a normalized energy. If you can identify a certain parameter as a loading and you can calculate the distance which this loading can potentially move, then you have an equilibrium equation. Alternatively if somehow you convince yourself that you have what is equivalent to a Lagrangian, then repeated scaling gives you the answer for equilibrium equation or equation of motion. Suppose I convince you that half of the inverse of the electromagnetic fine structure constant may be regarded physically and correctly as the numerical value of the Lagrangian. In this case repeated scaling using the golden mean will give you the solution of corresponding equilibrium equation or equation of motion. Again this sounds far more complex than when you really do it. Let us do it.
Half alpha bar is equal to 137.082039325 divided by two equal to 68.541020. Multiply this now repeatedly with the golden mean 0.618033989.
The result is then the following hierarchy 42.360680, 26.180340, 16.180340, 10, 6.180340, 3.819660 and so on. This is the well known Heterotic string hierarchy in the transfinite form. The first value is the inverse coupling constant for non-super symmetric unification of all fundamental forces. The second is the coupling constant for super symmetry or the number of bosonic strings. The 16.180340 relates to the additional dimensions added. The 10 are the super string dimensions. The 6.180340 are compactified dimensions and the final number is 4 minus k where 4 are the dimensions of spacetime and k is equal 0.18034 which appears in all other numbers. The explanation of all that is given in detail somewhere else but obtaining the result is less than elementary as far as computation is concerned. Let me show you first how we obtain the original equation. El Naschie showed long ago that the average curvature of Cantorian spacetime at the core is equal to 26.18033989. Squaring this you get the energy in normalized form which is 685.410197. If you remember, this is the dimension of Ray Munroe’s E12. Not quite but very near. It is also very close to the sum of the dimension of all the 17 two and three Stein spaces. Not quite but very near. Introducing a loading lambda index I, then we can define a potential distance equal to the square root of the sum of all theoretical values of the coupling involved in reconstructing the inverse electromagnetic fine structure constant. This is 60, 30, 8 + 1 = 9, and the quantum gravity coupling 1. Adding altogether comes to exactly 100. The square root is therefore 10. Lambda I is therefore equal to 685.410197 divided by 10. This gives us exactly the value we started with namely 68.5410197. When you scale it you get lambda 1 equal to 42.360680, lambda 2 equal to 26.180340 and so on. This is extremely simple isn’t it? It is simple but difficult to understand. It is only difficult to understand because of our habits of thinking which we do not want to get rid of easily. Remember we must free ourselves from all ties as Prof. M. B. once said. I am sure you have hundreds of questions. I can assure you if you are not shocked and if you have no questions, then could not possibly have understood anything.
Best regards,
E-infinity communication No. 9
Wyle scaling and deriving the spectral dimension 4.02 of Loll, Ambjorn and Jurkiewicz using E-infinity
There is no reason why we should not continue with our discussion and give further examples of the use of golden Wyle scaling in E-infinity. We have to take the opportunity first to draw the attention of the readers to the literature where details are given. The most important three papers which can be consulted on this subject are the following: From classical gauge theory back to Weyl scaling via E-infinity spacetime by M. S. El Naschie, Chaos, Solitons & Fractals (CS&F), 38, 2008, p. 980, A Feynman path integral-like method for deriving the four dimensionality of spacetime from first principles by L. Marek-Crnjac, CS&F, 41, 2009, p. 2471 and Density manifolds, geometric measures and high-energy physics in transfinite dimensions by S.I. Nada, CS&F, 42, 2009, p. 1539.
To explain the main idea in the most excessively simplistic terms you could say the following. There is a fundamental difference between changing direction in space and stretching a line in space. In the first case nothing really physical happened while in the second case something almost physical happened. That was the crux of Einstein’s objection against Wyle’s gauge theory. Lacking a natural scale, in a fractal setting a stretching can be set on the same footing as changing an angle. You read the rest please in the relevant literature. By the way many people know this trick and that is why fractal spacetime is becoming fashionable and will become even more fashionable as time goes by, so you can play it again Sam.
Let me now return to examples of Wyle scaling and we can do two things for the price of one. We give another example and derive the spectral dimension given for instance in the paper of Ambjorn, Jurkiewicz and Loll in Scientific American or the improved version they published in a book Edited by Daniele Oriti entitled Approaches to Quantum Gravity, published by Cambridge, 2009 a few years after the first derivation by El Naschie using Bose Einstein statistics. The excellent paper by the three authors who are world renowned for simplictic triangulation, in other words, tiling the space with simplexes which again means Regge calculus or finite element of John Argyris who was one of Mohamed El Naschie’s teachers in Stuttgart, Germany and Imperial College, London is entitled Quantum Gravity: the art of building spacetime on page 341-359. The important formula is on page 352.
Let’s begin at the beginning. In Heterotic string theory anomaly cancellation requires either O(32) or E8 E8. In this theory there are left and right moving sectors. The left is purely bosonic. We start with 26 dimensional strings where 16 have been compactified on a lattice, for instance E8 E8 lattice or spin 32 divided by Z2 lattice. The right movers on the other hand are super symmetric. Thus the right movers are explicitly spacetime symmetric. Recall that spacetime symmetry of super space is found until this moment by trial and error. Now in the right moving sector at the lowest level we have 8 plus 8 equals 16 states. In the left moving sector we have three distinct objects leading to 8 states plus 16 states plus 480 states. This makes altogether 504 states. To obtain the entire spectrum we follow Fock space rules of quantum field theory and multiply left movers with right movers. That way we obtain from 504 times 16 the well known 8064 states. This is a very well known result. You can find this result in popular literature like Scientific American or text books on string theory like Kaku’s book. Knowing that there are truly childish people hanging around on internet blogs with an enormous chips on their shoulders, we would like to give you the literature to check everything yourself so that these children do not write that we are telling you fibs. OK, here they are. Scientific American, Superstrings by M.B. Green, p. 183-203, in the caption of Figure 11.14. Reprinted in a book called Particles and Forces, Editor Richard A. Carrigan, published by W. Freeman, New York (1990). M. Kaku’s book Introduction to Superstrings and M-theory, published by Springer, 1999, see page 384. Now enter Mohamed El Naschie. He realized that the same result is easily obtained in an entirely different manner and in some respects a far simpler geometrical way by assigning the right amount of instantons to each site of Klein’s modular curve as a holographic boundary of string theory. Without going into details the instanton number in this case is 24. The number of triangles as you know from earlier communications is 336. Multiplying the two numbers you get the total number of instantons, namely 8064. Mohamed was under the false impression that this must be a well known method. He published it and mentioned the whole thing just as a marginal thing. He published it a few times without paying too much attention to any novelty. It was then after heated discussion that Nobel laureate Gerard ‘tHooft convinced him that it is an entirely new theory. I forgot to mention some of the extensions and the insight which El Naschie gave to this point also without realizing that it is an entirely new theory. First look at 504. This was realized by Mohamed as the summing of an exceptional Lie group. Here is a sum about which some people who are ignorant about the exceptional Lie group hierarchy rejected off hand and of course wrongly so. Add the dimensions of E8, E7, E6 and E5 together then you will have 248 plus 133 plus 78 plus 45 equals exactly 504. What most people did not know is the fact that E5 is nothing else but SO(10) of grand unification which is competing with SU(5). SO(10) is really an E5 based on its Dykin diagram. This fact was known to El Naschie as well as many other careful people which do not include of course John Baez who had too much to do on his shows on his diverse blogs for entertaining people. It was of course a mistake to stop at E5. In mathematics it would mean you are not using a complete set. To use a complete set you have to take all the exceptional groups. The sum of that as everyone knows in the meantime was found by El Naschie to be 4 times 137 equals 548. Similar reasoning would show that classical Heterotic strings are only approximation and the real numbers of massless states is not 8064 but 8872 when we ignore everything after the dot. Now El Naschie did not calculate it in this way. He reasoned differently using the holographic boundary. Taking compactification into account you do not have 336 triangles but 338.885438 weighted number of triangles. In addition he derived the exact instanton density and found that it is not 24 but exactly 26.18033989. Multiplying both exact numbers you find 8872.135951. Wonderful. This is now our numerical Lagrangian or potential or whatever you want to call it. Sixteen times differentiation is an E-infinity equal to sixteen times scaling with the golden mean. Multiplying our Lagrangian with the golden mean to the power of sixteen, you get 4.01999999 on a pocket calculator. This is Loll, Ambjorn and co’s result. For all practical considerations it is 4.02. Please do this calculation yourself.
It is interesting to ask why the pioneer of the holographic boundary did not find this result first. I do not know but we cannot all work on everything simultaneously. A reasonable explanation may be the following. The expert on the holographic boundary did not care about the 336 because they know once they compactify we get infinity and physicists do not like infinity and do not work with it. Mohamed El Naschie on the other hand took a gamma distribution weighted infinitely compactified Klein modular curve which added approximately 3 to the 336 to get approximately 339 which is a finite result and meaningful. He was also able to deal with a fuzzy K3 manifold and find the instanton number to change from the classical 24 to 26.18033. This is incidentally equal to the dimension of transfinite Heterotic strings as well as the corresponding Euler constant as well as the curvature which we discussed in an earlier communication. 26.18033 turned out to be an extremely important number. People interested in number theory knew that much earlier but did not know about any relevance in physics. Nobel laureate David Gross must be credited with super natural intuition to have invented Heterotic string theory. Witten and his friends were probably the first to introduce K3 to physics. So you can see string theory is not in any way as useless as some of its opponents want us to believe. Nothing is useless except underestimating people. Nothing is as harmful as belittling the achievements of other people. Nothing is as revolting as the yellow jokes of envious souls who ceaselessly inundate us with their superfluous comments, polluting the blogosphere with their inferiority complexes. You can find explicit calculations of the spectral dimension in several papers recently published in Chaos, Solitons & Fractals. For instance From Menger-Urysohn to Hausdorff dimensions in high energy physics by G. Iovane, 42, 2009, p. 2338 and On the Menger-Urysohn theory of Cantorian manifolds and transfinite dimensions in physics by Guo-Cheng Wu and Ji-Huan He, 42, 2009, p. 781. Also A Feynman path integral-like method for deriving the four dimensionality of spacetime from first principles by L. Marek-Crnjac, 41, 2008, p. 2471 and Density manifolds, geometric measures and high-energy physics in transfinite dimensions by S.I. Nada, 42, 2009, p. 1539.
I think by now anyone who has attentively read the last nine communications must be able to derive on his own the basic and fundamental dimensions of fractal spacetime, namely the topological dimension 4, the Hausdorff dimension 4 plus the golden mean to the power of 3, the scaling dimension 4 minus k equal to the golden mean square to the power of 10 and finally the spectral dimension which is 4.02. The work of Loll and her crew is remarkable in that they got the exact result without having the complete theory. They were right to think that only a computer can get this result right. I say only a computer or E-infinity theory and nothing in between. They may have wrongly led people to think that 4.02 is a Hausdorff dimension or a topological dimension for quantum spacetime. It is not. It is an exact dimension for the spectral dimension of quantum spacetime. In a sense this is an even more important dimension because it measures how spacetime unfolds. It looks at it as a dynamical process, not as a statical process. It looks at it like ink spreading in water. They should be commended for their handiwork. What a pity that they were not aware of all what we have done in the last 15 years using E-infinity theory and fractal spacetime and what a pity that people can sometimes be so individualistic that they cannot see the achievements of others. Cooperation is always much better than meaningless struggle against others. At the end of the day, at least in E-infinity, we have benefitted from all the sound ideas of well developed careful theories like string theory, loop quantum mechanics, holographic principles, Penrose tiling, noncommutative geometry and so on and so forth. I sincerely hope you agree with us, at least on this.
E-infinity communication 10
Continuation of previous discussion and comments
The best thing to start this section is to repeat the potent comments which were lodged on this very same blog by Anonymous. We agree with all that is said in this comment but would like to amplify it. Let us start by repeating here the comments of the respectable Anonymous person and then we will continue.
Start of Anonymous Comment: The comment regarding 137.082039325 is by far the worse. It shows total misunderstanding not of E-infinity only, but of science. It shows that the person in question is mixing between mingling experimental data with mathematics on the one side and the clear distinction between model building, mathematical that is and a fully fledged theory. If I understand ‘tHooft correctly he always laments that leading scientists who develop excellent mathematical models more often than not market it using false labels and sell it as a theory. Surely these are modern times of commercial internet and wheeler dealers, even on the level of prestigious prizes in physics. E-infinity is a theory. It is genuinely a fundamental theory. 137.082…. is from the world of theory. If you read at a minimum El Naschie’s famous review article, A review of E-infinity, then you know the experimental value is obtained by projection from 137.082… to find the almost exact experimental value of 137.03….. El Naschie was at pain to explain the difference and the projection formula is given explicitly. It was given explicitly in two dozen papers by El Naschie, his students and collaborators of which I am aware. You must understand the difference. A theory like string theory in ten dimensions or M-theory in eleven dimensions or F-theory in twelve dimensions or E-infinity in infinite dimensions cannot take any measurement in these higher dimensions. Measurements, whether we like it or not are taken in our 3 + 1 familiar spacetime. It is very doubtful that any man however superior, like Ed Witten, Penrose or Einstein himself could construct a laboratory in five or more dimensions to satisfy the theory through measurement. Of course we then translate everything from theory to experiment. That is why we have things like mathematics, imagination and intuition. To be sure, the ideal E-infinity value has not been postulated. It was deduced. How? Just read the papers. A mother, however kind, can do nothing more than feed the baby. Regardless of how careful and kind she is, and regardless how young and fragile the baby is, her job ends with giving the food on the spoon into his mouth. The baby must digest the food. I swear to God I am not being patronizing, big headed or arrogant. This is not at all my nature. I am a humble almost mediocre mathematician. I was just thrown out of equilibrium by the comment. I was disheartened that after so much, there is such a deep fundamental understanding of basic principles of scientific research. Let me explain to you how I understand El Naschie’s heuristic derivation of 137.082… He gave other mathematical derivations as well as physical derivations. However the present heuristic one is what I find best. It comes out of a heuristic derivation based on the renormalization group equation of quantum field theory. Take the experimental values of the electroweak and round them to a neat integer. It will not take much to find out that the inverse alphas are 60, 30, 9 and unity. In addition you replace the group theoretical fudging factor by the golden mean scaling. Thus the familiar Clebsch is replaced by 1.618033989. Pretty close are everything to the familiar values. Our reconstruction of the inverse electromagnetic fine structure constant becomes in this case the following. 60 multiplied with the inverse golden mean plus 30 plus 9 plus 1 where 1 is the coupling constant of the Planck masses to the Planck Aether. El Naschie says if you do the calculation correction you will find 137.082… comes out like magic. This shows that the correct theoretical values are 60, 30, 9 and 1. The first three were verified experimentally with the accuracy of measurement of our present day facilities at CERN and Fermi Lab. Some got the Nobel Prize for it. The unity requires Planck energy to verify experimentally which is probably for ever outside of our reach, experimentally speaking. However the present calculation and much of El Naschie’s reasoning are compelling proof that this is correct. The whole procedure is referred to in detail in El Naschie’s paper with the telling title Transfinite harmonization by taking the dissonance out of the quantum field symphony. Published in Chaos, Solitons & Fractals, 36, 2008, p. 781. Particle physics may be likened to a magnificent symphony….. We conclude by restating our conviction that nonlinear dynamics, chaos and fractals will revolutionize the way we think about our high energy physics problems. End of comment.
Now to dwell on what exactly is physical and what exactly is mathematical is not at all the important point. It may be important but it is not important for our purposes here. The vital point is that Einstein’s objection against Weyl is torpedoed and sank like the Bismarck in the North Sea once spacetime is a fractal. The fractal spacetime knows nothing called a priori scale. That is the crux of Bidharn’s intelligent observation forming his conjecture. That is why all theories using sophisticated scaling including what is relatively crude or brutal force computer methods using mesh or lattices along the lines of K. Wilson have all been successful in obtaining results reasonably close to the experimental verification. Again there are no experimental values at all for anything related to quantum physics. The whole idea is that every measurement depends on the energy with which you probe spacetime which means your observation, including the dimensionality of your space, are a function of your resolution. I am afraid we have to give up completely our old classical intuition and adopt a new intuition suitable for quantum physics but what am I saying here? This is really well known and the point is the following. People who work with quantum mechanics, including the Copenhagen version, got so familiar with their measurements that they intuitively do the right thing sometimes without knowing. So let me repeat again, the inverse electromagnetic fine structure constant is 137 or 127 or 42 or 26. It all depends on the energy at which you are probing things. The 26 for instance is where all fundamental forces, including electromagnetism, are exactly of the same strength and this is by the way a super symmetric theory. If you are talking about the Planck scale then we know of course that all the difference between everything disappears and you could say that the electromagnetic fine structure constant became unity. But then you never call it in this case the electromagnetic fine structure constant. You call it in this case the Planck mass coupling. So please be careful here. These are pitfalls but they are well known pitfalls.
Let me give you another two derivations of the inverse electromagnetic find structure constant at higher energy. Let us take the energy connected to Ed Witten’s eleven dimensional 5-Brane theory. The number of states, as is well known, is given by 528. This is more than our E8 E8 496. In this case it is easily shown that particle physics will be represented by our familiar 339 of the holographic boundary. However the graviton sector of Einstein is now a graviton sector of Kaluza Klein. I am not very careful with my terminology here. All what I really mean is that my dimensionality is given by D=5 and not D=4 of Einstein. Therefore I do not have 20 independent components in the Riemann tensor but rather 50. Just to remind you, the number of the independent components is given by n square multiplied with open bracket n square minus one close bracket divided by twelve. For n=4 you get 20 and for n-5 you get 50. The correct value can be shown to be 52 for technical reasons and not just 50 because we will have to use an exceptional Lie group called F4 to take care of the gravity section in this case. El Naschie’s fundamental equation now becomes 528 equals electromagnetism plus 339 plus 52. Trivial arithmetic then shows that electromagnetism must be 137 as it should be.
A final exercise for the reader is to derive 137 but this time from the much larger value of the sum over all 8 exceptional Lie groups family Ei. You remember that this is equal to 548. The analysis is very similar but now the gravity section will be represented again differently. It is not simply obtainable from the Riemann tensor nor is it an F4 group. It is represented by the roots of E6 or alternatively or equivalently by the kissing number of spheres in 6 dimensions. The value in this case is 72. Repeating the same elementary calculation of El Naschie’s fundamental equation you will find that the electromagnetic fine structure constant comes out as 137. There are a few subtle points here and subtle questions which I do not want to burden you with here. However I cannot resist the temptation of showing you a remarkable analysis in 11 dimensions where all the fundamental forces including electromagnetism reach the same magnitude of 339. It is extremely elementary to perform the calculation but the picture behind it is far from being trivial. You start by putting a quantum path in 11 dimensions. A quantum path has a Hausdorff dimension of 2. To put it in 11 dimensions you follow Connes or El Naschie’s bijection formula and raise 2 to the power of 11 minus 1. This is equal to 2 to the power of 10 which is equal to 1024. The number of states corresponding to that is 3 times 339. When you add to that the 7 compactified spacetime dimensions, it comes to exactly 1024 and you have from the 11 dimensions only 4 left. These are our spacetime dimensions which we are all familiar with. Next time we would like to discuss the Menger-Urysohn dimensional system and show you the wonder of the vacuum and how E-infinity has a handle on the vacuum which the great super string theory does not have. Until the next time, thank you for your patience.
E-infinity communication 11
Further elucidation and towards the general theory of transfinite dimension
We have a lot of work to do and we hope we can do it today but I doubt that we can seriously start explaining Menger Urysohn theory of transfinite dimension in this communication. Judging by past experience negative dimensions cause physicists a great deal of difficulty but let us start without much ado.
First we have to tie up loose ends from the last communication. In particular what is so special about 11 dimensions? Where did the 11 dimensions crop up for the first time? The first time it was super gravity. This is a theory in where there are nothing but gravitons and their super symmetric partners, gravitinos. Incidentally both have never been observed. Never the less this is a full blooded general theory. For a long time it was considered to be the only game in town, pretty much like super strings nowadays. The next 11 dimensional theory may be the M-theory of Edward Witten. This theory unifies the various competing super string theories but the exact form of this theory is not known. The theory is formulated in 11 dimensions. The 11 dimensions are normally justified in the following way: We need 7 dimensions for SU(3), SU(2) and U(1) of the standard model. In addition we have 4 spacetime dimensions and this gives us 11 dimensions. This is Ed Witten’s standard explanation stripped to the bones. Also from Ed Witten we have a 5-Brane model which requires 11 dimensions. Now the 5-Brane model has 528 states. We encountered this number before. Now it is time to tell you something about maximally symmetric spaces. This is mathematics connected to killing fields. Let us take n to be the dimension of a space, then if the space is maximally symmetric then the number of isometries will be equal to n multiplied with n plus 1 divided by 2. Taking n to be 32 you would have 528 isometries which is equivalent to the number of states in Witten’s model. If you write the model explicitly to obtain 528 you will notice that you have string-like objects as well as 2 and 5-Branes. The combinatorial formula is as follows: You have 11 over 1 plus 11 over 2 and 11 over 5. When you work out the combinatorial correctly you find that this is 11 plus 55 plus 462 making exactly 528. You see there are no zero objects, that is to say there are no points particles in the model. Also the 3 dimensional and the 4 dimensional objects are missing. You see number theory is trying to tell you something which is physical. If you get the hint, as Mohamed El Naschie did, then you will find that 11 over 0 plus 11 over 3 and 11 over 4 gives you exactly 496. Surely you remember this? This is the dimension or isometries of E8 E8. Now a second exercise into the simple addition of two numbers, 528 plus 496 gives 1024. This is exactly what we got when we put a quantum path in 11 dimensions and found the total dimensionality by the bijection formula of E-infinity which tells you that N0 is 2 to the power of the dimension minus 1. This is 2 to the power of 10 equals 1024. I hope at least some of you like it. Number theory is a powerful subject which physicists should take very seriously. A summary of elementary number theory in high energy physics was given by El Naschie and I recommend this paper strongly. It appears in Chaos, Solitons & Fractals, 27, 2006, p. 297. There are a few theorems given in this paper and let me mention here theorem No. 5 on page 310. This theorem shows that the theoretical value of the inverse find structure constant is given by certain expressions which lead to 137.082039325. So as long ago as 2006, which means four years ago, this value was given by a mathematical theorem and there are dozens of other derivations leading to the same. Again particle physics is a symphony. Everything has to work together harmoniously. This harmony principle is what leads to this value. Remember the comments of Sir Roger Penrose about his tiling. He said the tiling has a memory and when you violate the rules, the tiling remembers and takes it revenge when you suddenly find you cannot go on. Words which El Naschie always remembers to say in his lectures goes back to Niels Bohr…. Only in the multitude lies clarity. Believing that this is really the case, let me go on amplifying the preceding discussion a little bit before we turn to another subject. You remember how we got 528 from summing over exceptional Lie groups. It was E8, E7, E6 and E5. This led to the important section formed by 504 in string theory. Now let us add E4. Most people who do not know E5 will not know E4. For sure American mathematician John Baez, who calls himself a one man internet army, did not have enough time left to familiarize himself with E5 or E4. E5 turned out as we saw before to be SO(10) of grand unification. E4 on the other hand is the SU(5) proposal for grand unification. The dimension is 24. Adding the 24 to 504 we get Witten’s 528. In fact if you go on adding E3, E2 and E1 you arrive to El Naschie’s sum, namely 548. I will not discuss this any further here but ask a final question. Why stop with D equals 11 of M-theory? An Iranian born scientist who lives in America went on to invent a further theory with 12 dimensions for spacetime. Let me show you some marvelous things about this from a number theoretical point of view and I will leave it to you to think of the rest. Take SL(2, 12). The dimension is given by the well known formula and leads to 1716 isometries. On the other hand the Riemann curvature tensor leads for the same dimension to exactly the same number as independent components, namely 1716. Please try it yourself. This coincidence, which is not coincidence, is behind many of the wonderful things about F-theory. For instance we could do something similar to what we did much earlier. Take the fundamental 5 forces. Give each 339. The total would be 5 times 339. Think yourself in a bosonic 26 dimensional space of strings. You have to compactify 22. Add this 22 to the 5 times 339 and you find 1716 plus 4 dimensions left from the 26 to be our ordinary spacetime. I promise now to move to transfinite theory of dimensions.
The father of the inductive theory is an extraordinary Jewish Russian scientist, Paul Urysohn. This wonderful man died tragically while swimming in the Biscay at a very tender age. Hilbert knew the value of the young man from Russia. In a famous anecdote David Hilbert asked his assistant Richard Courrant what happened to the paper of Urysohn which was supposed to be published in Hilbert’s journal. Courrant said he sent it for peer review. Hilbert immediately snapped, I told you to publish the paper. I did not say you should send it for dam refereeing. We clearly live in different times where the thin veil of so called peer review is used and misused on a daily basis in high places. I do not think I want to go into that but I just wanted to tell you this because David Hilbert knew there is nobody except him who could appreciate and understand the work of this young Russian mathematical genius. Now I cannot give you the mathematical derivation of Urysohn’s work, not even in the improved and simplified version by the great Jewish Austrian mathematician Karl Menger who was forced like so many other great people to emigrate to the United States because of the ignorant unscientific Nazis who took over Germany and then marched into Austria before putting the entire of Europe on fire. What I will do here is an extremely, extremely simplistic way to come to the nitty gritty which theoretical physicists like ourselves can use to tame quantum mechanics.
Let us start with 3 dimensions, say a cube. What is the dimension of the boundary of a cube? Clearly this is an area which is two dimensional. That means it is 3 minus 1 equals 2. Apply the same thing to a 2 dimensional square. The border is a line which is one dimension so it is 2 minus 1 equals 1. Take a line. How can we separate the line or what is the border of the line? The border is a point which has a dimension equal to the line minus 1 equals zero. True mathematical genius or any kind of abnormally inclined human being never stop where almost everybody else stops and they ask questions which are unexpected although very simple and natural. Suppose we want to separate the point into two parts. What would the border of a point be? Formally and naively we just continue our implicit formula which we used so far, namely the border of n dimensional object is always n minus one. Applying this to the point we have 0 minus 1 equals minus 1. Mathematicians call the point with zero dimension the zero set and what we just calculated and found to be minus 1 is called the empty set. That is where everything in mathematics and set theory starts. I hasten to say that in physics the zero set is misunderstood on many occasions and as for the empty set, it is more than misunderstood at the majority of physical occasions. In his theory El Naschie did not even stop at this point. He continued asking the question, namely what is the border of an empty set? Formally this is minus 1 minus 1 equals to minus 2. This is a set which is even emptier than the empty set. Using certain relations between topological and Hausdorff dimensions, El Naschie continued this game to minus infinity or total nothingness. He was able to show that the Hausdorff dimension corresponding to this nothingness which is minus infinity is a simple 0. In addition it is a trivial result of his bijection formula that the zero set has a Hausdorff dimension equal to the golden mean while the empty set has a Hausdorff dimension equal to the square of the golden mean. The same result automatically follows from Connes’ theory of noncommutative geometry applied to the Penrose universe as explained in the book of Connes which we mentioned in an earlier communication. We will stop here and promise to give the details in the next communication. Best wishes,
E-infinity communication 12
The Menger-Urysohn transfinite theory of dimension as used in E-infinity theory
Before continuing our discussion we must give you some literature on the subject as well as give the various people who contributed to E-infinity their due by mentioning their achievements. The easiest way to start studying the transfinite theory of dimension is to look at the work of Nada, Crnjac, Iovane, He and Zhong. Prof. Shokry Nada is a professor of topology who got his Ph.D. from Southampton, UK and although he is originally Egyptian he is since many years part of the full time staff of the University of Qatar, Dept. of Mathematics. I strongly recommend contacting him personally but only on scientific questions. He has no patience for gossip or triviality and will not answer to things like those using the internet for entertainment on an unacceptable level. I give here without any particular order seven papers from these various authors that will help in studying the Menger-Urysohn theory tailored to E-infinity and physics. All papers are from Chaos, Solitons & Fractals. On the mathematical theory of transfinite dimensions and its application in physics, 42, 2009, p.530. The mathematical theory of finite and infinite dimensional topological spaces and its relevance to quantum gravity, 42, 2009, pp. 1974. Partially ordered sets, transfinite topology and the dimension of Cantorian-fractal spacetime, 42, 2009, pp. 1796. On the Menger-Urysohn theory of Cantorian manifolds and transfinite dimensions in physics, 42, 2009, p. 781. Density manifolds, geometric measures and high-energy physics in transfinite dimensions, 42, 2009, p. 1539. From Menger-Urysohn to Hausdorff dimensions in high energy physics, 42, 2009, p. 2338. From the numerics of dynamics to the dynamics of numerics and vice versa in high energy particle physics, 42, 2009, p. 1780. It is however recommendable to go back to the original contribution of Menger and Urysohn. The papers of Urysohn are most in French and German. Menger published mainly in German and English. There is an important classical book by Hurwitz, a Polish mathematician writing in German. You will find all of that referred to in the papers of Nada, Iovane, He and Crnjac. Regarding the golden mean in physics, we forgot to mention a few papers which were initially considered outlandish but in the meantime, that is no longer the case. They are important papers by Leonard J. Malinowski, Electronic golden structure of the periodic chart, CS&F, 42, 2009, p. 1396. The other papers you will find on Elsevier’s Science Direct.
To understand E-infinity fast we recommend that you take no short cuts. You must read at least one review paper by Mohamed El Naschie or Marek-Crnjac from the beginning to the end. At a minimum you should read El Naschie’s paper in David Finkelstein’s journal, Int. Journal of Theoretical Physics, Vol. 37, No. 12, 1998 from the beginning to the end. It is fair to say that after flirting with nonlinear sciences due to his background in the theory of stability and bifurcation as well as Rene Thom’s catastrophe theory, El Naschie started to seriously enter into nonlinear science and chaos due to the encouragement of his friend, the chaos pioneer, Otto Rössler. Much of what he knows about chaos was taught to him by his Israeli friend, the well known scientist Itamar Procaccia from the Wiseman Inst. as well as the legendary figure of chaos, Mitchell Feigenbaum. El Naschie was highly impressed by the overall personality of Mitchell although he did not share his passion for red wine and smoking. El Naschie is almost a vegetarian, unlike Mitchell who lives from red meat.
Maybe it is time here to correct some old mistakes and omissions. El Naschie in retrospect was always at pain to acknowledge that Indian meteorologist Marie Selvam may have been the first to notice the E-infinity theoretical value of the inverse electromagnetic constant. He mentioned that on many occasions before but he asked me to mention that again on this occasion. In addition we must acknowledge that Carlos Castro did some important work on E-infinity and it is with sadness that we note that he moved towards blogs and internet publications, leaving serious discussion with his old friends on science. However these few words are said because we cannot rewrite history as we like. The truth remains always the truth, no matter how painful it is. A person who is outside our group but works with incredible dedication on fractal time is a German American Susie Vrobel. We should mention her work like Fractal Time and the Gift of Natural Constraints, Tempos in Sci. & Nature, Structures, Relations and Complexity, Vol. 879, June 1999. Another person outside our group who works with dedication on fractals in cosmology is the South African Jonathan J. Dickau and we recommend his paper Fractal cosmology, CS&F, 41, 2009, 2102. Now we should return to our main subject.
The vital so called bijection formula of E-infinity theory basically says the following. If you want to know the Hausdorff dimension of a Cantorian set in n dimensions then all that you need is the inverse of the zero set raised to n minus 1. As we said earlier and we will prove it again, the Hausdorff dimension of the zero set is the golden mean. So if we want to know what the Hausdorff dimension is in two dimensions, we take the inverse of the golden mean and raise it to two minus one. That means we would have 1 divided by the golden mean which because of the nice property of Feigenbaum’s golden mean renormalization is exactly equal 1 plus the golden mean. For three dimensions you can easily work out the result to be 2 plus the golden mean. For four dimensions you obtain the famous formula which is 1 over the golden mean to the power of 4 minus 1. This is 1 over the golden mean to the power of 3 which is our famous number 4 plus the golden mean to the power of 3 equals 4.23606799…. What is interesting now is to look into the zero case and the empty set case. Let us see what the dimension for n equals 1 is. This would be the inverse of the golden mean to the power of 1 minus 1. This would be equal to the inverse of the golden mean to the power of 0. This means it is unity. In E-infinity this is called the normality condition. The one dimension is a special case. Here the Menger-Urysohn extension of the topological dimension and the Hausdorff dimension coincide and are equal to unity. Let us go one step further and ask what the Hausdorff dimension is in dimension 0. I mean now the Menger-Urysohn dimension 0. Then we have 1 divided by the golden mean to the power of 0 minus 1 equals minus 1. That means it is the golden mean to the power of 1 which is equal to the golden mean. Thus we have proven the assumption. Now we go to the second most vital step and ask what the Hausdorff dimension is when the Menger-Urysohn topological dimension is equal to minus 1. Remember this is the empty set as defined classically. The result is we have 1 over the golden mean to the power of minus 1 minus 1 which means to the power of minus 2. This means we have the golden mean proper to the power of plus 2. Now we have resolved indirectly the famous two-slit experiment. The most conservative explanation is due to the work of Mohamed El Naschie together with his late teacher, fatherly friend and mentor, Prof. Dr. Dr. habil Werner Martienssen who sadly died a few weeks ago. Martienssen and El Naschie decided to give the wave the Menger-Urysohn dimension of the empty set. This is a Hausdorff dimension equal to the golden mean to the power of two. The particles on the other hand get a Menger-Urysohn zero and a Hausdorff dimension equal to the golden mean. All these are of course interpreted as probabilities but I am running ahead of my theory. I will repeat all of that later on. El Naschie asked a trivial question but in doing so, he solved a major problem. He said why stop at the empty set? Why not ask if there are emptier sets? Being a naive engineer he said why should I integrate from minus 1 to plus infinity. From many engineering problems of struts on an elastic foundation extending relatively from minus infinity to plus infinity, some time referred to as full or half space in theory of elasticity, he is used to integrate from minus infinity to plus infinity. So he found emptier and emptier sets with Hausdorff dimensions equal to golden mean to the power of 3, then golden mean to the power of 4, then to the power of 5 and so on. At minus infinity you will have the golden mean to the power of infinity. Since the golden mean is smaller than one, to be raised to infinity you have an absolute 0. That way El Naschie discovered as it would turn out later, a unidirectional system giving a hint at the unidirectionality of time starting at a singularity. When he reached this result he wrote a paper in some Pergamon mathematical journal and dedicated it to his teacher and mentor, Ilya Prigogine who was rather excited about it and said so in correspondence. Related results were also communicated to K.F. von Weizsäcker who was very excited about it and wrote as much. Weizsäcker advised El Naschie to take the work of David Finkelstein seriously and told him that his own work was quite near to that of Finkelstein. My recounting of all these anecdotes may not be very accurate because I know them all second hand from someone who was told by someone that El Naschie told him. However the big paper dedicated to Ilya Prigogine is there for everyone to read and so is the complimentary letter of Weizsäcker to El Naschie. I am saying all these relatively unimportant facts because there is a whole industry now on the internet whose main reason d’être is to shed doubt and spread doubt and rumors and lies about E-infinity and its founders. El Naschie’s theory was not invented in 2008 when all this noise started with an article published in Scientific American. El Naschie has been working on his theory since the late 80’s. Only Garnet Ord and maybe Laurent Nottale were first chronologically speaking.
I think we have covered substantial ground on Menger-Urysohn theory and the extension of the classical empty set to the absolute empty set by El Naschie. The relevant papers which are cited in the pure mathematical literature quite a bit will be given to you in the next communication. It is important to understand now that we have subdued infinity. In Cantorian spacetime and fractal spacetime there is no ultraviolet or infrared catastrophe. Everything is regularized automatically because we are working in a naturally renormalized geometry. Everything has at least two major dimensions to describe it. Therefore the old fashioned uniqueness of dimension theorem does not apply and do not restrict us anymore. That is why Cantor’s theory is a paradise from which we should not be evicted as David Hilbert asserted. E-infinity is based on Cantor’s paradise. There is no room here for cheap jokes except from those uncorrectable philistines whose jokes just pollute every site in the blogosphere. I am sorry for using these harsh words but E-infinity or not, we are luckily all human. Until next time, all the best.
E-infinity communication 13
Transfinite dimension applied to the two-slit experiment with quantum particles
Before we start again we ought to give some literature about the extended Menger-Urysohn system. The first important paper here is El Naschie’s 1994 paper On Certain ‘Empty’ Cantor sets and their Dimensions, CS&F, Vol. 4, No. w, 1994, p. 293. Three related papers which may be useful are Is quantum space a random Cantor set with a golden mean dimension at the core?, CS&F Vol. 4, No. 2, p. 177, 1994. Statistical geometry of a Cantor Discretum and Semiconductors, Computers Math. Applic. , Vol. 29, No. 12, p. 103, 1995 and Time symmetry breaking, duality and Cantorian space-time, CS&F, Vol. 7, No. 4, p. 499, 1996. Finally a very important general paper is by two Romanian mathematicians Mircea Crasmareanu and Cristina-Elena Hretcanu called Golden differential geometry, CS&F, 38, 2008, p. 1229. This particular paper incorporates the golden mean in a fundamental way to produce golden differential geometry which is extremely valuable in understanding E-infinity theory.
Now there are a few realizations which one should draw from El Naschie’s E-infinity. First an infinite dimensional system is necessarily chaotic. This is really a theorem. We first knew about it from the experimental work of Ji-Huan He using a computer and drawing infinite dimensional cubes. He did not really make it infinite. You do not have infinity on a computer. He went as far as a 26 dimensional cube. Then we realized that the great mathematical engineer and mathematician David Ruelle presented and used this theory in some early work. I am not sure but I think Ruelle also gave a proof. You will probably know the name of David Ruelle if you work in nonlinear dynamics for his contribution including Ruelle-Taken’s scenario for turbulence. There is an anecdote to tell here. Some hated that a civil engineer should be such a good mathematician which David Ruelle is. He could not get his paper on turbulence published even though he was an Editor in Chief of a journal. At the end he got fed up, submitted the paper to himself and accepted and published the paper anyway. The scientific community therefore owes him two things, an apology and thanks. Of course it is a procedure which should not be generalized but there are until today those who are consumed by jealousy and who keep attacking Ruelle for publishing his own paper in his own journal. In his famous book he recounts how a certain person used to go from one library to another library to tear out his paper from the journal copy in that library. Unbelievable what jealousy and hatred does to people, even scientists. Mind you this was even before blogs and internet vandals masquerading as scientists. Any case enough of that.
An important other point if the role played by the zero set and the empty set in physics. Physicists seem not to have incorporated this in a systematic way and have not developed the intuition to work mathematically with nothing instead of something. Thank God for B. Mandelbrot, M. Barnsley and Otto Peitgen. They gave us some intuition about fractals. Thank God for material scientists who were the first to jump on fractals and use them. In high energy physics on the other hand things were and are still relatively very slow indeed. The reason is they do not know how to use mathematics properly for fractals and connected to physics. They only use the marvelous device of Hausdorff dimension. Alas the Hausdorff dimension is not a topological invariant. The marvelous thing about the new theory, as shown in the work of Alain Connes and Roger Penrose is that both, the topological Menger-Urysohn dimension and the Hausdorff dimension are connected in one formula. In the theory of Connes this is done implicitly. In El Naschie’s work this is explicitly as evident from his bijection formula discussed earlier on. For this reason we will have to discuss at length the work of Tim Palmer who came to similar conclusions but in a qualitative way and could not make quantative computation because he does not assign the right dimension to the empty set. You could say Palmer discovered fractals for quantum mechanics but only qualitatively. His biggest triumph and achievement in quantum mechanics was his ingeniously apt formulation – quantum mechanics is blind to fractals. However by not admitting Menger-Urysohn minus 1 dimension to the classical empty set which represents the vacuum at its first level, he could not resolve the problem of quantum mechanics except qualitatively. Palmer’s paper is on the ArXiv in a revised form since 2009. The paper was published in yet another revised form in the Proceedings of the Royal Society and we strongly suggest that you read it. Some profound work preceding the work of Palmer is due to the group around the very competent mathematician and nonlinear dynamics connoisseur Prof. George Nicholas. Three from the same family published a very influential paper on the two slit experiment in Chaos, Solitons & Fractals many years ago. Again this paper was motivated by the work of El Naschie on the same subject using a simple analogy to the three point chaos game. This chaos game as well as the four point chaos game is well explained in the relevant literature referred to in the papers of G. Nicholas and M. El Naschie all published in Chaos, Solitons & Fractals. What has passed unnoticed was several papers written around the same time or slightly after in which Mohamed El Naschie uses the theory of Menger-Urysohn to explicitly show that the two slit experiment with quantum particles and the wave collapse could be explained completely rationally, you could say almost classically, using the Menger-Urysohn theory of dimensions. This theory can be worked out in infinite dimensions only when spacetime has an infinite dimensional topology. This is the case with the Hilbert cube or El Naschie’s four dimensional cube inside another four dimensional cube and so on ad infinity. The idea which started with the work of El Naschie was refined considerably in its mathematical formulation first in Italy by Prof. Gerardo Iovane from the University of Salerno, Dept. of Mathematics and subsequently in several papers by the remarkable Russian mathematician A.M. Mukhamedov, from Kazan State Technical University, Russia. Maybe some will recall that hyperbolic geometry was invented in the University of Kazan. In this connection we would like to draw attention to a recent paper by Prof. Mukhamedov entitled Towards a deterministic quantum chaos, published in Problems of Nonlinear Analysis in Eng. Systems, No. 2(32), Vol. 15, 2009. There are many papers about the two-slit experiment published in CS&F including those of Prof. S. Nada who used density manifolds, Prof Marek-Crnjac who used quotient manifolds. Both ideas go back to work by El Naschie and for proper understanding you need knowledge of geometrical measure theory and/or string theory. Some very simple straight forward papers were published earlier on by El Naschie like The two-slit experiment as the foundation of E-infinity of high energy physics, CS&F, 25, 2005, p. 509. Let me give you an excessively simplified version but having the same flavor.
The probability of finding a Cantor point in a one dimensional Cantor set is a topological probability equal to the golden mean. The probability of finding no Cantor point is the complimentary probability, namely 1 minus the golden mean. This is the golden mean squared as a pocket calculator will assure you. Based on this elementary observation you can reason that the probability for a Cantor point to be at a point number 1 or a point number 2 must be the sum of two probabilities and it turns out to be the sum but where the golden mean square gets a negative sign. So the sum is really the difference. The difference between golden mean and golden mean squared is equal to the golden mean to the power of 3. On the other hand the multiplication theorem of the theory of probability teaches us that being at point 1 and point 2 simultaneously is the multiplication of the golden mean of the golden mean squared. This is again equal to the golden mean to the power 3. This simple analysis shows us that based on probability theory we cannot distinguish between a point at 1 or 2 or a point at 1 and 2 simultaneously. This is the indisguishability condition. That is why you cannot say what is a particle and what is a wave unless you prepare the experiment accordingly. In quantum mechanics, as in Cantorian spacetime, a point can be in two different places at the same time. When you take measurements by imposing yourself on the empty set, the empty set is no longer empty and the wave collapses. It is a simple as that. Simple as it may be, this is a hefty dose for this communication and after you have had a look at the literature, I will go through all that with you once again in far more detail. However you must work with me. I can repeat all this to you a hundred times but unless you do it yourself, and although it is very simple, it will slip through your fingers and you will shake your head in bewilderment wondering what it is all about. It is very simple. However it seems nothing is as difficult as simple, unfamiliar things. On the other hand, familiarity breeds contempt and you become careless. We promise you to be neither. Best regards
E-infinity Communication No. 14
Defining the probability in Cantorian spacetime, negative dimensions and the transfinite theory of Menger-Urysohn
Defining the probability in a Cantorian setting initially seems to be a hopeless undertaking. Some may be worried that such a task could lead to the same end station of George Cantor’s mental institute or was it only a nerve clinic. After all Anthony Eden ended up in something similar after his Suez Canal adventure. Let me explain. Suppose we have n distinct objects. Suppose we have an experiment where choosing a single object of the n is truly random and cannot distinguish between the different objects. Combinatorics says that the probability of haphazardly picking a particular object is equal to 1 over n. Suppose the ensemble element n is infinitely large, the probability is thus zero and combinatoric probability fails miserably to help us in a Cantorian setting because we have not only infinitely man Cantor points, but actually uncountably many. This uncountable infinity brought some people to despair. Cantor’s diagonal methods were labeled nonsense by nobody less than Kroniker. He stood in the way of Cantor’s promotion to professorship in a respectable German University like Berlin or Munich. Poor George was left in a small provincial German university. Finally he was mocked and teased by his colleagues to the extent that he several times suffered from a breakdown of nerves. It may also have been the intense contemplation of the Cantor set and its meaning. When you have taken out all the middle thirds out except for the n points iteratively and continued this process ad infinitum, then you have no length left. The set is measure zero. How on earth could you have not only something left which has the cardinality of the continuum, but you have something with a respectable and considering the situation quite sizeable finite dimension. OK it is a Hausdorff dimension but dimension never the less equal to ln2 divided by ln3 which is 0.63…. Not bad for something which is not there. Friedrich Hegel in his dialectic coined an ugly Greek/Latin half breed word coincidentia oppositorium. He must have had the Cantor set in his mind. Remember a line is a dimension 1 and is a continuum. Compare this with the Cantor set which is not really there because it is measure zero. It has a dimension 0.63 and thus a majority holder in the company and as if this is not enough, it has the cardinality of the continuum itself. In more earthly words, a Cantor set has as many points as a continuous line. No wonder Shakespeare said ‘there are things in heaven and earth Horatio’. No wonder that until this day many atheoretical physicists would rather not know anything about Cantor sets. This all would convince you that Cantor sets have nothing to do with reality and consequently it should have no place in physics. I call now to the witness stand Prof. Friedrich Pfeiffer from the University of Munich. This professor is probably one of the world’s most prominent professors of mechanical engineering. He was not only the Head of the Department of a Centre of Excellence, namely the University of Munich, Germany but also he was the Editor in Chief of the German Journal Ingenieur Archv. If you published a paper in this journal then you reached the promise land in Germany. The professor’s specialty is chaotic research of mechanical systems applicable directly in the industry. Before joining university again, the famous professor worked for an even more famous car producer in Bavaria, BMW. Some of his contributions were to take unnecessary noise from the motor and the clutch and let a BMW car be as silent as a Rolls Royce. No wonder BMW bought Rolls Royce. In a manner of speaking the famous professor was taking the Cantor set causing chaotic noise out of the BMW car. Cantor set, esoteric or not for physicists, are as real as hell for engineers and many other professions. It is strange that of all people high energy scientists and quantum physics theoreticians who deal with things which nobody has ever seen or experienced firsthand should consider Cantor sets esoteric and resist its integration into quantum physics as a basis of new micro spacetime geometry. There are of course many exceptions which we mentioned earlier, David Finkelstein, Heinrich Saller, Parisi, Ord, Nottale and many others. However the typical theoretical physicist neither appreciates nor most of the time knows anything about Cantor sets. Now let us return to our subject properly. When the combinatorial method fails this is no surprise. We have the geometrical method. You know in the darts game you also have infinitely many points but then you define probability geometrically by the quotient of different areas. Great. However calamity struck. A Cantor set has no measure. So we have no length. We are dealing with an esoteric phenomena measured zero. The geometrical method and geometrical probability goes out of the window. Of course there are sophisticated methods based on measured theory. Mark Kac described probability theory as a measure theory with a soul. We would like to keep this soul and remain in probability theory as much as we can. When all things fail, you rub Aladdin’s wonder lamp and ask the genie to bring you in some topology. A line, thin as it may be has a dimension 1. The Cantor set also has a dimension and from a well known theorem by Mauldin and Williams a random Cantor set of the triadic type also has a dimension when you assume uniform probability which is the simplest assumption you could possibly make and this Hausdorff dimension is equal to the golden mean. Now we can define a topological probability so to speak. Such quotient would be made of the dimension of the Cantor set divided by the dimension of the line. This is the golden mean divided by 1 which is equal to the golden mean. At long last we have at least something with which we can do some computation. If the simplicity of the solution confuses you and if the question you pose to yourself confuses you even further, do not despair. The great Poincare himself managed to confuse himself in an exam for mathematics and failed. He took the examination once more, succeeded and then invented topology. When he became so famous he failed once more. This time he did not realize that he was wrong. He failed to recognize the work of Cantor. He considered Cantor’s geometry a gallery of monsters. He agreed to a certain extent with Kronecker that Cantor’s ideas are maladies which inflicted mathematics and he was sure that mathematics would soon recover from it. You would rightly say nowadays that you could take all this nonsense from anybody, including Kronecker the arch enemy of the non-finite but for God’s sake, not Poincare. However we cannot rewrite history. Poincare did not recognize his three body problem. The geometry of non-integrability is the geometry of chaos and the limit set of chaos and the backbone of any chaotic system is as James Yorke taught us, a Cantor set. God invented the integer. Everything else is the work of man. This is of course total nonsense, typical for Kronecker on this subject. We have now so many infinities and hierarchies of cardinalities beyond Kronecker’s and even Cantor’s imagination. They are sufficient to make Kronecker turn in his grave, infinitely many times. I hope you got the right taste for the thing to come next to resolve the two-slit experiment which is the basis for the Cantorian proposal for quantum spacetime. And yes, a point, whatever we mean with this word can exist at two different locations at the same time in the infinite dimensional topology of Cantorian spacetime and similar spaces. One of the nicest people one could ever meet anywhere at any time is an English/Canadian physicist whose name is Garnet Ord. Ord is the man who coined the word fractal spacetime. He corresponded and discussed many things with Richard Feynman. Einstein is great but Feynman is something else altogether. Ord intuitively knew about the power of fractals and Cantor sets but if you see Ord and El Naschie discussing things regarding Kronecker and Cantor you would think these two extremely close friends are in two totally opposed sides as far as finite and infinite are concerned. I will keep many anecdotes about El Naschie and Ord for the next communication but I advise you for the time being to familiarize yourself with the greatest mathematical genius of all time, Georg Cantor by reading the wonderful book of Joseph Warren Dauben entitled Georg Cantor, His Mathematics and Philosophy of the Infinite, Princeton University Press, 1979. All the best. (23.3.10).
E-infinity Communication No. 15
Set language and probability language dictionary of E-infinity as a two-slit experiment with quantum particles
A philosophically inclined cleric in England invented diagrams which are quite useful to use to move from one language to another in E-infinity theory. I mean set theoretical language and the language of probability theory or events. This was important for El Naschie in developing E-infinity and may be helpful for some in understanding this part which is crucial. The classical language of events or probability language speaks of probability space, events, impossible events, not d or the opposite of d, either/or or both, both, mutually exclusive and if then. In set theoretical language and in the same order you could say universal set, subset, empty set, compliment of d, the union operation, intersection operation, operation of intersection for a totally disjoined set and a set being a subset of another set. The set notations are different from one author to another but are well known. El Naschie studied mathematics in Germany under Kaluza. The anecdote connected to examining a rebellious though peaceful member of the extra-parliamentary opposition is laid down in his reminiscence of his student years written on the occasion of his 60th birthday, A tale of two Kleins unified in strings and E-infinity theory, CS&F, 26, 2005, p. 247. I hope referring to these things is not interpreted draconically as cult which is way over the top to say the least. Everyone has his own style of writing. Some like these anecdotes as a welcome distraction from the boredom of too much formality. It is immodest to recommend one’s own taste but it is hypocritical to recommend somebody else’s taste. If we must we would rather be something, we would rather be immodest than hypocritical, so if our kind friends would bear with us and consider that we are doing all that for fun and free of charge to the benefit of everybody else, then at least be so tolerant to leave everyone express himself in the way he likes. If you do not like something, do not read it. We are not offensive to anyone and mentioning the outstanding achievements of people like Richard Feynman, Nottale, Ord, Rössler and Tim Palmer should not be provocative to anyone. We would like that the young people have the right examples to follow at least if they are serious about science.
Now let us suppose you are on the unit interval of which a Cantor set was made. The dimension of the interval before digging so many holes in it is unity. This is the same for the Hausdorff dimension as well as the topological Menger-Urysohn. If you are a member of the Cantor set then the dimension attached to you would be zero for Menger-Urysohn and the golden mean for Hausdorff. If you are not then you are definitely in the empty set. The corresponding dimension would then be minus 1 and the golden mean to the power of square. These simple facts follow from Connes’ formulation of Penrose universe. This is just another formalism of the bijection formula with a slightly different mental picture. You can have whatever mental picture you want as long as this helps you to have this mystical feeling of understanding. Now you have two basic operations from the set theoretical point of view, union and intersection from which you get two dimensions for two elementary manifolds. The quotient manifold is given by a dimension which is the quotient of the dimension of the sub manifolds. That way you find the elementary fact that the dimension of a manifold which can sustain the preceding set theoretical conditions is nothing else but the infinite dimensional Hilbert cube which is not identical but very similar in many aspects to the space studied by Ji-Huan He. It is the same space which you obtain from putting a four dimensional cube into another four dimensional cube and so on ad infinitum. Again I am probably going to fast now but everything you know from string theory and high energy physics can be obtained as a deformation of this infinite dimensional Hilbert cube which has a Hausdorff dimension four plus the golden mean to the power of three, a topological expectation value of exactly 4 and a formal dimension of infinity. It is infinity in a hierarchal sense. It is infinity because you are taking infinitely many concentric four dimensional cubes to reach this result. Do not forget, Ruelle’s theorem. A classical system is necessarily chaotic when the dimensionality is infinity, even when it is hierarchal. In a sense you are holding infinity in the palm of your hand as in the famous poem of William Blake. For a geometrical visualization and easy access to the connection to string theory, I strongly recommend that you carefully study the excellent paper entitled Twenty-six dimensional polytope and high energy spacetime physics by Ji-Huan He, Lan Xu, Li-Na Zhang and Xu-Hong Wu, CS&F, 33, 1, 2007, p.5-13. In addition we cannot stress enough the importance of reading the work by the exceptionally gifted young Italian professor of applied mathematics, Gerardo Iovane. One of Gerardo’s computer graphics representations of fuzzy K3 Kähler manifold of E-infinity was entitled ‘E-infinity Cantorian Universe: A fractal manifestation of love’. Many of Gerardo’s papers can be obtained free of charge either directly from Elsevier’s science direct who do not always charge for every paper when you come to it through Google Scholar or can be found on certain pirate blogs claiming to belong to E-infinity members which is not always true.
Mohamed El Naschie is a walking encyclopedia when it comes to certain anecdotes of famous people with whom he had the privilege of talking. Again I hope this remark is not taken as cult. He said quoting Sir Prof. Herman Bondi that a fool can ask more questions than a wise man can answer’. However he hastened to say following Weizsaker who was again quoting Heisenberg ‘one has to learn, sometimes the hard way, that asking the right question is normally half of the answer’. We are frequently perplexed because the questions we are asking are imprecise or meaningless or undecidable. Undecidability is not the work of the devil. In fact undecidability in the sense of Gödel implies chaos and chaos implies fractals and fractals imply E-infinity theory. Very frequently when one does not understand something, one should not try too hard. There are very frequently mental blockages caused by our very selves. It is best to go and sleep and not to try too hard. During sleep the subconscious work and it is frequently according to surrealistic artists far more intelligent than consciousness. That is at least the theory of the sleep walkers advocated by Koestler. Accordingly I am now going to sleep. It is one o’clock after midnight local time. (24.3.10)
E-infinity communication No. 16
Summing over paths, dimensions, exceptional Lie groups and knots in E-infinity and tidying some loose ends from previous communications. Part I
I am taking over from my colleague and will start by apologizing for various gaps in the presentation. The number of states which are of interest are 496 for superstrings, 528 for the 5-Brane model, 548 for summing over 8 exceptional Ei Lie groups, 685 for summing over 17 two and three Stein spaces and the same sum for 12 exceptional Lie groups some of which are not from the Ei family. Finally and most importantly 8872 for 219 three dimensional crystalographic group. The knowledgeable reader will remember that the 17 two dimensional crystalographic group, that is to say the 17 Alhambra Andalusian tiling corresponds to 230 three dimensional group. This is an error and a common one. The 17 correspond to 230 minus 11 making 219. These are the 3D crystalographic group which truly corresponds to the 17 two dimensional crystalographic group. This is all explained in details in the literature and the papers mentioned in the previous communication. The connection between Heterotic string theory and the 219 crystalographic group is truly remarkable. As far as I know it is Mohamed El Naschie who drew attention to this fundamental fact for the first time. Before I finish remind me to tell you about two missed opportunities. The first is connected to Nobel laureate Gerard ‘tHooft and the second is connected to Nobel laureate Steve Weinberg. It is not only an anecdote I will quote paragraph and verse hoping this will at least make you trust E-infinity a little because unless a certain amount of trust is assumed at the beginning, you cannot made headway easily.
Let me explain the last two lines in come details. When you explain Einstein’s special theory of relativity, what do you say? You simply say that you no longer think of our space as being 3 plus 1 space and time but as a fused four dimensional spacetime. You would also say that the simultaneousity is not possible because every point has four different coordinates in this spacetime. Finally you add that the speed of light cannot be exceeded. When you want to be more sophisticated you say that these conditions are not independent of each other as is obvious from the Lawrence transformation which preceded Einstein’s work. To give the impression of historical sophistication one would probably add that Poincare knew all of that long before Einstein and that in his work he spelt out indirectly that E is equal to m multiplied with C squared, where C is the speed of light. The general relativity is much easier to explain because you only say that in the very, very large spacetime is curved. This geometrical curvature is what we perceive as gravity. If you want to be sophisticated you just add that since the speed of light is the maximum speed in spacetime, then gravity’s effect cannot travel between two gravitating masses instantly and must travel at a maximum with the speed of light. We do not notice the curvature because it is noticeable only on very large scales. When you say all that everybody is happy. Why are things different in E-infinity? Why, despite so many explanations do people want to understand in a simple way what this E-infinity is? Here is a simple way. While classical spacetime is smooth and Euclidean at intermediate distance and curved at very large distance it is not smooth and not continuous at very short distance. That is all folks. To model the large scale geometry we use Riemannian geometry. The curvature tensor is the driving force behind Einstein’s equation. Similarly to model quantum spacetime at these very short distances we use fractal geometry in its simplest form. The simplest form of a fractal is a Cantor set. We take infinite numbers of Cantor sets to do the job. How on earth could anyone draw a conclusion from that that Cantorian spacetime is infinite. No it can be finite and have infinite dimensions. Even from elementary school mathematics we know that we can sum an infinite series and find a finite answer. That is the whole point behind fractals. When you hold a piece of fractal in your hand, you are de facto holding infinity in your hand. E-infinity theory is very similar to the no boundary proposal of Hawking. It is all extremely simple. Sometimes I am reminded with the heavy weight lifters who are used to carry this 300 kilos and go to lift something which is very light although it does not look it and in doing so, he hurts himself because he was not expecting it to be so light. It is similar to Sonny Liston when he directed a blow to Muhammad Ali’s head but Ali shuffled sideways and Sonny tore the muscle in his right arm. People are expecting E-infinity to be difficult. That is why they find it difficult. Just relax and have faith. It is far simpler than you think. I will not answer any question unless the concerned person gives me his word of honor that he has read at least one single review paper written by an expert on E-infinity theory. In the next communication I will go into slightly more detail and talk about the missed opportunities of ‘tHooft and Weinberg, two people for whom we have the highest possible regard a scientist can have for another scientist as far as science is concerned.
E-Infinity Communication no. 17
Some lost opportunities
What we have to say in this communication might seem at the first instance to be a diversion. Believe me it is not and I just ask you to bear with me a little. I will give two examples where E-Infinity could have been of a great help but it was not utilized and not even considered. We are all no matter how liberal and open minded we think of ourselves subject to prejudice which is deep rooted. Try as much as we want, we cannot escape from two things: The noise of our upbringing and the noise in our brain. Nobody is saved from this “condition humane, to use a term of John Paul Sartre. The two examples we will give are related to the work of two towering figures of the 20th century theoretical physics who are still with us and are still working and contributing vigorously to the literature. The two are Nobel Laureates Steven Weinberg and Nobel Laureate Gerard ‘tHooft. Let me start with a much simpler example of Weinberg. I don’t think there is a single person who has anything to do with theoretical physics who wouldn’t know the great man Steven Weinberg who has written the Definitive Treatise on Quantum Field Theory in three volumes published by Cambridge Press. In addition Steven is a great intellectual personality and his influence goes far beyond physics. For instance in his book facing up he presented in a courageous and logical way the point view of Zionism. This is of course a ticklish issue particularly nowadays and with regards to the Middle East and the rise of the Muslim religion in political form. But Weinberg’s Zionism has an undeniable human and logical face. He is right to warn from the rise of any religious discrimination. He is also right to warn from repetition of the holocaust in any form or guise. Some years ago I was told that the great man was invited to a Conference dedicated to transfinite physics. Weinberg did not hear this expression before. He declined the invitation politely being a responsible and courteous person. What a pity that he wasn’t there. Gaining such mega brain to transfinite physics would have completed the revolution which started with Richard Feynman and continued with the work of Garnett Ord, Laurent Nottale , Mohamed El Naschie, Goldfain and dozens of other scientists including Sidharth, Svotzel, Otto Rossler and yes Renate Loll, Jan Ambjorn and of course Fay Dawker in England who followed a slightly different line initiated by David Finkelstein in America and Heinrich Saller in Germany. Let me be now specific. Consider volume 3 of Steven Weinberg’s book on quantum field theory. Volume 3 is dedicated to super symmetry and the book was published by Cambridge in the year 2000. On page 188 to 192 of the book, Weinberg considers super symmetric unification of the strong and electro weak. He calculates the inv unification coupling constant and finds it by virtue of equation (28.219) to be 1 divided by 17.5. In other words the inverse coupling constant of unification is simply 17.5. Now to us working in E-Infinity we recognize immediately this result as wrong. We do not need to make many calculations to realize that somewhere a misfortune and probably trivial computational error was introduced to Weinberg’s analysis. Let me explain why: First, if you are dealing with super symmetric unification, then you are implying gravitational force whether it is explicit or not. Consequently, Weinberg’s analysis is not simply a grand unification but complete unification of all fundamental forces namely electro magnetism, weak force, strong force and gravity. Now we know for sure that the exact value approximated to the nearest integer of the inverse coupling in such case must be 26. The 17 and half is too far for 26 to be even remotely correct. For instance one of the past students of Weinberg and a leading string theoretician and a colleague of Weinberg in the meantime is Joseph Polchinski. In volume 2 of his book String Theory, he calculates the same value and comes to the result that the inverse unification coupling constant must be something near to 25. In fact on page 347 of Polchinksi’s said book, two values are given for the unification by virtue of equation (18.312). The first value is the unification energy of about 10 to the power of 16 Gev. The second value is the inverse unification of 25. Of course Polchinski says it is for grand unification. However and as we reasoned earlier since super symmetry is involved, it is for complete unification including gravity. Now how did we notice so quickly. The reason is that we have at our disposal, a set of very simple principles and even simpler sets of equations which leads us directly to the correct result. In detail we should say the following: We know the exact theoretical value of the four fundamental couplings of the electro weak theory which we need to reconstruct the exact theoretical value of the inverse electromagnetic fine structure constant namely 137.082039325. The value needed for that is Alpha bar 1 equal to 60, alpha bar 2 equal 30 and alpha bar 3 equal 9. In addition, we have the Planck coupling 1. Using the well known reconstruction of E-Infinity we obtain 137.082039325. As anticipated by E-Infinity theory the exact theoretical value 60, 30, and 9 are extremely close to the nearest integer approximation to the experimental value found in the literature. It is a very elementary business to find by averaging a value for the unification inverse constant. You simply take the geometrical mean of the 3 said values. In other words you take the third root of the multiplication of 60, 30 and 9 + 10. You could of course take 9 instead of 10. If you take the 9 you get 25.3 as an approximation. If you take the 10 you get 26.2 as an approximation. An E-Infinity exact analysis will give you of course the exact transfinite value namely 26.18033989. This is nothing but the inverse golden mean to the power of 2 and taken 10 times. It is interesting to note that the value of approximately 17 refers only to partial unification. This partial unification is easily obtained by averaging. You just take the square root of 30 multiplied by 10 and get 17.3. This is unification of strong force with electro weak alone without considering electromagnetism. You see how easily we can do calculations because the golden mean binary system makes cumbersome computations elementary. But this is also not all. We have conceptual simplicity. We know the building blocks of spacetime. They are the random Cantor set of the golden mean Hausdorff dimension. The great Dutch scientist Nobel Laureate Gerard ‘tHooft made the search for the building blocks of Nature and the title of his beautiful popular book: In search of the ultimate building blocks published by Cambridge University Press 1997. That brings us to the next example of a missed opportunity. In his book on page 2 in figure 1, Gerardus ‘tHooft plays with a wonderful idea namely making smaller and smaller kites from a sheet of paper. Miraculously and as if an invisible hand is moving ‘tHooft’s hand, he designed in figure 1 a logarithmic spiral connected to the golden mean without saying so. For practical considerations, which are totally justified, ‘tHooft stopped before making the ultimate theoretical conclusion that he is reaching an element of a wild topology with a Cantor set ramification which harbors the golden mean. Then on page 174 ‘tHooft finds at long last a unification theory which he could praise since he does not like string theory. This theory not surprisingly is Loop Quantum Mechanics of Lee Smolin and Rovelli. ‘tHooft writes “In this theory the only thing relevant is the number and kind of knots linking the loops………..Accidentally Knot Theory is one of the most difficult branches of modern mathematics”. In E-Infinity we beg to differ profoundly. It is a prejudice to think knot theory is difficult. Knot theory is simple. You can do experimental with the rope and nothing more. Most of the knots are in 3D and they become unknots in 4. There is of course more complex knots in 4 which become unknot in 5. Mohamed El Nastier used Knot Theory skillfully to point a connection between knots, Feigenbaum scenario and ramification at a Cantor set. In one of his very readable papers titled: Fuzzy muti-instanton knots in the fabric of space-time and Dirac’s vacuum fluctuation, El Naschie discussed ‘tHooft’s work in detail and points out how he can obtain ‘tHooft instanton as a volume of a symmetry group. This is a trivial consequence of E-Infinity theory. You see the special orthogonal group SO3 has a volume equal to 8 Pie square. Mohamed El Naschie gave the exact value in transfinite form as well as many other interesting points. You see this way, the instanton becomes more physical as a knot which has volume and becomes nearer to a particle like state or a collection of 16 particle like states. The schism between string theory calculation of the 8064 and the holographic calculation of the same disappear. In a certain way, it was ‘tHooft who pointed out to Mohamed El Naschie that he has a new theory. It was however a lost opportunity to combine forces and pull in the same direction rather than sit on the fence and have to bear what Shakespeare called: the slings and arrows of outrageous fortune. However at E-Infinity we are ready to follow Shakespeare as well. We will take arms against a sea of trouble and by opposing we will end them. And it is nobler in the heart to suffer.
E-Infinity
E-Infinity 18
Part 2 of Communication No. 17
Not only Confucius is advising restraint when faced with the awesome power of irrational hatred or the poisonous device of twisting facts. Aristotle finds no way to face the artificial wit of those who have nothing in their heads apart of comic strips and slapsticks except to fortify yourself in continuing serious discussion. That is what we will do here.
There is nothing called weird topology. Of course you can call it what you want but there is no such expression in use. The correct expression is wild topology. The only person who could be very upset about that in a professional way that is must be John Baez. In his n-Category café he made a meal out of wild topology only to find that he is of course wrong. That is what happened to you when you spend too much in cracking jokes, writing silly articles with ha-ha-ha instead of reading seriously to expand your horizon. So many people calling themselves anonymous spend unreasonable amount of time on worthless blogs achieving nothing except maybe getting rid of their internal frustrations with themselves. John Baez of Riverside University proclaimed loudly that there is no 8 exceptional Lie group. Of course he was wrong and his victim was right. We have E8 with 248 generators. Then we have E7 with 133 and then we have E6 with 78. That is not where it stops. Because most people know F4 with 52 and G2 with 14. But these last two are not E line. The correct E5 is somewhat surprisingly SO(10) with 45 generators. Then we have E4 and this is a counterpart of SO(10) namely, SU5 with 24 generators and you can go on that way until you exhaust the E Line. The sum was found by El Naschie to be 548 to the nearest integer. Next blunder of John Baez was regarding 2 and 3 Stein spaces. He never heard of them. What a blog master does not know about does not exist by definition. It is replaced systematically by silly jokes and ha-ha-ha. That is what Charlie Chaplin would call modern times, or theoretical physics a la blogs with café au lait. We could go on indefinitely like that but this will violate the rules laid down by Prof. Mohamed El Naschie about refraining from personal remarks and jokes that have nothing to do with science. Wild topology is a very important part of general topology. It is connected also to knot theory. The Russian literature is abounding with such examples. In particular, a Great Russian mathematician living in France made this connection and that is where Mohamed El Naschie learned his stuff about the connection between knot theory and cantor sets. He added a trick to that which he learned from Nicholas Hoff. Do not ask me who is Nicholas Hoff? However I know from Mohamed El Naschie that he is one of his teachers. He used to be the Head of Aeronautical and Astro-nautical Department in USA. When faced with big nonlinear problems, Hoff chopped the thing into two parts. He ignored the nonlinear terms and solved the linear part. That is mathematically acceptable. But then he did something unconventional based on intuition. He chops the linear part and somehow regards only the nonlinear end state. That is a bit unusual to say the least. It is unusual for pure mathematicians although I am not one. Hoff did not carry the ballast and regarded only in state. When he reached the top with a ladder, he kicked the ladder and onlookers wondered how he reached the top. Mohamed El Naschie did not follow the intricate knot doubling of an entanglement. He took only the end state. He took the limit set. This limit set is the Cantor set. Something similar was done though not quantitatively by Tim Palmer. Before them something similar was done by Michael Berry. People working in nonlinear dynamics have an engineering sense. This is a world apart from the algebraic computational approach of a great man like Nobel Laureate Gerard ‘tHooft. There is no doubt of the admiration of Mohamed El Naschie to Gerard ‘tHooft but the latter would be the first to acknowledge that a Nobel Prize is not a passé partout. It is a great pity that this great man hasn’t fully taken in the fact that a random cantor set has a golden mean Hausdorff dimension and that a random cantor set is the end state and by duality it is the very very beginning. As such you are started with a golden mean binary system. You have now a chance to solve things with unheard of simplicity similar to what John Wheeler has always proclaimed. At such level it is absolutely misguided to take the theories of Newton or even Einstein as a guiding light. It is totally wrong at this level to differentiate between physics and maths. It is completely naïve to think that there is really a distinction between reality as we think we know it and ir-reality whatever we mean by this word in our so called real world where our labs exist and measurements are taken. There are those religious fundamentalists. We regard them as fatal and misguided apart from being non scientific. But there are equally inclined fundamentalists who think that everything is only measurable in the laboratory. These things must be expressed in lengthy complicated equations. We call them scientific fundamentalists. They are just as misguided. Dawker continuous onslaught using political means on religion and anything resembling it is but one symptom of this narrow minded fundamentalism. One question these fundamentalists never ask themselves: why are they so fanatic about denying anything they don’t see including God, whatever this is? A fast more rational way is to analyze the brains of these people as well as the brains of the oppositely inclined people to find what all the fuss is about. A look into the mirror, a chat with a beautiful woman, or reminiscing about past time may help fundamentalists to know the real reason for why they are insisting on whatever they are insisting on particularly when we could not know the answer. E-Infinity starts where George Cantor started. It will be surprising that those who believe in chairs and tables as chairs and tables and we jolly good will build some could swallow E-Infinity immediately. But I think I may be wrong here. If you really understand fractals, and if you really did not just learn it by heart from a text book because others have accepted it, then you will understand E-Infinity. The wonder of E-Infinity is the wonder of fractals and the empty set. Gerard ‘tHooft once said: “I understand what negative dimension is, it is a Grassmanian variable”. That is fine but it is not fundamental. The fundamental thing is to say a negative dimension is the dimension of the empty set. In fact the empty sets. All empty sets are fractal like. The totally empty set is the incarnation of nothingness. For traditional physicists with or without the highest decorations, this is a difficult part to swallow. Tim Palmer understood part of the dilemma. He put it in the most eloquent way we know of. Far more eloquent than anything which Ord, Nottale or Mohamed have written. It is eloquent because it is short, sweet and simple. He said quantum mechanics is blind to fractals. This is the main problem. That is why wild topology is important. At this level philosophy is not decoration. At this level philosophy is part and parcel of the physical shebang.
What I wanted to clarify in this communication was initially the question of unification, Weyl scaling and the rest. Inevitably diversion took place. We have to stop and start again in another communication. I will call it if I am the one to write it: an equation searching for a Lagrangian. Or was it six characters searching for an author. If Barandello is correct then they have to be six equations searching again for a Lagrangian. For the time being, I bid you goodbye for I have to take the plane or was it the train to San Fernando. A last minute note, you need some literature for wild topology. I cannot recommend strongly enough the classical book of John Hocking and Gail Young “Topology.” It is republished in Dover, copyright 1961. Another Russian book on set theory is by N.J. Wilenkin. It is called Set theory for Entertainment. Finally there is a marvelous book by Christian Beck, an English Professor of Physics at Queen Mary College called: Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields. It is published in World Scientific, copyright 2002. Beck reproduced everything that Mohamed El Naschie has found but using a computer. The work is recommended to anyone who is deluded to think that El Naschie’s work is numerology. Beck calculated the possibility that this is numerology and found that the possibility is 10 to the power of minus 38.
Goodbye for now but not for long.
Communication No. 19
Miscellaneous comments, some corrections and continuation of Part 3 of Communication No. 17
Let me start with a minor correction. We said that Christian Beck in his famous book linking non linear dynamics to high energy physics: “Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields” has calculated the probability of an element of numerology and found it to be absurdly small. We quoted a number. The correct number is even far absurdly small than our memory had it at the time. The number quoted on page 247 by Beck is ten to the power of minus 60. Let me repeat. This is 1 divided by ten and another 59 zeros beside it. You have to be incorrigible fanatic to talk of any numerology in the work of Beck who worked directly with a computer as well as the work of El Naschie who knowingly or unknowingly was working with the golden mean binary system which can match any computer.
We all know that super string depends crucially on a number well known from number theory namely 496. Is this numerology? Not even the greatest enemy of string theory could make such claim. We know that loop quantum mechanic would not work at all unless we multiply everything with a mysterious numerical factor, the Imerze-barbiro number. I probably spelt the name wrongly but never mind is loop quantum mechanic numerology? I don’t think our Nobel Laureate Gerard ‘tHooft would like that at all, he believes reasonably well in loop quantum mechanic because it does not have extra dimensions. Why? Because extra dimensions cannot be seen. Is this prejudiced? I don’t think he believes it is prejudiced. Forgive me for being confused at this point. Did anybody see an instanton? Did anybody see the unification monopole of Polyakov and ‘tHooft? Don’t get me wrong, I believe they will be found or something similar will be found. However it is clear that beauty is in the eye of the beholder. Funding is a problem. Some say “Cherchez la femme.” In our community it is more the case that we should “cherchez la monnaie”. Continuing in the same vein, what about this quadratic equation which Mohamed El Naschie has given? The equation was expressed in terms of alpha bar. It reads alpha bar square minus 140 alpha bar plus 400 equal zero. It has of course two solutions: the first is alpha bar is 137 plus k0. This is 137.082039325. Note it is the theoretical value of E-Infinity. It is not the experimental value. The second value is 3 minus k0. This is a scaling of one of the dimensions of spacetime. Now take the mass of the proton. According to E-Infinity it has to be alpha bar square divided by 20. That way you get the real experimental value of the proton. Take the experimental value of alpha bar and do the same. You will not get the correct value for the mass of the proton. You will get a good approximation but not the correct value. To understand that, you have to read El Naschie’s paper on the Mass Spectrum. However before this you have to depart from the naïve belief of reality physics and maths. You cannot take Newton nor Einstein as a paradigm in a naïve way. I may come to this point later on when we have the time. For the moment I would like to give a second example. In fact I would like to give many examples to refute the naïve belief of dividing physics from maths in a simplistic way. I said previously that the real negative dimension is the empty set. The best example I know is El Naschie SU(n) hierarchy. I will not give now details but I give it to you as given in the papers of El Naschie. It is minus 1, zero, 3, 8, 63 and then finally 3968. This is not ‘tHooft’s beloved Yang-Mill theory. It is the super Yang-Mill theory. If in any doubt, please see Kaku’s book:”Introduction to superstring and M theory, page 385. El Naschie’s hierarchy just stated is connected to n equal zero 1238 and 63 in the same order. I will explain that later on in detail. I just want to say that those who think this is numerology know nothing about numerology or physics and Beck calculation is a numerical confirmation for that. In fact the work of Renate Loll and Jon Ambjorn lies squarely on the side of El Naschie and Beck. They use with great skill the number crunching efficiency of a computer. El Naschie on the other hand does the same using a number system made extra for that. The Great Russian academician Alexey Stakhov said in his recent book which was acclaimed on the level of a Nobel Prize that missing the use of the golden mean binary system was a disaster and nothing less for the development of mathematical physics. I have made many claims here and I did not give you details. I owe you a few. I will do this as soon as we can piece things together for you. I also promise to show you that the theory presented for quarks confinement using E-Infinity is absolutely correct. We had our doubts initially but now there is no doubt whatsoever. That also will be addressed very soon.
Communication No. 20
The Golden Mean in High Energy Physics before, during and after E-Infinity
We will have to leave it to the philosopher and historian of science to determine the complex history of the golden mean in high energy physics. As far as we are concerned, we feel that Mohamed El Naschie must be accredited with integrating the golden mean in high energy physics in a systematic way and on a grand scale. He did not do that intentionally. It just happened. The golden mean more or less manifested in the computation as fundamental for any minimal consistent and accurate quantum field theory formulation outside the rules of classical quantum field theory. Without any attempt to be historically correct we must draw attention to very important papers where the golden mean manifested itself. I must say that the authors which I am about to mention were initially not traditional mainstream. They are not renegades. They are somewhere in between. They are meantime part of the establishment but it was not always like that. The first is an exceptional
Russian mathematician who worked initially in turbulence, A. Polyakov. The second is a superb solid state physicist, mathematician and hobby engineer, Subir Sachdev. If my memory serves me right, although this is slightly on the gossip side, I think Sachdev’s American wife is the daughter or the grandchild of Dwight Eisenhower, the great President of USA and the hero of D-Day in the Second World War. The paper of Polyakov is entitled: Feigenbaum universality in string theory, published in Journal of Theoretical Physics (JETP), vol. 77, No 6/March 2003, pp. 260-365. Polyakov found the period doubling of Feigenbaum in quantum field theory. Please read Mohamed El Naschie’s paper on the connection between the hyperbolic region of period doubling and the Hausdorff dimension of fractal spacetime. A critical value in the hyperbolic region is his famous 4.23606799. When you talk Feigenbaum, you talk golden mean renormalization groups. In fact it was
Mitchell Feigenbaum, Otto Rossler, Julio Casati, Boris Cherekov and Itmar Proccaccia who initiated Mohamed El Naschie’s interest in nonlinear dynamics, KAM theorem, period doubling and thus the golden mean threshold. Mohamed El Naschie merely extended that to high energy physics. The second paper by Sachdev was published in Physics Letters B 309, 285(1993), Polylogarithm identities in a conformal field theory in three dimensions. You can find it free of charge published in arXiv: hep.th/93605131, 25 May 1993. An extremely instructive and neat summary of the application of the golden mean is a nice paper by the very versatile, Slovenian mathematician L. Marek-Crnjac. The paper is titled: The golden mean in the topology of four-manifolds, in conformal field theory, in the mathematical probability theory and in Cantorian space-time, published in Chaos, Solitons & Fractals 28(2006) 1113-1118. A wonderful paper by Professor Christian Beck from Queen Mary
University, London and Muhammad Maher from the same department is: Chaotic quantization and the mass spectrum of fermions, published in Chaos, Solitons & Fractals, 37 (2008) 9-15. This paper was refereed and recommended for publication in Chaos, Solitons & Fractals by Professor, Dr. Dr. Werner Martienssen from the University of Frankfurt. In this paper you can see the influence of nonlinear dynamic and cantor sets in modern physics and determining the mass spectrum of elementary particles in a similar but not identical way to E-Infinity. It was not always golden mean from the beginning. El Naschie used initially deterministic fractals. He started initially by using the classical triadic cantor set with the Hausdorff dimension ln2 divided by ln3. You do not get golden mean for deterministic cantor sets. Paradoxically it is randomness which introduced golden mean harmony. You can see that from a paper published in Vistas in Astronomy. The author is
Mohamed El Naschie. The title of the paper is: Quantum Mechanics, Cantorian Space-time and the Heisenberg Uncertainty Principle. This paper dated 1993, vol. 37, pp. 249-252, did not include the golden mean yet. Mohamed El Naschie rediscovered the average Hausdorff dimension of a quantum path. This is equal to 2. It is the individual Hausdorff dimension of a quantum path which is equal to the golden mean. The interplay between 2 and the golden mean produced the approximate value for the Hausdorff dimension of the core of quantum cantorian spacetime which is approximately equal to the exact value. To be specific, it is 2 divided by ln of the inverse golden mean which gives us an approximation to the exact value 4.23606799. There is a nice paper summarizing the application of the golden mean by El Naschie titled: The Golden Mean in Quantum Geometry, Knot Theory and Related Topics, Chaos, Solitons & Fractals, Vol. 10 No. 8 page 1303-1307 (1999). Another
paper which seems to have strong influence on groups working in the Parameter Institute in Canada is El Naschie’s Quantum Groups and Hamiltonian Sets on a Nuclear Spacetime Cantorian Manifold, published in Chaos, Solitons and Fractals, vol. 10 no 7, pp. 1251-1256 (1999). The golden mean as such and its connection to E8 became fundamental in the work of Mohamed El Naschie after one of his students, Dr. Ahmed Mahrus from Newcastle, Department of Physics, UK, drew the attention of Mohamed El Naschie to the golden mean binary system. Academician and Nobel Prize nominee Alexei Stakhov expanded this system in a recent magnificent work published by World Scientific entitled: The Mathematics of Harmony. Mohamed El Naschie started his adventure with period doubling and renormalization relatively early. He was at the time Director of Projects and one of the main editors of a prestigious Middle Eastern Journal. Later on he published a paper on the subject
titled: Order, Chaos and Generalized Bifurcation. The paper is published in the Journal of Engineering Sciences, King Saud University, vol. 14, no.2 (1988), pp 437-444. I did hear some good news for those who do not have easy access to expensive scientific Journals. I am told that a charity organization did put all the scientific papers of Mohamed El Naschie in a free access blog. I do not know where or when this was done, but those who will search will find it. I hope this will facilitate serious study of E-Infinity. Of course those who prefer other activities will not be deterred from following their natural inclinations. We hope however that the majority will follow their scientific inclination. We hope also this little contribution is helpful and we will be shortly returning with more.
E-Infinity
Communication No. 21
Why does the golden mean pop up everywhere in mathematical physics?
A true scientist aiming at scientific truth could not be afflicted by a worse malady other than prejudice. Some notable scientists who have done occasionally excellent work elude themselves in confusing prejudice with scientific skepticism. I am far too skeptical to believe anything easily, you will hear them say. Give him anything new and he will answer immediately: I do not need to read it. I waste my time on big names only. I know before I read that this is no good any case. Too many great people have tried before and failed. Who the hell could be this guy from Rumania to teach me a Caltech man about the quantum or anything at this level? When it comes to the golden mean things could be dozen of times worse. Ignorance is invariably covered up pure arrogance and never ever forget the witty jokes, the hallmark of a hole head. In what follows we would like to give well known elementary evidence that it is absolutely natural for the golden mean to be
the foundation of quantum mechanics and high energy physics and much much more. Many of these evidences have been discussed at length by Mohamed El Naschie, Marek-Crnjac and their students. For convenience summary, we recommend a paper titled: A short history of fractal-Cantorian space-time by L. Marek-Crnjac, published in Chaos, Solitons & Fractals 41 (2009) 2697-2705.
1. The golden mean is the solution of a simple quadratic equation with appropriate sign. A quadratic equation is the simplest non linear equation we know of. Only a linear equation is simpler. Einstein said if everything would be linear nothing would affect nothing. Therefore, for things to affect things we need at least a quadratic equation to describe physics of a minimal complexity. The simplest vibrational set fulfilling Einstein’s requirement is a 2 degree of freedom linear oscillator. The characteristic equation for such set when normalizing all constants is a quadratic equation with a golden mean solution. This is discussed at length in a paper by Mohamed El Naschie titled: On a class of general theories for high energy physics, published in Chaos, Solitons & Fractals. 14 (2002) 649-668. The Slovenian mathematician Marek-Crnjac suggested that fusing infinitely many but hierarchical sets of this type leads to E-Infinity theory. The connection
to the basic concepts of string theory is evident. It might be interesting for some to note that structural engineers habitually replace complex structures by systems of springs and masses. In some sense this is a finite element realization of a structure. In the same sense and taking a bird’s eye view this is the connection to Regg Calculus. No wonder that Mohamed El Naschie used this method when you remember he is a Structure Engineer and a past student of John Argyres, the inventor of finite elements who used it skillfully in the modern fuselage structure of airplanes at the dawn of the aerospace age.
2. The golden mean is the most irrational number. Its continued fraction expansion involves only unity. Consequently it is the backbone of the KAM theorem. There is no stability in a Hamiltonian system without a rational number. Since the golden mean is the most irrational, it is the threshold of the most stable periodic orbit in a dynamic system. The marriage between KAM and quantum mechanics resulted in Rene Thom’s VAK conjecture which has been re-generalized by El Naschie and used in high energy physics.
3. A random cantor set possesses a Hausdorff dimension equal to the golden mean. Of course there are many cantor sets which are random and have a Hausdorff dimension close to or different from the golden mean. However the most simple and generic random cantor set has the golden mean as a Hausdorff dimension. A wild topology always ramifies at infinity into a set of wild point, equivalent to a random cantor set. Generically this is equal to the golden mean. If you regard the final state and forget about the mechanism leading to it, then you have golden cantor sets geometry at ultra high energy corresponding to the wild topology. The basic mathematical work in this direction was done by an American topologist Alexander. The most famous examples are Alexander Horned spheres and Antoine Collier. Mohamed El Naschie merely carried these ideas to high energy physics and was probably inspired by S. Kaufmann in the U.S.A.
4. The fundamental theory of 4-manifold depends crucially on the Fibonacci and thus the golden mean. This has been considered at length in the corresponding mathematical literature. Many references to this work are given in El Naschie’s papers and elsewhere, for instance Crnjac’s work.
We must stress that the excellent work of T. N. Palmer depends crucially on an understanding of number theory. In fact classical quantum mechanics depends on number theory. You must always remember the trivial fact that without complex numbers, there is no classical quantum mechanics. We think enough for today as my hands are getting heavier and we hope to be back as soon as possible.
Communication No. 22
The inverse problem of quantum field theory in E-Infinity theory and symplectic tiling
We mentioned few very good reasons why the golden mean should pop up at so many different places and so unexpectedly in quantum high energy physics. We reasoned that quadratic equation with golden mean roots is the simplest non trivial algebraic equation that there is. We mentioned the maximal irrationality of the golden mean and the role it plays in KAM theorem and the stability of dynamic systems. We alluded to wild topology and its connection to the simplest form of random cantor set which by a well known theorem due to American mathematician Mauldin and his student William will have a golden mean as a Hausdorff dimension. There are far more reasons than what we mentioned. The reasons are sometimes very subtle and none is so subtle and so important than the relation to Penrose Tiling. I remember vividly attending a lecture by Professor El Naschie in the Einstein Institute for Gravitation in the Max Planck Institute near Berlin, Germany. A very
imminent and famous German astrophysicist was present when El Naschie was talking about the Penrose Tiling and E-Infinity theory. The imminent German scientist became very agitated and said: “This is all a simple tiling, how could you scale things denying the existence of a natural scale and how could use that for high energy physics”? El Naschie was equally agitated but remained calm. Of course it was El Naschie’s mistake. He thought everyone has seen the wonderful example for non commutative geometry presented in the book of Alain Conne. El Naschie explained to the imminent astrophysicist that of course he should have said Penrose fractal tiling. El Naschie meant that every tile in Penrose Tiling could be tiled again using Penrose tiling and so on ad infinitum. The great German astrophysicist and he was definitely a great astrophysicist was not familiar with fractals. He belonged to a generation which worked decades before Andre Linde, the
Russian astrophysicist, moved to America and introduced fractals to the big bang. Talking to Mohamed El Naschie later on, the irony became even bigger. El Naschie learned much about fractal geometry in astrophysics and quantum mechanics from a remark in a paper published in the 60’s by the very same German astrophysicist. The remark concerned the famous paper of Carl Menger which he dedicated to Einstein at his birthday. Many of us and I am not an exception refer to papers without reading them attentively. Let us return to Penrose tiling. Without the golden mean, there is no Penrose tiling but why is it like that? El Naschie gave a naive example which I find very instructive. Being a structural civil engineer, his example comes yet again from engineering. Suppose you are building a wall. You are using bricks. Watch a master mason performing his job. He fits the bricks together. These are the integers in number theory. Now and then he takes a smaller
or a larger brick so that things fit a little bit better. These are the rationals. However no matter how clever our master bricklayer is, he will never get a smooth monolithic wall without resorting to mortar. The mortar between the bricks helps to produce a smooth monolithically connected wall. These are the irrationals of number theory. If you take the golden mean 0.6180339 you notice that it is almost equal to half i.e. 0.5 + 0.1180339. It is a rational plus an irrational tail which is easily expressed again in terms of the golden mean. In this case it is simply 1 + k all divided by 10. The k is the famous golden mean to the power of 3 multiplied by 1 minus the golden mean to the power of 3. We met this number frequently when we discussed transfinite corrections. It is this technique of writing things in the most convenient form using the simplest and most efficient binary system that there is which allows E-Infinity to produce exact results with
simplicity which is difficult to comprehend when we use the old mentality of algebraic manipulation brute force, patching and approximating at different stages until things become approximately correct but truly ugly and cumbersome to handle.
I am too young to have been together with Heisenberg, Paul Dirac and Neils Bohr at the Tate Gallery in London when they visited it. Mohamed El Naschie told me however the following story which he again heard it from Heisenberg directly. The story is too beautiful not to be true even if it is true and just ingeniously invented by El Naschie to impress us. Paul Dirac loved perfection. When he presented a draft paper to Neils Bohr and the latter made his usual correction, Dirac becomes extremely sad. One day the three mean were in London. Paul suggested taking the opportunity to visit the Tate Gallery. I knew the old location of this gallery because Mohamed El Naschie took me there when I visited him in London. They were standing all in front of a large picture either by Turner or Claude Monet. I cannot remember. Dirac stood silent for a while then he moved forward and pointed to a spot at the bottom of the painting and said in his calm voice:”This point
is wrong”. I do not think I can say anymore. That must be one of the best signs of Paul Dirac’s conviction that it must be beautiful or it is not true or correct. If you look to the classical form of the renormalized equation of unification of fundamental action in high energy physics, you realize that it is extremely involved, clumsy and cannot be described as being beautiful. However it is approximately true. To use the same language of Dirac, there is a point there which is wrong. Not so with the same equation which E-Infinity produces. The equation of E-Infinity is perfection per excellence in this case. We will discuss it in detail. But I wanted to introduce the idea first. I know from El Naschie that he traveled to a country in Northern Europe known for its beautiful tulips, in order to introduce his equation to a man whose opinion he values above every other mortal. To make a long story short, the great man told Mohamed El Naschie:”Yes,
it is amazingly simple but there are too many things before that. Your equation comes from where? It just comes out of the blue”. I cannot reproduce the sense of disappointment which Mohamed El Naschie felt at this point. I could almost hear what is going in his mind. Of course there are many things before that. I thought you know that I know that. In fact how a great man like you could think that I could not know that. The problem is Mohamed El Naschie knew what the great man knows but the great man did not know what Mohamed El Naschie knows mainly fractals, chaos and complexity theory. You could write any simple equation if you can. If you are clever you can always guess the answer and idealize it. Next you ask yourself: What do I need to have? Normally a Lagrangian in order to get this result. This is the well known inverse problem. El Naschie worked on many problems in the calculus of variations. He knew from his classical engineering work
the power of the inverse problem as a method. The inverse problem is always more difficult. For instance, relative to multiplication division is more difficult. Similarly we have many opposed mathematical procedures where the inverse is always more difficult. It is not impossible to find whatever you want when so many clever people have worked so hard to give you a theory which is almost but not completely perfect. Classical quantum field theory is such theory. It is almost but not completely perfect. Those who have studied it carefully will be led to a simple solution of the inverse problem as will be explained later. To be able to complete the job, you have to free yourself of all prejudice and have no hiccups about fractals, transfiniteness and golden means. The result of this effort will be evident from the following equation of E-Infinity theory. In general the equation reads as following: the inverse unification coupling is equal to the
following sum: alpha bar 3 plus alpha bar 4 plus the natural logarithm of the ratio of the unification mass or energy divided by a reference mass or energy. This logarithm is multiplied by the factor which has to do with super symmetry. In the classical form this equation is not much different but it is clumsy and filled with numerical factors which a beginner could not make heads or tails of. It takes a long time to familiarize oneself with it. The E-Infinity theory version is by contrast perfect. Let me mention first that the discovery of the Logarithmic scaling decay was a major step in high energy physics. I forgot who discovered it first and I forgot a lot of things about its history which I read once upon a time. I do not remember where I read it but I did. We all would be very grateful for readers of these comments and I mean readers who are only interested in science could draw our attention to this logarithmic law and give us its background
and history. That would be very nice indeed and would save us a lot of work to dig in old papers and textbooks. E-Infinity took this logarithmic law and changed it de facto to a golden mean scaling of E-Infinity hierarchy.
In the next communication we will show you these things step by step. We hope also that you will notice that the inverse problem becomes tractable and lead to a simple solution of quarks confinement using E-Infinity modification of the original classical solution. Hopefully without wanting in any sense to sound pompous and give our enemies fuel to burn more wood, let me say that if we in E-Infinity could see so far then it was because we stood on the shoulders of giants. These giants are: Gerard ‘tHooft, David Gross, David Politzer and Frank Wilczek, to mention only a few of the key players in this field.
Communication No 23
Unification and Confinement
Although the negative coupling constant of strong interaction related to confinement was discovered long ago by the young Gerard ‘tHooft in an unpublished work and was rediscovered several times later we hold different views about confinement. In E-Infinity theory we take the view proposed by El Naschie that an exact solution to confinement must be related to the structure of space time as well as the nature of the Planck scale. Similar to quantum gravity E-Infinity believe that the key to understanding confinement thoroughly is the Planck scale. We are of course aware that some regard experimental phenomena involving jets as an indirect observation of free quarks of a sort. However we find the picture a little bit involved using experimental and theoretical reasoning in a much interwoven way and therefore a clear cut mathematical derivation is not there. We are almost there. With almost I mean 95% there. However theoretical physicists are
perfectionists. They could not leave a single spot anywhere without worrying about. This attitude is classical and ‘tHooft himself is a classical example for this way of thinking. Having said that, we must stress that El Naschie is by no means the only one who thinks that confinement should be linked to Planck energy scale physics. A British physicist working in Hungary, Lawrence B. Crowell, holds very similar views but we will not go into there in this communication. Let us go back to our exact renormalization unification equation.
Let us probe things at the Planck scale. This is 10 to the power of 19 GeV. As an energy reference scale, we take this time not the Z0 scale but the electron scale. To be precise we take a Cuba pair. The mass is twice that of an electron which is roughly 10 to the power of – 3 GeV. The natural logarithmic term in this case is ln of the ratio of the above mentioned scale. In other words it is ln 10 to the power of 22. School mathematics teaches us that this is 22 multiplied by ln 10. Anyone looking at this number and is also familiar with E-Infinity transfinite corrections will guess immediately in a split of a second that 22 should be 22 + k and ln 10 should be 2 + 2k. Here k is our famous number 0.18033989. More accurately it is the golden mean to the power of 3 multiplied by 1 minus the golden mean to the power of 3. Our logarithmic term is therefore exactly equal to 52 plus 2k. Assuming super symmetry we have to divide by 2 and obtain 26 + k.
This is exactly 26.18033989. Our equation is now as follows. On the left hand side we have the inverse unification coupling. On the right hand side we have the inverse strong coupling alpha bar 3 added to alpha bar 4 added to the value of the logarithmic term divided by 2. On the other hand, we know very well that super symmetric unification takes place at the inverse coupling 26 + k. This term on the left hand side cancels the logarithmic term divided by 2 on the right hand side. Therefore we conclude that alpha bar 3 plus alpha bar 4 must be equal to 0. Now we know for sure that alpha bar 4 is equal unity. It follows therefore that the inverse strong coupling alpha bar 3 must be equal to minus 1. This is exactly the point which people searching for confinement missed again and again until young ‘tHooft found it and spoke about it in a discussion which took place in a conference in Marseilles, France many years ago. Our derivation is however exact.
El Naschie gave some eccentric elucidation using analogies from visitors in outer space. You can read this in his papers which are now available free of charge in a blog financed by a charity institute. Physicists love trivial eccentric analogies to illustrate the main points of what they are proposing. Einstein loved to ride over a photon and fly with a speed of light. Others have even more outrageous ways of illustrating their point. To make a fuss about these things shows only how shallow the person in question is. I am really putting it here very politely. After all we are assuming that all readers of this blog are scientists. The odd 2 or 3 do not really count. We will return to confinement and harmonization in the next communication.
Communication No. 24 (Part 1)
The Fundamental Equation of Unification of E-Infinity Theory as a harmonization of the corresponding renormalization group equation of classical high energy physics.
After considering many fundamental problems in non linear dynamics and its possible connection to quantum physics, a paper entitled: Complex Dynamic in a 4D Peano-Hilbert Space appeared in Il Nuovo Cimento in 1992. This was a famous journal where Einstein published some of his work but after a short period of stagnation the Journal was re-launched and renamed: The European Journal of Physics. The author of the paper is Mohamed El Naschie and he wrote it during his Cambridge time. It can be found in Vol. 107 B, N. 5 – Maggio 1992 pp.583-594. In this paper Mohamed El Naschie made an explicit connection though in general terms between quantum mechanics and fractals. Sub section 8, page 593 is titled: Quantum mechanics, Dirac’s vacuum and foam space-time. He was not accurate at that time by thinking that Wheeler’s foam space time is a proper fractal. Nevertheless the point was made. In fact he mentioned explicitly that lifting the Menger foam
to 4 dimensions leads to something very similar to our space time. He was clearly not yet aware of the fundamental role played by randomness and was still working with a deterministic cantor set. In a paper published at about the same time in the Journal of Franklin Institute, Mohamed El Naschie expanded the idea, discussed in more detail the Menger sponge and expressed his hope that fractal cantorian spacetime can easily resolve all the paradoxes associated with quantum mechanics. Mohamed El Naschie continued his effort until he determined the exact expectation value of the topological and Hausdorff dimension of quantum space time. He perfected the theory mathematically after discovering the relevance of Jones’ Invariant and quantum groups to his work. About 7 years after his the mentioned Nuovo Cimento paper, the mathematical basis and the connection to hyperbolic geometry and KAM theorem became crystal clear. Just as two examples for his work
in this period, I may quote the following two papers: 1. Jones’ Invariant, Cantorian Geometry and Quantum Space-time. 2. Quantum Groups and Hamiltonian Sets on a Nuclear Spacetime Cantorian Manifold. Both papers are published in the 1999 volume of Chaos, Solitons & Fractals. In other words latest by 1999 the following fundamental facts were proven by Mohamed El Naschie and elaborated upon extensively by J. Huan-He, Marek-Crnjac and Erwin Goldfain. First the building blocks of quantum spacetime are random cantor sets with a golden mean Hausdorff dimension. Second, the expectation value of the topological dimension of the space time core is exactly 4. On the other hand, the corresponding exact Hausdorff dimension is exactly 4 plus the golden mean to the power of 3. That is to say, it is 4.23606799.... In 2008 the paper by Jan Ambjorn, Jerry Jurkiewicz and Renate Loll appeared in Scientific American. The paper uses computer simulation extensively. Well
you could say it is not exactly computer simulation but computer computation of a Regg calculus approximation of a quantum gravity model. You could also view it as an improvement of Regg calculus using Ambjorn’s triangulation. Professor Jan Ambjorn from the Neils Bohr Institute in Copenhagen is a world authority on triangulation technique of this kind. Leaving details aside and to make a long story short, the main thrust and results of the paper are the following: Quantum space time is best modeled using cantor sets as building blocks. Second, assuming Prigogine’s arrow of time on the quantum level, things work out perfectly. Third and most importantly, a space time dimension of 4.02 was worked out from first principles for the first time. To see how much emphasis the authors put on this fact, let us quote verbatim what they wrote on page 29 of their July 2008 Scientific American Paper. They wrote “Imagine our elation when the number of
dimensions came out as 4 (more precisely as 4.02 plus minus 0.1.)It was the first time anyone had ever derived the observed number of dimensions from first principles. “ We in E-Infinity theory group can imagine their elation but we ask you at the same time to imagine our alienation when no reference whatsoever was made in this paper or any of similar papers published at the same time in Physics Review Letter and the Journal of the Institute of Physics to our work. The first derivation of the dimensionality of quantum space time from first principle was made as you can check yourselves at least 9 years earlier. In fact the value found by Ambjorn et al namely 4.02 was found to be a spectral dimension and obtainable using other methods. One of these methods is the Bose Einstein statistics of El Naschie was found approximately 16 years earlier. When you see the structure of the paper in Scientific American you see that the logic expressed in the
fractal figures on page 28 and 29 including the Menger Sponge and the Serpinski Gasket follow the same logic of El Naschie’s paper in Il Nuovo Cimento and the Journal of the Franklin Institute. Rediscovering things again and again was not uncommon in the past. We mean no disrespect to anybody when we point out these facts. What is important is how we react or others reacted to these facts. Something positive came out of this any case. Cantor sets are in quantum mechanics to stay. Random cantor sets with golden mean Hausdorff dimension are meantime an experimental fact. No amount of propaganda could possibly change these facts. Should we have unintentionally compromised anyone then we as a group apologize collectively as long as the issue of priority is restored. For science priority is unimportant. For scientists it is important. The Yang-‘tHooft know that better than anyone else. He wrote a great deal about something similar which happened to
him in connection with the strong interaction. This is the subject which we will discuss shortly. Before doing that however let us recall something extremely important in various respects which involves Nobel Laureate Gerard ‘tHooft. In an excellent book on quantum gravity, edited by Daniel Oriti published in Cambridge this year but dated 2009, Gerard ‘tHooft makes the following answers to questions on page 155. The question was by L. Crane about non integer Hausdorff dimension in quantum gravity. ‘tHooft answered as follows: “We thought of such possibility. As far as the real world is concerned.............................................Weltman once thought there might be real physics in non integer dimensions but he never got anywhere with that............................................ I do know what negative dimensions mean................................it is anti-commuting coordinate.” Those in the know must be exhilarated to see how
far ahead of anybody E-Infinity group was. With all the due respect and it is a genuine respect we have for Gerard ‘tHooft, his statement could not remain unchallenged. Mohamed El Naschie derives the exact value of the Hausdorff dimension of quantum spacetime namely 4.23606799 from ‘tHooft’s dimensional regularization and he gets real physics out of it. We use the word real physics with large doses of salt. Talking about real physics in a simple way in the realm of quantum gravity can be the most misleading thing which one can do. We do not intend to dwell on the illusive nature of reality. However those who are philosophically inclined to use words like real physics should know about dozens of incidents where inclination to reality made reality disappear. The paper in question was entitled: On ‘tHooft dimensional regularization in E-Infinity space, Chaos, Solitons & Fractals 12 (2001) pp 851-858. This paper was preceded by another paper
which was entitled: “ ‘Thooft dimensional regularization implies Cantorian Space time.” The paper was presented at a conference which the great man ‘tHooft himself attended. We sincerely hope that this closes the subject which Oh! So many lesser mortals try to keep artificially alive again and again on certain blogs. When you multiply 4.23606799 by ten you get 42.3606.........and this is the non super symmetric inverse grand unification coupling. Let us discuss how this value as well as the super symmetric value namely 26.18..........could be obtained from the fundamental equation of unification. This is what we will do in part 2 of this communication.
Communication No 25 Part 2
In Part 1 of this Communication we discussed the fact that the basic Hausdorff dimension of E-Infinity follows from ‘tHooft’s dimensional regularization and we hinted at several other things, for instance negative dimension as well as the connection between E-Infinity and certain types of Regg triangulations. We will return to negative dimension, fractal and the so-called physics as distinct from mathematics. Here however we will concentrate on the unification equation.
In E-Infinity the ideal value of the electro weak, electro magnetic, and strong coupling are given by inverse forms and they are remarkably integer values. We use them to reconstruct the inverse electro magnetic constant. We have done this several times before in these communications. Let me remind you. We have alpha bar 1 equal exactly 60. Alpha bar 2 is half this value namely 30. Alpha bar 3, we divide in 2 alphas. It is equal to 8 + 1 = 9. All these values are remarkably close to the experimentally found value. Please note these values belong to an idealized world. It is the world of Plato. It is the world of ideals. It is the world of the supreme entity which organizes this world. It is the world of the theoretical and mathematical physicists who are constructing general theories emulating the work of the Supreme Being and hoping at the end to find some indication that these theories are right. The indication comes out from the messy real world. It
comes out from the messy laboratories. It comes out from billion of datum measured with different accuracies subject to human scatter, fallibility and noise. In addition we have alpha bar 4 which is equal unity. This value denotes the coupling between the Planck masses to the Planck ether, something which we most probably will not discover directly. The square of the Clibsch factor is also given in its ideal transfinite form namely the inverse of the golden mean which is pretty close to the classical value namely 5 divided by 3. Try it out. 5 divided by 3 is equal to 1.666.........The inverse golden mean is on the other hand 1.618033......Now we can write our first renormalization equation result for which we obtain the exact theoretical E-Infinity inverse electro magnetic fine structure coupling constant. The equation reads: alpha zero bar equal 60 multiplied by 1.618033.....plus 30 plus 9 plus 1. This adds together to exactly 137.082039325........
Next we would like to derive the non-super symmetric unification inverse coupling. We follow almost the same procedure. This so called coupling constant which is of course not a constant but a function of the resolution which means a function of energy is equal to alpha bar 3 plus alpha 4. That means equal to 9 plus 1 = 10. Then we have the logarithmic term. The exact theory does not have a logarithmic term. The exact theory replaces logarithmic term with golden mean scaling. However let us work first with an approximation using the logarithmic term because it is educational and helps the beginner to see the light. Grand unification takes place roughly at the ‘tHooft-Polyakov monopole. This is a mass of about 10 to the power of 16 GeV, when expressed in energy unit. Of course we cannot take a natural logarithm of a figure with dimension. We have to resort to a trick to make it dimensionless. The trick is not arbitrary. We more or less take a
reference energy which is the energy at which we take measurement. In this case we take the electro weak. More precisely let us take, as in the classical theory which you can find in any textbook, the energy scale of Z0. This is 91 GeV. When you take the natural logarithm of the ratio of both energies and we implore you to do so with your computers or pocket calculators, you will find it 32.3305...... We can assure you that the exact value is 32.3606...... and that this value is nothing but a scaling of the exact inverse electro magnetic fine structure constant of E-Infinity. In other words the logarithmic term is in reality nothing more and nothing less than 137.082039325........ multiplied by the golden mean to the power of 3. Adding our earlier 10 to this value you will find that the inverse unification constant becomes 42.3606...... This is exactly as we anticipated ten copies of the Hausdorff dimension of E-Infinity spacetime. What we have
done here needs more elaboration. We will do that later on in detail. For now it is sufficient to know that 42.3606.... is the non-super symmetric inverse unification coupling constant. To obtain the super symmetric value, we contemplate in the following manner. The minimal super symmetry requires that every particle should have a super symmetric partner, a so-called sparticle. Since we are working with inverse value, the doubling enters as an inverse value. In our equation this is expressed via a factor 1 divided by Roh. In this case Roh is 2. Our factor is thus a half. This half has to be multiplied by the natural logarithmic term. This term we just calculated to be 32.606.... Half of that is exactly 16.18033.....Adding to this value our 10, the final result is 26.18033.......This is a delightful result. With minimal effort and incredible accuracy we obtain the two fundamental coupling constants of unification of all fundamental forces. The result
agrees with most of the approximate results existing in the literature using various methods. These results vary in the first case between 40 and 45 and in the second case between 24 and 27. We will continue our calculation in the next Communication but we would like to close the discussion here by returning to ‘tHooft’s elucidation of negative dimensions. On such occasion it is quite in order to make some remarks on the nature of reality.
You remember Gerard ‘tHooft said in the discussion reproduced on page 152 of the book of Oriti which we cited earlier on: “A negative dimension could be understood as an anti commuting coordinate”. In other words, we think ‘tHooft means a Grassmanian value, something which is routinely used in super symmetric theory. We note parenthesically that ‘tHooft is normally skeptical about super symmetry and does not believe that it really exists in Nature. Then ‘tHooft continues by saying that we could think of an anti commuting differently and understand it as a negative dimension which replaces integration by differentiation. He said explicitly “differentiation is the inverse of integration.” A pure mathematician would find this language absurd. However from our E-Infinity viewpoint, we agree with the imaginative language of ‘tHooft and refute the pure mathematician allegation of absurdity. Of course we know that 1 divided by integration is
not equal differentiation. Such a sentence has no meaning. Nevertheless a theoretical physicist, we know, what ‘tHooft is trying to tell us. We find it a great pity that the truly great Nobel Laureate Gerard ‘tHooft is not familiar with the mathematical theory of dimension and the work on E-Infinity and particularly El Naschie’s theory about negative dimension. You see ‘tHooft could reformulate himself by saying that he knows that a negative dimension namely a minus 1 dimension is the dimension of the empty set. The beauty of the whole thing is that we could use a more tangible language and say that something which has the dimension of minus 1 could be modeled by a fractal. As fractals become thinner and thinner, the dimension becomes more and more negative and when the fractal totally disappears, its dimension becomes infinitely negative. Many could say that fractals are just geometric figures and not physics. In such case I just pass. If
someone feels that Grassmanian variables are more physical than fractals, then I have to give up and I have nothing anymore to say. The point is the following: To encompass the whole world we have to integrate from zero to infinity. Such world could be classical. However a quantum world has a ground state. A ground state could only be understood when we go behind the Zero. From this point of view to encompass the entire world, we have to integrate from minus Infinity to plus Infinity. That is what E-Infinity solved. In the same time this is one of the major problems with string theory. Nevertheless it was string theory which taught us how to deal with this problem in principle. It was on the other hand, the mathematical theory of Menger-Urhyson which gave us the tools to solve it. Our solution is what we call E-Infinity theory.
Communication No 26
From Dirac ‘s Sea to the Density Manifold of E-Infinity
Mathematics in some philosophical view is nothing but a highly efficient notation. Novel mathematics depends crucially on modern notation. Some would go further and could say that far more important than notation is the number system which we employ. Imagine what could have happened if the Europeans would not have had the good sense to accept Fibonacci’s advise and adopt the Arabic number system instead of the Roman number system. In previous communications we discussed extensively the role of the golden mean binary in E-Infinity. In particular we drew attention to what academician Alexey Stakhov named harmony mathematics or the mathematics of harmony to use a less Slavic and a more English expression.
In the present short communication we aim at expanding on few things. First we show how to drive the dimension of a manifold combining the virtues of non-commutative geometry and topological quantum field theory. Second we outline the way from Dirac’s theory of holes to Hilbert cube density manifolds and E-Infinity using simple random Cantor sets construction.
Let us start from Connes’ non-commutative dimension named Index of N in M where N is a sub manifold of M and M is explained in non-commutative geometry. It turns out that this index is equal to 1 divided by L plus 1 divided by 1 – L. Here L is a trace of E as explained in the excellent book of Connes referred to in previous communications. This is our first dimension. Now let us go to the theory of sub factors. The sub factor dimension corresponding to the previous dimension is given for instance in the excellent book of Saunders as 1 divided by 2d – 1. Remembering that quantum topological field theory is derivable from sub factors, we could imagine intuitively without much ado that equating both dimensions would result in a theory of manifolds combining the two concepts. Thus equating the inverse of L multiplied by 1 – L and the inverse of 2d– 1, we find an equation which is easily solved by setting L equal to d or d equal L. Expressed in term of d, we get the following quadratic equation ...............
d square plus d minus 1 = 0. The two-solution is d equal to the golden mean and d equal to the negative inverse square of the golden mean. Inserting the positive value into our index dimension we get the classical value 4 + the golden mean to the power of 3. We see that the most important topological invariant of a manifold combining non-commutativity with topological quantum field theory is the Hausdorff dimension of the core of E-Infinity theory.
Now let us move to Dirac’s theory of holes. Consider the construction of a classically random Cantor set. We take a unit interval. It starts at 0 and ends at 1. For later reference remember that the inverse of the beginning is Infinity. While the inverse of the end remains 1. These peculiar inverses of the boundary play a crucial role in the mathematics of the theory. Now the random Cantor set as is by now well-known has a dimension equal to the golden mean and a measure 0. But what is the dimension of the infinitely many holes left by construction in the Cantor set initial unit Interval? Since the Menger-Uryhson of the initial interval is obviously 1, then the dimension of all these holes is 1 – the golden mean which is equal to the golden mean to the power of 2. On the other hand it is by now well-known from our previous communications that the golden mean to the power of 2 is the dimension of the classical empty set. Now let us look in a purely formalistic way to the dimension which is formed by the union of the Cantor sets and the holes. It seems intuitively obvious and mathematically it is of course correct that the union of both dimensions would restore the situation to a unit interval. This is the trivial result of saying that the golden mean plus the golden mean square is equal to unity. When you are on this line of union you are either on a point or on an empty space. And the probability for that is the highest possible namely Unity. We talk of probability because the Hausdorff dimension could be interpreted in this case as a topological dimension as explained in previous communications. Now we ask another question namely what is the dimension of point belonging simultaneously to empty space and Cantor set? Such a requirement is classically impossible from a classical physical point of view. However formally it is the multiplication of both dimensions namely the intersection of sets. We have thus the dimension golden mean multiplied by the golden mean square which is the golden mean to the power of 3. The next interpretation is crucial. The golden mean to the power of 3 refers naturally to a much emptier set than the classical empty set. This set spans a sub space described by quotient space. This quotient space is given by the quotient formed from the union of the two sets divided by the intersection of the two sets. In other words, the dimension of the quotient space is the division of the golden mean plus the golden mean square divided by the golden mean multiplied by the golden mean square. Working out the simple arithmetic, we find yet again that the dimension is equal to the inverse of the golden mean to the power of three. This is essentially our famous 4 plus the golden mean to the power of three. In a nutshell, our Hilbert cube is a model for the core of E-Infinity theory and implies that it is a quotient manifold or a density manifold akin to what is known in measure theory as Gaussian manifold. The preceding explanation is a short version of a lecture given to post-graduate students by Professor Mohamed El Naschie in the Faculty of Science, University of Cairo five years ago. We thank him for permitting us to reproduce here though in a very concise version.
Note added: In his lecture El Naschie emphasized the reciprocity between what is known as quantum dimension in the theory of quantum group and E-Infinity. For instance the quantum group dimension is given by the inverse of a dimension d to the power of 3 minus d to the power of 3 all divided by the inverse of d minus d. Setting d equal to the golden mean we find that the quantum group dimension is exactly equal 4. This is the exact topological dimension of space time in E-Infinity as well as in reality. From that alone we see that we have a strong indication that the golden mean calculus as used in E-Infinity is a natural quantum calculus. Thus without saying it explicitly, E-Infinity employs tacitly quantum calculus disguised as a golden mean calculus.
E-infinity Communication No. 27
The Lee Symmetry Group Bonanza in high energy physics – A remarkable hierarchy.
Shadowing is a powerful method in nonlinear dynamics. Two decades ago some of us attended a lecture on this technique by the remarkable scientist, James York who incidentally gave chaos its name. An engaging melange of theory and demonstration ended with applause for arriving at Mars in a precise orbit using chaos. In general one could always or let us say most of the time shadows a transfinite scheme using an integer approximation. Let us take the plunge and give a remarkable example for what we mean without much ado. There is of course the danger for the uninitiated to confuse our reasoning with numerology. However those who have persisted until now in reading our communications are necessarily sufficiently sophisticated not to jump in such trivially wrong conclusions. We consider only SU(n). The standard model is based on SU(1)SU(2)U(1). This is not one but a combination of three groups. In general the dimension of SU(n) is equal to n square minus 1. Starting by n = 0 one finds the dimension of the zero set. However, wait a minute! If we are familiar with the exact mathematical theory of dimensions of Menger-Uryhson then we should start at least from the classical empty set. The classical empty set is of the dimension minus 1. Taking SU(-1) we find the dimension zero. Remarkably when we take SU(0) we find the dimension minus 1. In a sense we have a limited cycle confining us to what in physics corresponds to the beginning of the fractal world. Let us leave this and move to the next stage which is n = 1, then n = 2. Thereafter we follow the hierarchy which is self-explanatory. In other words, we take the dimension of SU(2) which is 3 then calculate the dimension of SU(3) which is 8 and then the dimension of SU(8) which is 63 and so on. This and so on refers to one step further only namely SU(63). For this SU(63) the dimension is exactly 3968. Look at this figure carefully. 3968 is 8 times 496. Remember 496 is the dimension of E8 E8. Consequently the dimension of SU(63) is nothing but the number of states of super Yang-Mill theory. You can ascertain yourself of this fact by consulting the relevant text books particularly the excellent book of M. Kaku. To understand the relevance of the result look at the spin reduction. The number of states should be divided in this case by 2 to the power of 6 which is 64. Our reduction equation would be in this case the dimension of SU(63) divided by 64 equal to 62. Doubling of this number leads directly to the familiar standard model representation for instance 124 as well as El Naschie-Lisi representation namely Dim E8 = 248. There has been several attempts by El Naschie and Lisi as well as Ray Munroe to give interpretation to this number as number of particles. As far as we are concerned one of the best interpretations is the following: The 124 could be split into 114 + 10. The 114 is twice 57, where the 57 is the dimension of the modular space M(19,3) which has a dimension 57 as is well known. The 10 on the other hand could be interpreted as 1 graviton with up and down equal to 2 plus 8 dimensional Higgs field. Some suggested the 2 graviton states plus 5 Higgs plus 3 space dimensions. We do not intend to discuss this in detail here. The main point was to show why Lee groups can come very near to exact solution. In general we may have two equivalent alternatives in high energy physics. Either we keep adding new exotic particles or we keep a good eye on the fine structure of space time. The theory of technicolor which was developed by various remarkable Israeli scientists is an example of the first strategy. On the other hand E-Infinity theory of Goldfain, Nottale, El Naschie, Ord, Crnjac, He and their associates as well as Alain Connes’ theory of non-commutative geometry is an example for the second strategy connected to a meticulous study of the fractal fine structure of space time.
We thank Mohamed El Naschie for giving permission to reproduce his Lee symmetry group hierarchy leading to super Yang-Mill particle-like states which were presented for the first time three years ago at various occasions.
E-Infinity
E-infinity Communication No. 28
Gödel’s theorem, Banach-Tarski theorem in E-infinity quantum mechanics
There is no doubt that skepticism is a valid attitude of many conservative scientists and particularly in physics. Never the less skepticism is one thing and name calling is another. Mudslinging is neither a legitimate nor an effective response in any event. Unfortunately some main stream scientists particularly mathematical physicists could easily cross the line between open minded skepticism and inflexible certainty which could easily be confused with ideological commitment. They close legitimate debate by insinuating moral deficiency in those expressing dissenting views. This sounds as if we are talking about politics in one of last century's banana republics; however we also have in mind conventional quantum field theory. Ironically conventional quantum field theory faced the same skepticism of the invalid type a few decades ago in a way not dissimilar to what some proponents of E-infinity are facing today. Reading some of the pseudo arguments used against E-infinity you immediately feel that the main aim is not so much to refute our arguments but rather to discredit our motives. Branding your opponent as a crack pot or pseudo scientist is a convenient way of ducking real questions and possibly novel answers. Of course those clever character assassins based their falsehoods on minimal truths to confuse the unsuspecting reader. For instance the laughable claim that Obama was the illegitimate child of Malcolm-X is so absurd that is had no affect what so ever. On the other hand any falsehood can acquire currency if it is published in a reputable journal such as say for instance, Nature particularly when it is supported by people who have a vested interest in spreading the falsehood.
In E-infinity theory issuing a flat out denial that the theory is wrong is not an option. In this case you turn to manufacturing doubts. A factory of spreading doubts is put in place and it works wonders when financial interests and funding are in question. At the end you have the depressing tale of a systematic rejection of a consistent mathematical theory based on prejudice and hearsay attributed to well established scientists who never really expressed their opinion directly and who names are used by sock puppets to spread the rumors.
Take fractal spacetime as advocated by Ord, Nottale, El Naschie, Goldfain, Ji-Huan He and their followers. Superficial criticism tries to make the unsuspecting reader believe that this idea of a fractal spacetime is total quackery. No attempt will ever be made to show that the idea is based partially on the work of leading scientists such as Richard Feynman and G. Parisi. Those who doubt these words should consult the excellent paper by G. Parisi called Hausdorff dimensions and gauge theories, published in Physics Letters, Vol. 81B, number 3,4, February 1979, p. 357. When El Naschie and L. Crnjac determined the mass spectrum of the standard model with unheard of precision the work was scornfully criticized as numerology. However when Frank Wilczek in a paper entitled Mass by Numbers published in Nature, Vol. 456 27 Nov. 2008 presents a very similar view of the mass spectrum, no one dared to question the profundity of the truly profound work not because they understood it in a different way but because the author carries the Nobel laureates hat. We have many more examples which are worth studying in details. For instance in a paper by F.J. Dyson entitled Divergence of Perturbation Theory in Quantum Electrodynamics, published in Physical Review, Vol. 85, No. 4, February 15, 1952, p. 631 the famous mathematical physicist does not shrug from indicating a connection between the number of terms in an expansion namely 137 and the number of particles created spontaneously as well as hinting towards a deeper connection to the inverse electromagnetic fine structure constant. Similar results based on P-Adic expansion as well as exceptional Lie symmetry groups which was obtained by El Naschie a couple of years ago is on the other hand rejected off hand as crack pot for a very deep reason, namely that El Naschie is an engineer and does not have a very inspiring first name. If the reader does not believe it he should scrutinize a certain dissenting voice which is disproportionately reported on the World Wide Web. And yet another example: Goldfain and later on El Naschie found a connection between high energy physics and Feigenbaum’s universalities. This work was disregarded or made fun of. However in December 2002 D. Polyakov and I. Kogan reported similar results in an Archiv paper entitled Ghost-matter mixing and Feigenbaum universality in string theory, arXiv:hep-th/0212137 v3 23 Apr 2003. This paper was well received because Polyakov enjoys in the meantime the good will of the establishment in the West. It is interesting to note that Polyakov came to his result before ‘tHooft regarding the famous monopole because of his knowledge of fluid turbulence. The great Russian born physicist was proud to note his expertise in turbulence. On the other hand the profound paper by Alfred M. Mukhamedov communicated to Chaos, Solitons & Fractals by M.S. El Naschie which was entitled Towards a gauge theory of turbulence, 29, 2006, p. 253 was almost totally disregarded. If you ask why, the sad answer is because it was published in a journal regarded by the establishment as a non main stream journal. This is enough for many to ban publication of this kind. What can we say? Yet another example: A paper published by Subir Sachdev called Polylogarithm identities in a conformal field theory in three dimensions, Physics Letters B, 309, 285, 1993, arXiv: hep-th/9305131 25 May 1993. In this paper the golden mean features as a fundamental quantity of conformal field theory in three dimensions. The paper was of course well received because the author is a prominent Yale University professor. On the other hand our generalization of the theory to a fully fledged golden quantum field theory is regarded by the same single critic as crack-pottery, a word with which this doubtful critic is in love. Again, what can we say? This word says about the man who is using it more than the man who is accused. We would like to mention three papers by El Naschie involving Banach-Tarski theorem and Gödel’s theorem and its relation to E-infinity quantum mechanics. These papers are Banach-Tarski theorem and Cantorian Micro space-time, Chaos, Solitons & Fractals, Vol. 5, No. 8, p. 1503, 1995, Statistical mechanics of multi-dimensional Cantor sets, Gödel theorem and Quantum spacetime, The Franklin Inst., Vol. 330, No. 1, p. 199, 1993, Gödel, Cantor and modern nonlinear dynamics, First Int. Symposium on Gödel’s Theorems, Edited by Z.W. Wolkowski, World Scientific, May 1991. The papers were relatively well received, for instance by the legendary Martin Gardener. However the establishment remained silent, if not hostile. Interestingly Prof. Lawrence B. Crowell from Alpha Inst. of Advanced Study in Hungary expressed similar views in his book published in 2005 by World Scientific entitled Quantum Fluctuation of Spacetime. In the next communication we will discuss these things in more depth as well as the role of Fuzzy platonic spaces as a model for quantum physics which was published in the journal Mathematical Methods, Physical Methods & Simulation Sci. & Techn, Vol. 1, No. 1, June 2007, p. 91.
In the meantime we hope you will read carefully not only what is written explicitly in this communication but also what is written between the lines.
E-infinity Communication No. 29
Gödel’s theorem, Banach-Tarski theorem in E-infinity quantum mechanics
There is no doubt that skepticism is a valid attitude of many conservative scientists and particularly in physics. Never the less skepticism is one thing and name calling is another. Mudslinging is neither a legitimate nor an effective response in any event. Unfortunately some main stream scientists particularly mathematical physicists could easily cross the line between open minded skepticism and inflexible certainty which could easily be confused with ideological commitment. They close legitimate debate by insinuating moral deficiency in those expressing dissenting views. This sounds as if we are talking about politics in one of last century's banana republics; however we also have in mind conventional quantum field theory. Ironically conventional quantum field theory faced the same skepticism of the invalid type a few decades ago in a way not dissimilar to what some proponents of E-infinity are facing today. Reading some of the pseudo arguments used against E-infinity you immediately feel that the main aim is not so much to refute our arguments but rather to discredit our motives. Branding your opponent as a crack pot or pseudo scientist is a convenient way of ducking real questions and possibly novel answers. Of course those clever character assassins based their falsehoods on minimal truths to confuse the unsuspecting reader. For instance the laughable claim that Obama was the illegitimate child of Malcolm-X is so absurd that is had no affect what so ever. On the other hand any falsehood can acquire currency if it is published in a reputable journal such as say for instance, Nature particularly when it is supported by people who have a vested interest in spreading the falsehood.
In E-infinity theory issuing a flat out denial that the theory is wrong is not an option. In this case you turn to manufacturing doubts. A factory of spreading doubts is put in place and it works wonders when financial interests and funding are in question. At the end you have the depressing tale of a systematic rejection of a consistent mathematical theory based on prejudice and hearsay attributed to well established scientists who never really expressed their opinion directly and who names are used by sock puppets to spread the rumors.
Take fractal spacetime as advocated by Ord, Nottale, El Naschie, Goldfain, Ji-Huan He and their followers. Superficial criticism tries to make the unsuspecting reader believe that this idea of a fractal spacetime is total quackery. No attempt will ever be made to show that the idea is based partially on the work of leading scientists such as Richard Feynman and G. Parisi. Those who doubt these words should consult the excellent paper by G. Parisi called Hausdorff dimensions and gauge theories, published in Physics Letters, Vol. 81B, number 3,4, February 1979, p. 357. When El Naschie and L. Crnjac determined the mass spectrum of the standard model with unheard of precision the work was scornfully criticized as numerology. However when Frank Wilczek in a paper entitled Mass by Numbers published in Nature, Vol. 456 27 Nov. 2008 presents a very similar view of the mass spectrum, no one dared to question the profundity of the truly profound work not because they understood it in a different way but because the author carries the Nobel laureates hat. We have many more examples which are worth studying in details. For instance in a paper by F.J. Dyson entitled Divergence of Perturbation Theory in Quantum Electrodynamics, published in Physical Review, Vol. 85, No. 4, February 15, 1952, p. 631 the famous mathematical physicist does not shrug from indicating a connection between the number of terms in an expansion namely 137 and the number of particles created spontaneously as well as hinting towards a deeper connection to the inverse electromagnetic fine structure constant. Similar results based on P-Adic expansion as well as exceptional Lie symmetry groups which was obtained by El Naschie a couple of years ago is on the other hand rejected off hand as crack pot for a very deep reason, namely that El Naschie is an engineer and does not have a very inspiring first name. If the reader does not believe it he should scrutinize a certain dissenting voice which is disproportionately reported on the World Wide Web. And yet another example: Goldfain and later on El Naschie found a connection between high energy physics and Feigenbaum’s universalities. This work was disregarded or made fun of. However in December 2002 D. Polyakov and I. Kogan reported similar results in an Archiv paper entitled Ghost-matter mixing and Feigenbaum universality in string theory, arXiv:hep-th/0212137 v3 23 Apr 2003. This paper was well received because Polyakov enjoys in the meantime the good will of the establishment in the West. It is interesting to note that Polyakov came to his result before ‘tHooft regarding the famous monopole because of his knowledge of fluid turbulence. The great Russian born physicist was proud to note his expertise in turbulence. On the other hand the profound paper by Alfred M. Mukhamedov communicated to Chaos, Solitons & Fractals by M.S. El Naschie which was entitled Towards a gauge theory of turbulence, 29, 2006, p. 253 was almost totally disregarded. If you ask why, the sad answer is because it was published in a journal regarded by the establishment as a non main stream journal. This is enough for many to ban publication of this kind. What can we say? Yet another example: A paper published by Subir Sachdev called Polylogarithm identities in a conformal field theory in three dimensions, Physics Letters B, 309, 285, 1993, arXiv: hep-th/9305131 25 May 1993. In this paper the golden mean features as a fundamental quantity of conformal field theory in three dimensions. The paper was of course well received because the author is a prominent Yale University professor. On the other hand our generalization of the theory to a fully fledged golden quantum field theory is regarded by the same single critic as crack-pottery, a word with which this doubtful critic is in love. Again, what can we say? This word says about the man who is using it more than the man who is accused. We would like to mention three papers by El Naschie involving Banach-Tarski theorem and Gödel’s theorem and its relation to E-infinity quantum mechanics. These papers are Banach-Tarski theorem and Cantorian Micro space-time, Chaos, Solitons & Fractals, Vol. 5, No. 8, p. 1503, 1995, Statistical mechanics of multi-dimensional Cantor sets, Gödel theorem and Quantum spacetime, The Franklin Inst., Vol. 330, No. 1, p. 199, 1993, Gödel, Cantor and modern nonlinear dynamics, First Int. Symposium on Gödel’s Theorems, Edited by Z.W. Wolkowski, World Scientific, May 1991. The papers were relatively well received, for instance by the legendary Martin Gardener. However the establishment remained silent, if not hostile. Interestingly Prof. Lawrence B. Crowell from Alpha Inst. of Advanced Study in Hungary expressed similar views in his book published in 2005 by World Scientific entitled Quantum Fluctuation of Spacetime. In the next communication we will discuss these things in more depth as well as the role of Fuzzy platonic spaces as a model for quantum physics which was published in the journal Mathematical Methods, Physical Methods & Simulation Sci. & Techn, Vol. 1, No. 1, June 2007, p. 91.
In the meantime we hope you will read carefully not only what is written explicitly in this communication but also what is written between the lines.
E-infinity communication No. 30
From Gödel and Banach-Tarski theorem to randomness and Undecidability in physics
Prof. Karl Svozil from the Dept. of Physics, University of Vienna is one of the early pioneers of fractal spacetime. Svozil worked closely with Otto Rössler as well as the leading experimental physicist Anton Zeilinger. In addition he made many contributions of Chaos, Solitons & Fractals. Randomness and Undecidability in Physics is the title of his modus operandi as a book writer which was published by World Scientific in 1993. In this book Svozil drew attention probably for the first time to the connection between Banach-Tarski paradox and nonlinear dynamics. El Naschie acknowledged that this led him to his work on the Banach-Tarski paradox in cosmology. An account of El Naschie’s work on this subject is given on pages 179-190 of a book entitled The Pea and The Sun by Dr. Leonard M. Wapner. The book was published by A.K. Peters Ltd, Massachusetts, USA in 2005.
In excessively colloquial terms Gödel’s theorem implied that you cannot lift yourself by pulling yourself by the toes. The Banach-Tarski theorem on the other hand represents the ultimate free lunch that is getting something from nothing. We promised in our last communication, No. 27, to discuss these two theorems in connection with high energy physics in some detail. However we feel that we should postpone such a discussion to a later communication and concentrate for the moment on something else which is deeply connected with randomness and Undecidability.
When you cannot know for sure anything you normally invoke probability as the method of last resort. Probability in physics means statistical mechanics. In this communication we would like to present this subject in connection with quantum field theory and E-infinity theory from a slightly unconventional point of view. It is said that good scientists always search for analogies between theories while the best of them search for analogies between analogies. Encouraged by such a philosophical attitude we would like to present some illuminating analogies which are of relevance to what we intend to demonstrate. Let us first start with the analogy between quantum field theory and statistical mechanics. Imagine a table where on the left hand side we have quantum field theoretical concepts and on the right hand side we have statistical mechanical concepts corresponding to the former. Let us start with the most obvious and most simple correspondence and analogy. In our view this is Feynman’s path integral. In statistical mechanics the corresponding notion is the partitioning function. The second simplest concept in quantum field theory is the change in the vacuum. The corresponding statistical mechanical concept is phase transition as well as bifurcation. It follows then that quantum fluctuation of quantum field theory corresponds to statistical fluctuation in certain expectation values. Next we know that the Lagrangian corresponds to the H function of statistical mechanics. Slightly less obvious is that gauge theory of quantum field theory corresponds to intersection and union of hyper areas in n dimensional spaces which is similar to what we do in E-infinity theory. The quantum h corresponds to the inverse of beta which is equal to k Boltzmann multiplied with T. The final analogy is probably the least understandable at first sight to those not familiar with statistical mechanics. The mass m of quantum field theory corresponds to the inverse of the length of correlation. To gain an intuitive feeling for the meaning of this analogy we have to remember the relation between the correlation length and the mean free path known from atomic physics. This mean free path measures a probability of collision. On the other hand we know that the smaller the distance we are probing, the larger the energy involved because of Heisenberg’s uncertainty principle. Since mass is directly related by relativity to energy, this mass must be inversely proportional to the correlation length and therefore m is equal to one divided by L. In E-infinity theory this explains the relation between the mass and the inverse of the electromagnetic fine structure constant and the geometry and topology of spacetime as will become evident later on.
Next let us attempt to write down a similar table for the analogy between quantum field theory and E-infinity theory. The most obvious thing here is again the Feynman path integral. This path integral is effectively summing over all paths which a quantum particle could take. In E-infinity we sum over all weighted dimensions. The corresponding expression is thus the expectation value of the Hausdorff dimension which is equal to four plus the golden mean to the power of three as is well known. Next quantum fluctuation of quantum field theory will correspond to deterministic chaos in E-infinity theory. More precisely it corresponds to an average symmetry as explained elsewhere in more detail. Gauge theory of quantum field theory is the intersection and union of Cantor sets in E-infinity formulation. The Lagrangian is replaced in E-infinity theory with a scaled energy where differentiation and integration is replaced by golden mean scaling up and down as explained in previous communications. The mass of quantum field theory appears in E-infinity theory as an average equal to the dimensions of certain symmetry groups all related to the inverse electromagnetic fine structure constant which is regarded in E-infinity theory as more fundamental than any other so called constants of nature. The change in vacuum in quantum field theory corresponds to the universal fluctuation of E-infinity which is equal to the golden mean to the power of three. Finally our classical quantum h corresponds to various powers and combinations of the golden mean as explained in detail in E-infinity theory. We admit that the preceding account is quite condensed but we are hoping that the reader has read at least some of our earlier communications and is therefore in a position to fill in the gaps of such a condensed summary.
E-infinity communication No. 31
The quantum gas model of E-infinity – the work of L. Marek-Crnjac
Over the years many of the E-infinity researchers introduced by way of analogy physical pictures for the mathematical structure of E-infinity Cantorian fractal spacetime. For instance El Naschie drew analogy to polymers. Somewhat related is the fractal sphere packing of Marek-Crnjac. However this could not be said to be a physical but rather geometrical topological interpretation of E-infinity. To be sure a physical interpretation for E-infinity by way of analogy is somewhat scarce.
On the other hand and rather recently a Slovenian mathematician L. Marek-Crnjac noticed a most instructive and useful analogy between the mathematics of ideal quantum gases and that of E-infinity. The model presented by Crnjac is as beautiful and charming as the picturesque mountain landscape near to Maribor where this Slovenian scientist lives and works. The idea is published in a formal paper but we may outline the gist of it.
Building on work by Mohamed El Naschie related to conformal field theory, the golden mean and its relation to polylogarithmic identities, Crnjac was able to show that a vital quantity related to a chemical potential could be seen as being analogous to the Hausdorff dimension of the core of Cantorian spacetime. In this case the familiar Hausdorff dimension is obtained from a polypseudolog to be divided by the square of one minus d. Substituting the golden mean for d the polypseudolog gives 4.236067899 and that for a negative dimension equal minus one. Recalling the dimensional theory of Menger-Uhrysohn as discussed by El Naschie, Nada, Iovane and He as well as L. Crnjac we see that this value in this presentation explicitly admits its indebtedness to the empty set for which the Menger-Urysohn dimension is equal minus one.
There are many highly interesting aspects of this new discovery of Crnjac. For instance the Bose-Einstein and Fermi-Dirac statistics are unified in a completely novel manner. This unification gives considerable support to the idea of super symmetry. In other words we have an argument for the existence of unified theory for both fundamental statistics apart of super symmetry. The work of Crnjac is still at the very beginning and is tying many fields together.
However we decided to mention it because it is new and very credible evidence that the basic tenants of E-infinity theory are correct.
At the beginning one may be surprised at this new result of Crnjac. However deep reflection shows that this is completely normal and expected on the basis of conformal field theory. To explain this point we have to recall a few facts. The concept of quantum field theory is almost 75 years old but it is only in the last 25 years that quantum field theory was taken seriously and used in high energy physics. However there are problems, the main one being that quantum field theory still uses the Lagrangian approach. This is the mainstream approach. It is also an insightful approach. Alas it is the least precise approach to quantum field theory. Mainstream scientists normally omit this point and even criticize E-infinity theory for not making use of the conventional Lagrangian approach. They omit mentioning that the Lagrangian approach, although useful for perturbative analysis, masks the genuinely non-perturbative aspects. Here comes conformal field theory to save the day. Conformal field theory like E-infinity theory lends itself easily to non-perturbative treatment. This aspect stems from the fact that infinite dimensional conformal symmetry can be presented in two dimensions. This is very familiar to people working in E-infinity. You recall that Klein’s modular curve has 336 isometries. The compactified Klein modular curve of E-infinity on the other hand has infinitely many isometries equivalent to infinitely many progressively smaller and smaller triangles. These triangles ramify at infinity into a Cantor set yet the whole thing is in two dimensions. That is to say a two dimensional projection and what is more interesting, by taking a gamma distribution the infinitely many isometries can be weighed statistically to be equivalent to almost 339 isometries. In other words we have three isometries more than the classical 336 isometries of the classical Klein modular curve. These simple but deep mathematical facts help those working not only in E-infinity theory but also in string theory. The connection comes from the mathematical relation to Affine Lie algebras, Monster groups and the modular invariance related to modular manifolds. The relation to Kac-Moody algebra, 3-point functions and fusion algebra was hinted at by El Naschie in many of his earlier writings.
We are deeply indebted to Prof. L. Marek-Crnjac for allowing us to publish this extract and we would be grateful if anyone using this condensed summary ensures they refer to the original work of Prof. L. Marek-Crnjac.
E-infinity group.
E-infinity Communication No. 32
Polylogarithms, the golden mean and elementary particle confinement
There has recently been renewed interest in the polylogarithmic approach to high energy physics. The work bears strong resemblance to E-infinity’s approach. The resemblance is obvious from very recent papers by Cvijovic as well as Lee and Sachdev. It was due to one of our group, Prof. L. Marek-Crnjac that we noticed that there is a synthesis connecting these superficially disparately different approaches. The existence of the relationship which we are about to briefly outline, yet again confirms the belief expressed by Mohamed El Naschie in his first Editorial in the first issue of Chaos, Solitons & Fractals almost eighteen years ago. El Naschie said that interdisciplinary research holds the key for the future. It is clear that E-infinity is the best example for this belief. In what follows we would like to show in an extremely brief form the most important connections between the polylogarithmic approach and the E-infinity way of looking at things. Here are the most important points:
1. E-infinity assumes spacetime to be a very special form of fractals, namely a Cantor set albeit an infinite dimensional and hierarchal Cantor set. One of the most important attributes of a Cantor set or more generally a fractal is self similarity. Self similarity is a form of gauge theory. In this sense it is extremely striking to know that the polylogarithm is self similar. It is possible to prove that under certain conditions polylogarithms become self similar. This was used by Lee to prove that bosons cannot be confined. Interestingly the very mathematics of E-infinity is self similar. I mean the very number system used in E-infinity is self similar because the golden mean is self similar under various elementary operations. For instance the inversion of the golden mean is a translation by one of the golden mean. You can ascert yourself of this obvious fact by dividing one by 0.618033989 and find that the result is 1 + 0.618033989. Even the continuous fraction of the golden mean is simply 1 repeated indefinitely in the usual manner. The square of the inverse golden mean is a translation of the golden mean by two. In other words one divided by the golden mean and squared is equal to 2 + the golden mean and so on.
2. As is well known on the holographic boundary of E-infinity or more general in the two dimensional Klein modular curve conformal field there is no difference between bosons and fermions. It is in a natural way ultra super symmetric. Similarly the Fermi-Dirac as well as the Bose-Einstein statistic are unified in describing the statistical thermodynamics of ideal quantum gas when expressed in terms of the polylogarithmic function. This fact seems to be little known to theoretical physicists, at least before the work of Lee, Sachdev and Cvijovic to mention only a few of those pioneers.
3. The most surprising thing may be the fact that Sachdev’s famous result, namely that the central charge is a rational number equal to 4 divided by 5 for a three dimensional conformal field theory, can be derived only when the fugacity z is set equal to the golden mean as well as the golden mean square and the negative golden mean and combining these results. It is conjectured that the fugacity in this sense represents a type of reduced geometrical density of Cantor points. Consequently one can see the relation between the random Cantor sets of E-infinity which have a Hausdorff dimension equal to the golden mean and this very special role which the golden mean plays in the corresponding polylogarithmic theory.
4. There are various formulas found in the mathematical literature on polylogarithms which can easily be transferred to E-infinity theory. Take for instance the sum from n equal 1 to infinity of li subscript minus n of z divided by n factorial. This well known formula gives 1 plus z all divided by 1 minus z and then the result is divided by 2. For z equal to the golden mean we have the well known isomorphic length of Conway which was explained in many interesting articles by Martin Gardener. The value of this isomorphic length is as well known, equal to half the inverse golden mean when raised to the power of 3. In other words this is equal to (4.23606799) divided by 2.
5. One of the most important results is the fact that log li subscript minus 1 of z is equal to z divided by (1 minus z) power 2. Again if you put z equal to the golden mean you get 4.23606799 which is the Hausdorff dimension of the core of E-infinity Cantorian spacetime. The interesting point is that this result is inherent to the dimension of minus 1. This is a negative dimension. Initially the people working in polylogarithmic theory were confused regarding the physical relevance of a negative dimension. At the end Prof. Lee in the USA understood that it is at least of theoretical value and that it is related to confinement. Based on these ideas he was able in a paper published in Physics Review towards the end of the last century to prove that bosons cannot be confined. Of course Prof. Lee was not aware of the Menger-Urysohn theory of transfinite dimension which Prof. Mohamed El Naschie transplanted in high energy physics and used it to show that a dimension minus 1 is the classical empty set. In geometrical terms such empty sets are inherently fractal. In fact you can let things grow into the negative dimension to minus infinity and get the absolutely empty set. In other words a negative dimension is an important physical device in a physical theory despite the fact that it is totally anchored in an abstract mathematical theory of dimension. Needless to say this negative sign is essential for explaining the two slit experiment with quantum particles in a rational way. This negative sign is also important for the work of Garnet Ord who extended the Bernoulli binary 0 and 1 to the Bernoulli triply as referred to by El Naschie, namely 0, 1 and minus 1. Prof. Garnet Ord referred to this as anti Bernoulli. In addition Mohamed El Naschie regards the negative sign as an anti-commuting Grassmannian coordinate. This view was taken over by Gerard ‘tHooft in recent times. Also recently El Naschie considered the possibility of the Möbius function to obtain the three fundamental numbers of the anti-Bernoulli, namely 0, 1 and minus 1. For instance the Möbius function for n equal 4, 6 and 2 assumes the values 0, 1 and minus 1 of the anti-Bernoulli of Garnet Ord. Finally we should not forget to mention employing special relativity to motivate anti particles and consequently the minus 1 of Ord as proposed for the first time by Garnet Ord and elaborated by Mohamed El Naschie using his conjugate complex time.
We are aware that every point of the above needs pages of explanation and comments but we will stop at this point and will come to it later one when time permits.
Communication No 33
Motivation for the negative sign in quantum mechanic, high energy physics and the two slit experiment
Part 1
The Ord-El Naschie proposal using special relativity
In addressing the audience on the occasion of Sir Roger Penrose’s birthday one of today’s most remarkable mathematicians Allen Conne said: “The space of Penrose tilings is a prototype of a non-commutative space.....” Of course Conne means Penrose’s fractal tiling. He found it so obvious that he did not mention the word fractal. E-Infinity theory as developed by El Naschie is in effect a sweeping generalization of this prototype of Penrose which did go beyond Allen Conne’s non-commutative spaces. Needless to say that without the contribution of Allen Conne and non-commutative spaces, E-Infinity could not have been developed at least not as fast and as far as it happened.
In a previous communication we discussed various proposals to motivate the vital negative sign involved in explaining the destructive interference in the two slit experiment with quantum particles. In what follows we discuss at some length a proposal originally due to Garnett Ord which was further developed and elucidated by Mohamed El Naschie.
Let us look at the signature of the Lorentz matrix of special relativity. This is x squared equal to x1 square plus x 2 square plus x 3 squared minus t squared where t is the time. For two-dimensional-space time the matrix is simplified to x square equal to x 1 squared minus t squared. Suppose we are stuck on the light cone. Consequently x square is equal nil. Therefore it follows that x 1 squared equals t squared. Solving for x 1 we find that x 1 is equal plus or minus square root of t square. In other words x 1 is equal to plus or minus t. This is exactly the special relativity interpretation and origin of the needed negative sign as proposed by Garnett Ord and elucidated by Mohamed El Naschie. In further discussion El Naschie emphasized that this negative sign implies in addition to many other things a rotation as indicated in the work of many researchers like Marc Levy-Leblond as well as recently by Mitchell Feigenbaum. In quantum field theory this rotation is known as Weck rotation and implies imaginary time namely minus it leading to a 1 sign Euclidean metric ( +, +, +,+,) instead of the Lorentzian metric (+,+,+,-). In other words and as discussed in many recent papers by Mitchel Feigenbaum which were published partially in the New Scientist, the negativity of time has a deep geometrical meaning connected possibly to the Bloch sphere of quantum mechanics. This is all apart from its connection to the adjoint system method and the conjugate complex time of El Naschie as well as the empty set of Menger-Urhyson dimensional theory which was introduced to high energy physics also by El Naschie and our group notably Ji Huan-He, L. Marek-Crnjac, Nada and G. Iovane.
E-Infinity
Communication No. 34
Application of chaos and fractals in fundamental physics and set theoretical resolution of the two-slit experiment and the wave collapse
M.S. EL NASCHIE
Abstract
We present a convincing resolution of the wave collapse and the two-slit experiment based on random fractal sets and the extended Menger-Urysohn transfinite theory of dimensions. We use two component dimensions starting from A. Connes dimensional function of noncommutative geometry
D = a + bϕ ; a, b ∈ ; = ( − 1)/2 .
We demonstrate the equivalence of Connes function to that of E-infinity bijection formula
dc(n) = (1/dc(0))n-1
where n is a real integer and dc(0) = = ( − 1)/2 .
The golden ratio = ( − 1)/2 = 0.618033989 which was experimentally confirmed a few months ago this year in the Helmholtz Centre in Germany and University of Oxford, England as basic to quantum mechanics arises naturally from the requirement of a random fractal horizon of a noncommutative geometry akin to that of Penrose fractal tiling and the related compactified Klein modular curve. This curve with 336 + 3 = 339 hierarchal degrees of freedom or isometries is also equivalent to the holographic boundary of E-infinity spacetime.
One of the most important results of the present work is the identification of the empty set with wave-like quantum probability and the vacuum while quantum-like particles are described as a zero set. Consequently the two dimensions relevant to the particle are
dim particle = P(dMU , dH)
= P(O , ϕ)
where dMU is the Menger-Urysohn dimension and dH is the corresponding Cantorian or Hausdorff dimension. For the quantum wave on the other hand we have
dim wave = W(dMU , dH)
= W(-1 , ϕ2) .
Since we have identified the wave which is devoid of energy matter and momentum with the empty set, it follows then as an almost trivial result that any attempt to observe the two-slit experiment while in progress will render the empty set non-empty and instantly lead to what we perceive as a wave collapse, leaving the zero set of the particle as the only observable. Using the preceding conclusion it is then a relatively straight forward and technical analysis which leads us to the form of semi manifold which supports the preceding requirement. It turns out that this semi manifold is a fractal quotient manifold of the Gaussian type with a Hausdorff dimension given by
DH =
= 4 + ϕ3
= 4 + __1___
4 + ___1___
4 + ……
= 4.236067977 ,
and a Menger-Urysohn dimension
DM =
= ∞
as well as an average topological dimension equal to
<> =
= 4 .
In other words this manifold is nothing else but the core of E-infinity Cantorian spacetime which may be envisaged as an infinite hierarchy of concentric four dimensional cubes.
The present lecture is divided into two parts. We first give a general review of all previous work on the application of chaos and fractals in physics. The second part is subsequently devoted to the resolution of the paradox presented by the two-slit experiment and the associated wave collapse.
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