This is a very important communication

E- infinity communication No. 90

This is a very important communication

Date: 16/12/2012

The E-Infinity Group would like to report briefly that El Naschie's Ultimate F theory has culminated into an incredibly powerful mathematical machinery which is essentially an extension of Witten's topological quantum field theory to a topological theoretical theory of physics. The theory is based on Ultimate L, the K-Theory, Penrose fractal Tiling, E-Infinity, Hardy's Quantum Entanglement, Witten's M-Theory and Topological Quantum Field Theory. The result is a startling simplification of computation reducing day's work of super computer to lines of calculator aided hand computation.

E-Infinity Group

E-Infinity Communication No. 89

E-Infinity Communication No. 88

Lates News

Heisenberg uncertainty, quantum entanglement and von Neumann’s entropy from the view point of fractal spacetime and E-infinity theory

14th May, 2011
E-infinity Communication No. 86
Heisenberg uncertainty, quantum entanglement and von Neumann’s entropy from the view point of fractal spacetime and E-infinity theory

There has recently been a flurry of papers reassessing the validity of Heisenberg’s uncertainty in view of quantum nonlocality and maximal quantum entanglement. Two of the many recent publications which are worth mentioning are first an article in Science, 19 November (2010), Vol. 330 by Oppenhein and Wehner entitled “The uncertainty principle….”. The second is published in Nature–Physics-Letter (2010), 6, pp. 659-662 by M. Berta et al entitled “The uncertainty principle in the presence of quantum memory”.
Readers of E-infinity communications may be interested and slightly surprised to know that the new relations indicated in the mentioned recent research are a straight forward consequence of the fractal nature of quantum spacetime. For E-infinity experts and seasoned researchers the connection between entropy, quantum entanglement and Heisenberg’s uncertainty are natural and obvious facts. For the benefit of those not that familiar with fractal spacetime and E-infinity theory we offer the following few remarks, comments and elucidation.
Let us start by recalling two well known facts. First that one of the eigenvalue of Arnold’s mixing cat map or the two degrees of freedom unit oscillation is the golden mean. Consequently the topological entropy of these two fundamental models is the natural logarithm of 1.61803398 which is equal to 0.4812118252. The second well known fact is that the Hausdorff dimension of the path of a quantum point particle is not one, but rather unexpectedly 2. This fact was used by Abbott and Wisse as well as Ord, Nottale, El Naschie and Parisi. To find the entropic content of the quantum path we divide 2 by the natural logarithm of the golden mean and find 4.1561738141. This is a good approximation to 4 plus the golden mean power 3 namely the exact value of the Hausdorff dimension of the core of E-infinity manifold, the well known value 4.23606799. To obtain this exact value from the preceding entropic consideration we just expand the natural logarithmic term and retain only the linear terms. That way we find the familiar formula of E-infinity theory (1 + ϕ) divided by ( 1 ̶ ϕ) which gives (1/ϕ) to the power 3 or in decimal form 4.236067977… From this simple derivation we see how a quantum path is linked to the Hausdorff dimension of a peano curve which is D(H) = 2. Since a point on the peano curve is fundamentally resolution dependent and fuzzy it is clear that it can explain nonlocality. Also the dimension D(H) = 2 is essential for the derivation of the Heisenberg uncertainty principle as shown by Ord, Nottale and El Naschie (see for instance M.S. El Naschie – Quantum mechanics, Cantorian spacetime and Heisenberg’s uncertainty principle. VISTAS in Astronomy, Vol. 37, pp. 249-252 (1993). It should be noted that the ultimate explanation of nonlocality is the Lebesgue measure zero of the building blocks of spacetime which according to E-infinity are elementary random Cantor sets with a Hausdorff dimension, namely the golden mean as per a well known theorem due to Mauldin and Williams. This is the ultimate geometrical justification for the appearance of the golden mean in all quantum probabilities of entanglement as explained long ago by M.S. El Naschie. See for instance:
1 Chaos, Solitons & Fractals:
i. Vol. 1, No. 5, pp. 485-487 (1992).
iii. Vol. 9, No. 6, pp. 975-978 (1998).
iii. Vol. 26, No. 1, pp. 1-6 (2005).
iv. Vol. 9, No. 3, pp. 517-529 (1998).
2. Journal of the Franklin Institute:
i. Vol. 330, No. 1, pp. 183-198 (1993).
ii. Vol. 330, No. 1, pp. 199-211 (1993).
3. Il Nuovo Cimento, Vol. 107B, No. 5 (1992).

E-infinity Group.

Recent important papers on E-infinity

24th March, 2011.
E-infinity Communication No. 85
Recent important papers on E-infinity

The following are some important papers on E-infinity which have just appeared in various international journals. Further details may be found from the internet.

1. M.S. El Naschie: Quantum collapse of wave interference pattern in the two-slit experiment. N.S.L.A., Vol. 2, No. 1 (2011), pp. 1-9.
2. Ciann-Dong Yang: Trajectory-based quantum chaos. N.S.L.B., (1), (2011), pp. 7-10.
3. L. Marek-Crnjac: The physics of empty sets and the quantum. N.S.L.B. (2011), pp. 13-14.
4. Lan Xu and Ting Zhong: Golden ratio in quantum mechanics. N.S.L.B., (1), (2011), pp. 25-26.
5. R. Murdzek, Daniela Magop and Roxana Stana: A fractal universe in Brane world scenario. N.S.L., (1), (2011), pp. 41-44.
6. Ji-Huan He, Tin Zhong, Lan Xu, L. Marek-Crnjac, Shokry Ibrahim Nada, Mohamed Atef Helal: The importance of the empty set and noncommutative geometry in underpinning the foundation of quantum physics. N.S.L.B., (1), (2011), pp. 15-24.
7. Ji-Huan He: The importance of the empty set underpinning the foundation of quantum physics (Editorial). N.S.L., (1), (2011), pp. 11-12.
8. M.S. El Naschie: Application of chaos and fractals in fundamental physics and set theoretical resolution of the two-slit experiment of the wave collapse. N.S.L.B., (1), (2011), pp. 1-3.
9. M.S. El Naschie: On the philosophy of being and nothingness in fundamental physics. N.S.L.B., (1), (2011), pp. 5-6.
In forthcoming communications we will give a brief discussion of the above mentioned papers as well as more helpful references and literature.

E-infinity Group.

Quantum scale, small world and nanotechnology

23rd March, 2011.

E-infinity Communication No. 84

Quantum scale, small world and nanotechnology

The macro world is abound with an incredible diversity of shapes, chemical and physical properties of living and inanimate material forms each distinct from the other in almost infinite ways at least on the phenomenological level. However as our observational scale becomes tinier and we descend to the micro and quantum scale, this diversity becomes highly restricted. At the level of the smallest building blocks of nature, namely the so called elementary particles the only diversity left is that of mass, spin and electrical charge. At ultra high energy we are left in theory with nothing more than numbers of state-like particles or the dimension of global symmetries. For instance in one of the presently most popular theories of everything, the so called superstring theory, all that we know is that we have 496 massless indistinguishable gauge bosons.

The problems thrown up by quantum physics of the micro world are solved traditionally using classical lattices (you may think of the finite difference or finite element method used in engineering and called in physics Regge calculus). However a relatively recently deciphered phenomena in social sciences called small world introduced a new kind of random transfinite lattice similar to the E-infinity Cantorian spacetime proposed by Mohamed El Naschie. In fact a small world network is behind the incredible connectedness which we witnessed in social networks such as Facebook and Twitter. It is for instance shown in this theory that for a population as large as that of the earth we only need 24 friends for each chosen two people to reduce the entire population to one compact net. In such a net only two randomly chosen persons are connected through 6 people at most. This is what lies behind the by now very famous phrase of only six degrees of separation. Note that 2 + 24 = 26 and that this is the dimension of the bosonic string theory which is supposed to be a theory of nearly everything. Note also that the exact value of El Naschie’s transfinite version of string theory is 26.18033989 and 6.18033989.

In future communications we will dwell on the connection between nanotechnology, high energy physics and small world networks. It is argued that Nano drugs delivery could benefit from understanding the body as a small world so that side effects are drawn into the healing processes. The same applies to social problems and the breakdown of the social fabric in a political uprising for instance.

It is conjectured that singularity theory (catastrophe theory), deterministic chaos and transfinite small world networks could be used under the auspices of nanotechnology to change the world and solve problems hitherto thought to be unsolvable such as stock market crash, material complex hair cracks propagation, fluid fully developed turbulence, earthquake predictions and climate disasters forecast.

For literature on the subject the reader may consult the Prof. El Naschie’s homes pages at: www.msel-naschie.com and/or www.el-naschie.net.

E-infinity Group

On El Naschie’s small world of infinite dimensions

22nd March, 2011.
E-infinity Communication No. 83
On El Naschie’s small world of infinite dimensions

In the following we give a number of observations, interpretations and comments on the relationship between E-infinity theory and the theory of small world. The copyrights of the material published here, some for the first time should be respected by quoting the source, namely E-Infinity Communication Publications. We have not attempted to mention each of the sources of the information given here which is largely due to the papers as well as formal and informal talks and lectures by Prof. Mohamed El Naschie, Prof. Ji-Huan He, Prof. G. Ord, Prof. L. Marek-Crnjac and Prof. G. Iovane.
1. Ergodicity or global chaos as well as complete order may be assigned both the same zero complexity indexes. Both complete order and complete chaos are consequently structurally unstable in some vague topological sense. By contrast a KAM system which encompasses both chaos as well as order on all scales may be seen as relatively structurally stable and robust. This property of robustness to perturbation is shared by small world networks. In a sense robustness to perturbation is a substitute for friction in dissipative systems. Hamiltonian systems have no physical friction. However the irrationality of the winding number is what replaces physical friction in Hamiltonian systems such as quantum physics according to KAM theorem. That is how the golden mean comes in as being the most irrational number. From this view point E-infinity and small world theory seem to have common roots if not much more than that.
2. E-infinity’s quasi manifold is probably one of the most amazing geometrical and topological constructions which unite the un-unitable. It is infinite dimensional yet it has a resolution dependent finite expectation value for all its topological invariants including dimensions. It is infinitely large yet it is in more than one sense compact and so is its holographic boundary. It is fuzzy but within this fuzziness everything is probabilistically exact. It is completely discrete but due to the transfiniteness of its geometry it resembles the continuum. It is infinitely large but because it reproduces itself latest after moving a maximal distance equal to the isomorphic length multiplied with an arbitrary radius, it is semi-finite and resembles a semi-small world.
3. There are clear applications of both small world theory and E-infinity theory to any transfinite network such as neurons in the brain as well as complex fracture systems such as seismic fault structures relevant to earth quakes. The application in sociology may be among the most profound applications to things which may be shaping the future right this minute.
4. With regard to high energy physics Mohamed El Naschie’s theory de facto proposed the replacement of the classical lattice of the large world with the transfinite Cantorian lattices of the small world theory.
5. It is frequently argued that the six-degrees separation does not apply to a set of people alive at different times. The classical example for that is that Alexander the Great is separated from Albert Einstein by more than six-degrees. However if we take the degree of a degree into account, i.e. we take the weight of a degree (or a dimension) into account then we could still end with 6 or less degrees of separation. For instance we know that Alexander was interested in the art of knode. On the other hand Lee Smolin showed that Stuart Kauffman’s knode theory is relevant to quantum gravity. That means it is relevant to gravity and this connects Alexander to Einstein albeit it is a very weak connection.
6. Between two random people we have 24 acquaintances according to an application of Erdös-Rényi theorem. Thus we have 24 + 2 = 26. Similarly we have a world string sheet with two dimensions and when we add the 24 instantons of a Kähler to it we can simulate 2 + 24 = 26 degrees of freedom corresponding to the 26 bosonic dimensions of string theory. This is of course an extremely loose argument for the obvious but mathematically still terse connection between stringy networks and small world networks.
E-infinity Group.

The small world of ‘t Hooft-Susskind holographic boundary – An E-infinity view

20th March, 2011.
E-infinity Communication No. 82
The small world of ‘t Hooft-Susskind holographic boundary – An E-infinity view
The diffeomorphic kinship between the Penrose fractal tiling universe and the compactified Klein modular curve is well known from El Naschie’s work on the holographic boundary theory. Following this theory a fundamental equation was established stating that the total numbers of state-like particles or massless gauge bosons may be taken to be the dimension of E8E8, namely 496 and that this number must be equal to certain isometries and dimensions pertinent to all elementary particles living on the holographic surface of the 496 dimensional bulk as well as pure gravity and finally electromagnetism. Since in four dimensional Einstein gravity as well as eight dimensional pure gravity the number of the corresponding independent components of the Riemannian tensor and the number of isometries is the same, namely 20, then it follows that 496 must be equal to 20 plus electromagnetism plus particle physics. Following E-infinity theory the number of particles on the holographic boundary are equal to the number of isometries of the classical Klein modular curve, namely 336 plus the compactification effect taking the boundary to infinity as in projective hyperbolic geometry, namely 3. Thus the total number of particle-like isometries is 336 + 3 = 339. Using the equation the inverse coupling of electromagnetism 137 is found exactly and given a topological meaning, namely a dimension of an electromagnetic manifold determined by the fundamental equation 496 – 20 – 339 = 137.
There is something even more astonishing about this holographic boundary which relates it to the theory of small world in an unexpected way which upon reflection should have been expected. The so called isometric length of E-infinity theory applied to the holographic boundary is given by half of the E-infinity core Hausdorff dimension which is half of the famous value 4.2360679, that is to say 4 plus the golden mean to the power of 3. Half of that is exactly 2.118033989. Now there is an approximate value to the Hausdorff dimension found using the classical continuous gamma distribution which was given long ago by El Naschie as well as S. Al Athel, namely 2 divided by the natural logarithm of the inverse golden mean which leads to 4.156173841. Dividing this by two we find an approximation to the isometric length, namely the (inverse) natural logarithm of 1.618033989 which means the isometric length is equal to the inverse of the natural logarithm of the inverse golden mean. There are two important points which we have to consider at this point. First the isometric length is the distance which we have to maximally travel in order to find our surroundings replicated almost exactly as if we had not moved at all. That means that our compactified holographic boundary which describes an infinite universe in all directions is still a finite and in fact small world-like universe. Let us call it small world-like holographic and Penrose universe. Second a small world non-transfinite ordinary network is typically characterized by a distance given also by a logarithmic value. Nor N nodes the distance between the two randomly chosen nodes is proportional to the logarithm of N, namely Log N. This logarithmic relation is behind the relation between social networks like Facebook and Twitter and the transfinite neural network behind quantum mechanics such as E-infinity transfinite networks as proposed for the first time by Mohamed El Naschie and his student Dr. Mahrous Ahmed as well as several other members of the E-infinity Group.
E-infinity Group

14th March, 2011.
E-infinity communication No. 81
One road to quantum gravity and E-infinity as a transfinite social network
Lee Smolin wrote a nice little book some time ago entitled Three Roads to Quantum Gravity. We discussed this book in an earlier communication where Smolin touched upon and really only touched upon fractal spacetime.
In the present very short communication we propose to reduce the number 3 of Smolin to only 1. In our opinion there is only one road to quantum gravity. This road is based upon the skeleton of a very old idea namely that of a network. Such a network may be for all we know the same network of a small world or even Lee Smolin’s favorite approach of loop quantum gravity. However to be the one and only road such a network must be a self similar grid. In addition this grid must have an element of randomness. When the reader ponders these basic requirements he or she will immediately realize the equivalence of E-infinity theory of Mohamed El Naschie with that of a transfinite social network. Incidentally this is not a new insight. It is only a neglected relatively old insight of El Naschie about which he wrote a paper or two in Prof. Ji-Huan He’s journal. Thus we may recall the following article which the reader may find quite useful: M.S. El Naschie: Transfinite electrical networks, spinoral varieties and gravity Q bits. Int. J. of Nonlinear Sci. & Numerical Simulation, 5(3), (2004), pp. 191-197.
In conclusion it should be noted that the American engineering scientist A.H. Zemanian seems t be the first to propose transfinite networks in electrical engineering and wrote many excellent papers on the subject. In fact Zemanian was in direct contact with El Naschie discussing this exceedingly interesting subject and in view of the events in the Middle East revolutions is also timely because of the role played by social networks such as Facebook and Twitter.
E-infinity Group

Small world E-infinity spacetime, Facebook and who wrote more papers, Mohamed El Naschie or Paul Erdös?

13th March, 2011
E-infinity Communication No. 80
Small world E-infinity spacetime, Facebook and who wrote more papers, Mohamed El Naschie or Paul Erdös?
Maybe rightly or wrongly the uprisings sweeping the Arab countries and the Middle East including Mohamed El Naschie’s beloved homeland Egypt is attributed to social network sites such as Facebook and Twitter. We were not really fully aware of the possible connections between E-infinity, small world social networks and even less that Mohamed El Naschie had written any papers on this subject. Quite honestly we were mildly surprised when we became aware of several articles which he wrote on the subject in his daily column in the semi official Egyptian newspaper Rose Al Yusuf. Upon translating these articles into English we realized that he and some of his students published several papers in Chaos, Solitons & Fractals and elsewhere on the small world theory and its connection to super string theory and his E-infinity Cantorian spacetime. For more details the reader is referred to two papers:
1. N. Ahmed: Cantorian small world, Mach’s principle and the universal mass network. Chaos, Solitons & Fractals, 21 (2004), pp. 773-782.
2. M.S. El Naschie: Small world network, E-infinity topology and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals, 19 (2004), pp. 689-697.
The fact that although we worked rather closely with Mohamed El Naschie and are part of his group and yet were unaware about his paper prompted some of us to ask how many papers did El Naschie author and on how many subjects? The honest answer is that we still do not know and when we asked him it was clear that he does not know either and furthermore, does not care. A wild guess is that it is in the region of one thousand and some of us recall having seen that written by someone. Initially we thought that given his age, this must be a world record. However it was not long before we realized that among real scientists of international status in mathematics and physics, the record holder is a man admired by El Naschie, namely Paul Erdös with more than one thousand four hundred papers mostly with co-authors. Needless to say the number of papers nor the number of pages of written papers is no indication of any quality or excellence. It is ridiculous that some agencies use such trivial indexes to evaluate scientists.
Let us conclude this short communication by mentioning the most important common thing between small world and El Naschie’s Cantorian world. Similar to all quantum network approaches to quantum physics both theories are based on a network. However in E-infinity it is a self similar network on all scales and is thus a transfinite network. Second a true small world network is neither orderly nor disorderly. It is in between. This is exactly the KAM deterministic chaotic geometry of El Naschie’s E-infinity theory.

E-infinity Group

Fractal Time

Fractal Time

We would like to draw attention to a magnificent new book constituting a tour de force on the philosophy and science of fractal time. The author is Dr. Susie Vrobel who is the main pioneer of the wider implications of fractal time in philosophy. We enclose her Preface. The book is published by World Scientific 2011.

E-infinity Group.

Cobordism in E-infinity and negative (anti)space

27th February, 2011
E-infinity Communication No. 78

Cobordism in E-infinity and negative (anti)space

In this communication we draw attention to an important mathematical-physical note by Prof. Ji-Huan He regarding anti space and the empty set in quantum set theory. The paper was published in 2010 (copy attached) in the Int. Journal of Nonlinear Sci. & Num. Simulation, No. 11(12), 2010, pp. 1093-1095. There are of course those notable scientists, some with distinguished titles and prizes who think that mathematics is not physics at this very deep level of the quantum world. To those we can only repeat the famous words of Vincent Price in some of his horror films “How wrong…., how terribly wrong!”

E-infinity Group

A moonshine conjecture from E-infinity (number theoretical motivation)

26th January, 2011.
E-infinity communication No. 77

A moonshine conjecture from E-infinity (number theoretical motivation)

One of Alexander Grothendieck’s greatest insights was to follow Andre Weil’s hint at the deep connection between topological characteristic of a variety and its number theoretical aspect, i.e. its diplomatic aspects. Topologizing physics within a number theoretical framework seems to be an obvious characteristic of El Naschie’s E-infinity theory.
In the present communication we discuss a surprising relation between the totality of all Stein spaces, the compact and non-compact Lie symmetry groups on the one side and super string theory, path integral and the summing over dimensions procedure of E-infinity theory as well as the inverse fine structure constant = 137. The relation seems at first sight so bizarre and unreal that it is justifiably called the moonshine conjecture. In fact it has some similarity with the original moonshine conjecture and it is best to start by introducing the relation between the monster symmetry group and the coefficient of the j-function. The story starts when it was noticed that the minimal dimension for the monster is only one less than the first coefficient in the j-function. Thus we have D(min monistor) = b ̶ 1 = 196884 = 196883. The relation was clarified and the conjecture proven by Borcherds, a student of Conway (see El Naschie’s paper on the subject, CS&F, 32, (2007), pp. 383-387 as well as his paper “Symmetry groups prerequisite for E-infinity”, CS&F, 35, (2008), pp. 202-211 as well as “On the sporadic 196884-dimensional group, strings and E-infinity spacetime”, CS&F, 10(6), (1999), pp. 1103-1109.
We start by observing that the sum of the dimensions of the 17 two and three Stein spaces is exactly 686. This is equal 5 times 137 plus one. On the other hand the sum of the dimensions of the 12 compact and non-compact Lie symmetry groups is 1151. This is one short of 1152 which is 9 times 128, the electroweak inverse coupling of electromagnetics. This value (9)(128) = 1152 plays an important role in calculating the quantum states spectrum of the Heterotic string theory as can be seen in the excellent book of M. Kaku. Adding 686 to 1151 one finds 1837. Next we consider the total number of dimensions of the 12 non-compact Lie groups which comes to 1325. On the other hand the total number of the 8 non-compact 2 and 3 Stein spaces is given by 527. This is one short of Witten’s 528 states of a 5-Bran theory in 11 dimensions. Adding 527 and 1325 one finds 1852. The grand total is thus 1837 + 1852 = 3689. Now we embed 3689 in the ten dimensions of super strings and find that 3689 + 10 = 3699. Here comes the first incredible surprise because 3699 = (27)(137) = 3699 where = 137.
The second surprise in when we consider the “energy” stored in the “isometries” of the symmetry groups. Starting with the curvature of E-infinity spacetime = 26 + k we see that ( )( ) = (26 + k)(26 + k) which comes to 685.5. This is almost equal to 686 of the sum over all two and three Stein spaces. This is one of the best and simplest justifications ever for the theory of summing over symmetry group dimensions. Next we consider the intrinsic dimension of E7. This is dim E8(intrinsic) = 57. The transfinitely corrected compactified value is 57 + 1 + 3k 58.5. The energy is thus given by (58.54101966)2. This gives us 3427.050983. Here comes our next and final surprise for this communication. Dividing the energy by 25 one finds = 137.082039. The numerics indicate that there is indeed a deep connection between energy, symmetry and the electromagnetic fine structure constant. Members of the E-infinity group may like to think about a water tight proof for the above as well as pointing to more intricate relations.

E-infinity Group.

‘t Hooft-Veltman dimensional regularization implies E-infinity Cantorian spacetime (we told you so!).

25th January, 2011.
E-infinity communication No. 76

‘t Hooft-Veltman dimensional regularization implies E-infinity Cantorian spacetime (we told you so!).

At least for the last ten years or to be exact since his paper “’t Hooft dimensional regularization implies transfinite Heterotic string theory and dimensional transmutation”, Mohamed El Naschie maintained that the ‘t Hooft-Veltman method is not a mere mathematical trick but something physically more profound, namely a strong indication that our real physical quantum spacetime has a Cantorian-fractal geometry and topology. The paper which is included in the Proceedings of a conference also attended by ‘t Hooft is also published in various refined and modified versions in CS&F and other international journals. For ten years ‘t Hooft seemed unconvinced and as more often than not, skeptical which is a healthy scientific attitude in general although in this particular case not so clear why because the situation is rather straight forward and clear at least to those familiar with nonlinear dynamics and fractals. It was about three to four years ago that the great French mathematician A. Connes came to the same realization of El Naschie but of course in a far more mathematical and stringent way, namely that dimensional regularization is a clear indication of noncommutative geometry. In less abstract mathematical language, this means it is an indication of Cantorian-fractal spacetime geometry and topology.
In what follows we give a very short outline of El Naschie’s papers and direct the reader to the relevant literature where one can also find a copy of a letter from Richard Feynman talking about fractal curves in quantum mechanics to Prof. Garnet Ord. Rather than assuming like ‘t Hooft did that spacetime dimensionality is slightly less than 4 and approached the exact topological value from below, El Naschie de facto assumed it to be slightly larger than 4 and approached it from above. Interestingly he recovered two dimensions, namely 4 for the Menger-Urysohn topological dimension and a corresponding effective Hausdorff dimension slightly larger than 4, namely the famous number 4.23606799… He starts by assuming an unknown theory M with a spacetime dimension D(M) which may be used to replace the pole terms so that 1/(D ̶ 4) goes to → D(M). Replacing D by D(M) everywhere we find a quadratic equation (D(M) ̶ 4)D(M) = 1 with two solutions D(M) = 4 + ϕ3 = 4.236067977 and D(M) = ̶ ϕ3 = ̶ 0.236067977. Added together one finds the topological 4 of which 4.236067977 is the Hausdorff dimension of E-infinity spacetime. For details and discussions the reader is referred to the work of M.S. El Naschie and Alain Connes. Here we give first the classical literature on this extremely important evidence for the correctness of E-infinity and noncommutative geometry as being truly physical theories.

References:
1. M.S. El Naschie: ‘t Hooft Dimensional Regularization implies transfinite heterotic string theory and dimensional transmutation. Frontiers of Fundamental Physics, 4, p. 81-86. Edited by B.G. Sidharth and M.V. Altaisky. Kluwer Academic Publishers, New York (2001).
2. M.S. El Naschie: On 't Hooft dimensional regularization in E-infinity space. Chaos, Solitons & Fractals, Vol. 12, Issue 5, 4 January (2001), P. 851-858.
3. M.S. El Naschie: Dimensional Regularization implies transfinite heterotic string theory. Chaos, Solitons & Fractals, Vol. 12, (2001), P. 1299-1303.
4. A. Connes and M. Marcolli: Renormalization and motivic Galois theory. Int. Math Research Notices, (2004), No. 76, P. 4073-4091.

Now we come to the highly interesting new development regarding where Nobel Laureate Gerard ‘t Hootf currently stands vis-à-vis this fundamental and crucial connection between the method which he invented together with his thesis supervisor (Nobel Laureate M. Veltman and for which they shared the Nobel Prize) and fractal noncommutative spacetime geometry..
In 2009 in a book entitled “Approaches to Quantum Gravity” published by Cambridge Press and edited by D. Oriti, ‘t Hooft was asked by L. Crane the following question: “Do you think of dimensional regularization as a particularly effective trick or do you believe that it is a hint as to the fine structure of spacetime? In particular have you thought about the possibility of quantum spacetime having a non-integral Hausdorff dimension distinct from its topological dimension?” ;t Hooft answered as follows: “We thought of such a possibility. As far as the real world is concerned, dimensional regularization is nothing but a trick…. Veltman once thought there might be real physics in non-integer dimension, but he never got anywhere with that”.
Of course we, following El Naschie, beg to differ. Rather than quoting El Naschie to counter balance ‘t Hooft’s statement, we will quote A. Connes verbatim on this subject. Connes said: “We show that a careful investigation of dimensional regularization leads us to an interpretation that it is not just a formal procedure, but is an actual geometry….”. Somehow we hope that ‘t Hooft will validate the insight of El Naschie and Connes to the benefit of the development of unified theory of quantum gravity which we think is substantially complete in E-infinity, noncommutative geometry. Knowing ‘t Hooft’s genius, modesty and his ability to concede an error, we are very hopeful indeed.

E-infinity group.

Fake R(4) and exotic Milnor seven Spheres S(7) in the fuzzy or average knot Yang-Mills instantons of E-infinity

22nd January, 2011

E-infinity Communication No. 75

Fake R(4) and exotic Milnor seven Spheres S(7) in the fuzzy or average knot Yang-Mills instantons of E-infinity

Donaldson fake R(4) was considered in the work of El Naschie in E-infinity quite early on. A little later he considered the exotic Milnor seven spheres. In a paper published in CS&F6, 19 (2004), pp. 17-25 influenced by the work of El Naschie entitled “On Milnor seven dimensional sphere, El Naschie E-infinity theory and energy of a Bianchi universe” by Gamal Nashed of Ainshams University in Cairo, Egypt the particular relation between exotic geometry and E-infinity was discussed and an interesting summary was given in a very nice illustrative form in Fig. 1 on page 23. Also following El Naschie, Nashed made important use of the maximum sphere surface area and maximum sphere volume given in his figures 2 and 3 on page 24. El Naschie remarked that Nash formula gives a seven sphere for an Euclidean embedding of a one dimensional object because D = (0.5)(n)(3n ̶ 11) = 14/2 = 7. In addition he introduced the fractal seven dimensional sphere with the dimension 7 plus phi to the power 3, i.e. 7.23606799 which played a role in his fractal black hole theory. We recommend reading the paper entitled “Fractal black holes and information” by M.S .El Naschie, CS&F 29, (2006), pp. 23-35 and consider the explanation of Fig. 1 on page 25 and Fig. 3 on page 27. The most important conclusion of all these attempts for E-infinity research was the deep realization that the idea of moving from the factorial function to the gamma function should be generalized as done in moving from a topological dimension to a Hausdorff dimension. In fact doing this systematically one moves from classical quantum field theory to K-theory which is the mathematical realization of E-infinity theory. El Naschie proclaimed that Nottale’s idea of giving up classical differentiability and replacing it with Robinson’s non-standard analysis should be considered much deeper. El Naschie was familiar with non-standard analysis from his work on the canard of catastrophe theory. Therefore he was convinced that moving to Nottale’s frame work is a first step. The second step was to move to exotic ‘differentiability’. However this was not sufficient in his view and that is when he moved to point set geometry with cardinality equal to that of the continuum and that is how he arrived at Cantor sets and Cantorian spacetime of E-infinity theory.
It follows then that Yang-Mills theory must be modified to account for the true transfinite nature of high energy particle physics. This modification is what most probably inspired ‘t Hooft recently to include a dilaton field in his quantum field theory while hoping to refine classical calculus which is in principle of course possible when accepting some difficulties as the price. On the other hand random Cantor sets with their golden mean Hausdorff dimension offers natural quantization coupled with incredible computational ease due to the inbuilt golden mean number system which we explained in many previous communications. The usual mathematical way of thinking about fiber bundle theory is that we start with point set then move to a topological manifold, then smooth manifold, then geometric manifold, then bundle. We may start before point set and end beyond bundles. E-infinity is both the prior point set and the beyond bundle. Let us argue the case for an E-infinity action which is far more physical than ‘t Hooft’s S = 82 and at the same time much easier to hand, all apart from the unexpected fact that using E-infinity, hidden connection which would have passed totally unnoticed become obvious and trivially visible.
We reconsider again 82. This is obviously exactly 16 four dimensional sphere volumes. The volume of a four dimensional sphere with unit radius is as is well known, vol S(4) = 2/2. Consequently (16)( 2/2) = 8p2 = S, the action of ‘t Hooft’s Yang-Mills instanton. In E-infinity however we make a much richer relation when we take average everything. This average is a transfinite average. You could call it fuzzy values if you want. First we replace the volume of the spheres with the fuzzy hyperbolic volume of knot. We take K(82). For this knot the hyperbolic fuzzy volume is 5 ̶ ϕ4 where ϕ is the golden mean 0.618033989. Instead of taking 16 spheres we take an average of 16 + k = 16.18033989 knots of the 82 type. That way the total volume is exactly SF = ( /2) + 10 = 78.5419966. This is the value corresponding to 8p2 = 78.95683521 of ‘t Hooft. However we see here relations which we cannot see when using the classical analysis of ‘t Hooft. In particular we see that SF ̶ 10 when multiplied with 2 gives the exact theoretical inverse electromagnetic fine structure constant namely = 137 + ko = 137.082039325. From that a plethora of other relations follow, for instance (3 + ϕ)( ) = E8 E8 and remembering that E6 = 78 is already an integer approximation to S = (3)(26) = 78 we see that the net of interrelations with the exceptional Lie symmetry groups and not only SO(3) where we noted in a previous communication that volume SO(3) = 8p2. In general we can say the El Naschie fuzzy K3 is a K-theory K3 and that the transfinite Feynman diagrams of E-infinity are the equivalent of Feynman motives which was developed recently. Thus E-infinity could be called K-infinity and El Naschie’s fuzzy golden field theory is nothing else but a Grothendieck motives applied to the theory behind the standard model of high energy physics. In this sense Mohamed El Naschie was deadly right that ‘t Hooft’s dimensional regularization implies E-infinity spacetime which means noncommutative physical spacetime. The same conclusion was recently made by A. Connes. It is interesting to note that ‘t Hooft did not agree initially but he may reconsider the situation in view of the compelling E-infinity results.

E-infinity Group.

NAMING E-INFINITY:

20th January, 2011.
E-infinity communication No. 74

NAMING E-INFINITY:
Some aphorisms, quotations and remarkable historical events connected to the science of the infinite transfinite set theory and the theory of dimensions as well as the relation to religion and God.

1. In the beginning was the word and the word was with God and the word was God (The Gospel according to St. John, first verse).
2. “I have no need for this hypothesis”: This was the answer of Laplace to Napoleon’s question about why God did not appear in his work.
3. Philosophy is a battle against the bewitchment of our intelligence by means of language. (Wittgenstein – Philosophical Investigation).
4. Heisenberg “discontinuous” quantum mechanics was a giant leap forwards. Schrödinger’s continuity illusion of his differential equation was effectively a step backwards. (Mohamed El Naschie rephrasing words of A. Connes).
5. Not how the world is is the mystical but that it is. (Wittgenstein – Tractatus).
6. The name that can be named is not the eternal name. (Tao Te Ching).
7. In a fractal spacetime setting we must replace differentiation and integration with Weyl- Suslin scaling. In E-infinity the main scaling as well as the main renormalization semi group is the golden mean scaling and the golden mean renormalization group of M. Feigenbaum. (Mohamed El Naschie about E-infinity).
8. Dimension is a scale dependent phenomenon. (B.G. Sidharth about fractal spacetime).
9. Neither K. Gödel’s proof of the consistency of the continuum hypothesis with the axioms of set theory, nor mine of its independence from them was the final answer….but I think there is no answer other than the answer that it is undecidable. (P. Cohen).
10. The Cantor space c is the unique up to homeomorphism perfect non-empty compact, zero-dimensional space. (Text book definition of the Cantor space 2 to the power of Baire space used by El Naschie in E-infinity).
11. E-infinity theory is a weighted Borel hierarchy. (El Naschie following an idea due to J.A. Wheeler).
12. In Batz we walked by the sea …. It was there that Pavel Urysohn wrote his famous paper on countable connected Hausdorff spaces …. On August 17 out for another swim …. Urysohn was catapulted by a wave directly onto the rocks …. Urysohn was buried in Batz-sur-Mer… (Prof. Pavel Alexandrov remembering Pavel Urysohn, the discoverer (or inventor) of the transfinite deductive theory of dimensions and that the empty set has a topological dimension equal minus one ( ̶ 1)).
13. The Moscow School of Mathematics founded by Egorov and Luzin is one of the most important sources of information and inspiration for E-infinity theory of high energy physics. On the deep level of trying to understand quantum mechanics it is at best misguided and at worse childish and naïve to make a separation between mathematics and physics. (Mohamed El Naschie in one of his recent lectures 2009-2010).
14. For my work the most important members of the Russian (Moscow School of Mathematics) are Urysohn, Kolmogorov, Suslin, Gel’fand, Arnold, Alexandrov, Shinchin, Pontryagin and Sinai. (Mohamed El Naschie 2009).
15. Prof. Pavel Florensky was convinced that the nineteenth century was an intellectual disaster …. because of the concept of ‘continuity’ …. because of the strength of differential calculus with many practical application problems that could not be solved this way were ignored – essentially discontinuous phenomena (such as quantum mechanics). Florensky (member of the Moscow School of Mathematics) wanted to restore discontinuity to its rightful place in the “Weltanschaung”. (Graham-Kantor-El Naschie 2010).
16. Everything visible is connected to the invisible … the sensible to the nonsensible. Perhaps the thinkable to the unthinkable. (Novalis – fragment).
17. Anaxagoras conceived the infinite in the same way as Anaximander. He called the infinite (Apeiron) which is the primodal mixture of chaos. (P. Zellini).
18. von Neumann’s formulation of the question was adopted by Kurt Gödel. There, as in E- infinity, we distinguish between set and class. (Mohamed El Naschie).
19. In E-infinity we solve a measure problem by summing over infinitely many but countable Cantor sets. However every Cantor set has uncountably infinitely many points. That is how we arrive at finite expression for a completely wild situation as far as computations are concerned. (Mohamed El Naschie 2009).
20. If we went back to the most perfect image of the word soaring to the level of an invisible being, it would mean recalling the Hindu Vāc. In E-infinity the VAK is the vague attractor of Kolmogorov which is conjectured by R. Thom to be the stationary states of quantum mechanics and used by Mohamed El Naschie to calculate the mass spectrum of elementary high energy particles. The Logos of the Greeks is analogous to the Hindu Vāc. (P. Zellini and Mohamed El Naschie).
21. There is strictly speaking no such thing as mathematical proof. (G.H. Hardy).
22. It is obvious then that the mathematical problem of the infinite is automatically projected into the moral sphere …. This is clear in the work of F. Nietzsche and R. Musil. (P. Zellini and Mohamed El Naschie).
23. My fuzzy K3 as well as all the transfinitely corrected Betti numbers, Euler invariant and instanton numbers and curvatures as well as invariant dimensions are intuitive extensions of cohomology to the theory of fiber bundles. Without knowing, not being a mathematician, I extended cohomology in the same way as Grothendieck and Atiyah extended cohomology using K-theory. In other words my K3 and all the fuzzy manifolds are a product of a non-declared K-theory which was christened transfinite E-infinity theory. (Mohamed El Naschie 2010).

E-infinity Group.

K-theory (of Grothendieck) as a theoretical framework of E-infinity and extension of the Riemann-Roch index theorem

19th January, 2011.
E-infinity Communication No. 73
K-theory (of Grothendieck) as a theoretical framework of E-infinity and extension of the Riemann-Roch index theorem

As is well known, K-theory as well as the work of M. Atiyah are extensions of the celebrated Riemann-Roch index theorem. K-theory is largely due to the mathematician (almost cult figure) Alexander Grothendieck (born 1928, Field Medal 1966) who presently lives as a recluse and who declined receiving the Crafoord Prize or in fact any prizes since 1988 because he was disillusioned with the scientific community and society at large. He even resigned his prestigious position at the very prestigious IHES in France. Those interested in the very deep mathematical foundations of E-infinity theory should consider the role of the Riemann-Roch index theorem in the work of Mohamed El Naschie. During his Cambridge time, El Naschie published in 2000 a paper entitled “On the unification of Heterotic strings, M-theory and E-infinity theory”, CS&F, Vol. 11 (2000), pp. 2397-2408. In this paper El Naschie dealt with various mathematical aspects including the Riemann-Roch theorem which he evaluated on page 2406 for n = 4 and q = 1 + ϕ where = is the golden mean and found that the index is
Index = (2 n ̶ 1)(q ̶ 1)
= (7)(ϕ)
= 4.3262.
Numerical evaluation taking 1200 times by Castro and Granik using gamma function to find the average Hausdorff dimension of E-infinity spacetime gave 4.32 which is very close to the above exact theoretical value. Now the exact Hausdorff dimension of E-infinity spacetime core is well established and is given by 4.236067977 which is quite close to both the Riemann-Roch index as well as the gamma function approximation using these functions as weights in analogy to the golden mean weight of the E-infinity derivation. The reason for this proximity between 4.23606799 of E-infinity and 4.3262 of the Riemann-Roch index and the 4.32 of the gamma function numerical approximation is explained as follows by Prof. El Naschie: As observed by André Weil there is a deep connection between the topological characteristic of a variety and its number theoretic aspect and this blend between numbers and topology is a main line in E-infinity theory. The same is true for K-theory as we will discuss shortly. In a second paper published a year later in 2001 El Naschie returned to Riemann-Roch, this time in connection to moduli spaces. The paper is entitled “Remarks to moduli spaces, virtual dimensions and Heterotic strings”, CS&F, 12 (2001), pp. 1607-1610. He considers on page 1607 moduli spaces of bundles and derives the dimensions of super strings in integer form. Subsequently he derives the exact transfinite values including the Euler characteristic of E-infinity, namely 26 + k = 26.18033989. We repeatedly stressed that El Naschie showed that Penrose tiling is the prototype example for both Connes’ noncommutative geometry as well as E-infinity theory.
Let us look a little closer at the K-theory connection of Penrose tiling. E-infinity researchers are familiar with the construction of a Cantor set. However only those working in deterministic chaos will be familiar with Steve Smale’s horseshoe. From this construction of a horseshoe one will realize the relation between numerical sequences and a Cantor set. The Smale construction as a mapping between a Cantor set and infinite sequences of zeros and ones. There is an important result permitting to parameterize a Penrose tiling with a set K of infinite sequence of zero and one satisfying certain conditions which will turn out to describe the Connes dimensional function or El Naschie’s bijection formula. This K is a compact and of course totally disjoint space homeomorphic to a Cantor set and we also have a relation of equivalence on R. The space x of Penrose tiling is the quotient space x = K/R. In E-infinity theory, it is easily shown that the dimension of K is given by the dimension of the real line plus the dimension of the resulting Cantor set while the dimension of R is equal to that of the real minus the dimension of the Cantor set. Assuming a random Cantor set with a Hausdorff dimension equal ϕ as per Mauldin-Williams theorem we find that
dim x = dim K/dim R
= (1 + ϕ)/(1 ̶ ϕ)
= 4 + the golden mean to power 3
= 4.236067977.
This is the well known Hausdorff dimension of E-infinity and corresponds to a 4 Menger-Urysohn topological dimension. For further more technical discussion of the relation of Penrose tiling, noncommutative geometry and K-theory, the reader is referred to the nice mathematical papers of D. Bigatti for instance. El Naschie was frequently asked ‘what is the simplest most direct way to start making contact with E-infinity theory?’ Here is his answer. The deep problems are the continuum, the denumerable infinity and the not denumerable infinity which is related to the cardinality of the continuum. In E-infinity we have our cake and eat it so to speak. We have countably infinite numbers of Cantor sets which we sum over. In each Cantor set we have uncountably infinite Cantor points. Thus we are counting not over points but over equivalence classes. In his latest work to salvage quantum field theory ‘t Hooft adopted this view point and introduced a dilaton field akin to that of El Naschie’s compactified Klein modular space. This space has an inbuilt dilaton field and is homeomorphic to Penrose fractal tiling, i.e. to the x = K/R space. For a new theory connecting string states with the theory of instantons using x = K/R, the reader is referred to A. Elokaby’s paper “On the deep connection between instantons and string states encoded in Klein’s modular space”, CS&F, 42 (2009), pp. 303-305.
E-infinity Group.

Category theory in El Naschie’s E-infinity

19th January, 2011.
E-infinity communication No. 72

Category theory in El Naschie’s E-infinity

Category is mentioned explicitly in connection with ribbon category in El Naschie’s important paper on quantum groups. Utilizing results connected to tensor categories he arrives at a quantum dimension which takes the exact value of 4 when q is set equal to the inverse of the golden mean. The theory is strongly connected to knot theory, Hopf algebra and of course tangle category. The close connection to noncommutative geometry and E-infinity becomes trivially obvious when setting the Eigenvalue q = ½ in El Naschie’s expectation value of the Hausdorff dimension which is identical to A. Connes’ noncommutative dimension and finding that the dimension in this case is exactly 4 which corresponds to exactly 4 for the quantum dimension provided q = ½ is replaced by q equal to the inverse golden mean. The duality between the two dimensions becomes apparent when one realizes that for q equal the golden mean the noncommutativity Hausdorff dimension becomes equal to the famous dimension of an infinitely fractal four dimensional space, namely 4.23606799. This situation is explained in detail in “Quantum groups and Hamiltonian sets on a nuclear spacetime Cantorian manifold”, CS&F, Vol. 10(7), (1999), pp. 1251-1256. In general however El Naschie does not make explicit reference to to n-categories nor its coffee shop direct by Dr. J. Baez. The fact is E-infinity results are all obtained by elementary methods but could have been obtained using categories theory. It is also a fact that Prof. El Naschie did not refer to many relevant papers in category theory which may have ignited the unprecedented anger of Dr. John Baez of Riverside University, California. In fact we in E-infinity feel that we should consider n-categories in future despite the claim of some that it is simply too abstract to be physics. This is a claim which we do not accept in principle because at the deep level of trying to understand the building blocks of nature and figure out the meaning of the quantum, any separation between physics and mathematics is artificial and in general mathematical, logical reasoning is paramount.
E-infinity Group.

Fractal spacetime – some historical remarks to fractal spacetime deniers

18th January, 2011.
E-infinity communication No. 71
Fractal spacetime – some historical remarks to fractal spacetime deniers
We occasionally hear that fractal spacetime has nothing to do with quantum physics. These people may even go as far as saying that fractals have nothing to do with physics. Of course everyone is free to think what he likes but the unpleasant surprise comes when one finds out that most of those who make these statements are mentors of mainly those who work on fractals in physics although they are extremely inventive when it comes to giving different names to these things which we call scale relativity, Cantorian spacetime and fractal spacetime. This point could not be explained by mathematically based science. It needs the tool of social sciences and psychoanalysis. It could of course be far more straight forward than that and is mainly related to science policy and science funding coupled to the very scarce resources available to scientific research in general and theoretical and mathematical physics in particular. This subject is not particularly the source of joy to most of us and we will not dwell on it any further. However it is important to recall some important facts about the history and substance of fractal spacetime theory.
The first comprehensive paper published in an international journal with the title Fractal Spacetime was by the English-Canadian Garnet Ord who discussed this subject with Nobel Laureate Richard Feynman and was strongly influenced by Feynman’s views on the subject. Ord’s paper was published in 1983 in one of the journal of the Institute of Physics, namely J. Phys. A: Math. Gen., 16 (1983), pp. 1869-1884 and was entitled “Fractal space-time: a geometric analogue of relativistic quantum mechanics”. A little later and seemingly independently, a young but well known French astrophysicist Laurent Nottale published in 1989 a paper entitled “Fractals and quantum theory of spacetime” in Int. J. Mod. Physics !, 4, (1989), pp. 5047-5117. This was a sequel and generalization of his 1984 paper with J. Schneider entitled “Fractals and nonstandard analysis”, J. Math. Phys., 25, (1984), pp. 1296-1300. It seems however that Nottale had the same ideas as Ord at nearly the same time but for some reason he could not publish his paper except later on. His famous book “Fractal Space-Time and Microphysics” published by World Scientific in 1993 made up for this delay. Mohamed El Naschie on the other hand is at least ten years older than Nottale and Ord and came to fractals in quantum mechanics via nonlinear dynamics. El Naschie obtained his Ph.D. in engineering and started working in physics much later after having reached the position of Full Professor of structural engineering. All the same he was keenly interested in philosophy and science in general and quantum mechanics in particular. An encounter as a student of engineering in Hannover, Germany with the work and personality of Werner Heisenberg and K. von Weizacker changed his scientific interests completely. However serious work had to be postponed until he became aware of the mathematics of Cantor sets and their relation with number theory, topology and symmetry groups. El Naschie moved from applied mechanics to applied nonlinear dynamics, chaos, singularity theories and fractals to quantum mechanics and finally high energy physics. His first efforts were in understanding turbulence via fractals and Cantor sets in a way similar to the work of Kolmogorov. He was also familiar with stability theory of Poincaré, Köiter and Andronov as well as R. Thom’s catastrophe theory which helped him to move from engineering to physics. Turbulence was the first problem which Heisenberg tackled but could not solve. El Naschie used turbulence as a paradigm for vacuum fluctuation following Wheeler. The first paper by El Naschie which was relevant to fractal spacetime indirectly was the 1991 paper entitled “Multi-dimensional Cantor-like sets and ergodic behavior” in Speculations in Science & Technology, Vol. 15, No. 2, pp. 138-142. This was followed by two papers with direct relevance to quantum mechanics. First “Quantum mechanics and the possibility of a Cantorian spacetime” published in Chaos, Solitons & Fractals, Vol. 1, No. 5 (1991), pp. 485-487. This was followed by “Multi-dimensional Cantor sets in classical and quantum mechanics”, CS&F, Vol. 2, No. 2, (1992), pp. 211-220. After that an important paper in computational and applied mechanics was published entitled “Physics-like mathematics in four dimensions – implications for classical and quantum mechanics”, Computational and Applied Math II. W.F. Ames and P. van der Houwen (Editors), Elsevier (North Holland), (1992), IMACS.
Since these efforts by Ord, Nottale and El Naschie to establish a new field, namely the field of quantum-fractal physics, many papers were published intermittently. The author of these papers seems to be unaware of previous similar publications by the trio Ord-Nottale-El Naschie or for some reason or another chose not to make reference to these contributions. Whilst this was maybe understandable before the world wide web, in the meantime it is not understandable at all. In what follows we list some of the interesting work on fractal quantum and high energy physics which shows that this field is vibrant and is becoming unusually combative in a not very pleasant manner, at least occasionally.
There are many papers in quantum mechanics which mention the concept of the Hausdorff dimension as well as the Hausdorff dimension of a quantum path. These papers are relatively well known from the work of scientists like Parisi and will not be mentioned here. We may start by mentioning the paper of M. Wellner “Evidence for a Yang-Mills fractal” in Physical Review Letters, Vol. 68, No. 12 (1992), pp. 1811-1813. An earlier similar paper is “Onset of chaos in time-dependent spherically symmetric SU(2) Yang-Mills theory”, Physical Review D, Vol. 41, No. 6, (1990), pp. 1983-1988. A 2010 paper by S. Carlip entitled “The small scale structure of spacetime” (arXiv: 1009.1136VL[gr-qc]6 Sept 2010 makes the usual remark about Wheeler foam, mentions Loll and Amjborn’s work and completely overlooks the work of Ord-Nottale and their associates. Similar remarks apply to the work of Fotini Markopoulou. The paper with the almost identical title of that of Nottale and Ord “Fractal space-time and black-body radiation” published in Astrophysics and Space Science, 124 (1986), pp. 203-205 by A. Grassi, G. Sironi and G. Strini is also oblivious to the work of Ord and Nottale. Strangely Benedetti’s paper which uses the same quantum group concept of El Naschie did not mention the work of Ord, Nottale or El Naschie. The paper on fractal spacetime by O. Lauscher and M. Reuter seems to be totally unaware of the work of Nottale and Ord let alone El Naschie. Equally very strange is the absence of any reference to Nottale or Ord in the paper “Fractal geometry of quantum spacetime ar large scales” by I. Antoniadis, P. Mazier and E. Mottola although the first author works in France where L. Nottale is well known. We have not mentioned more than one percent of the large body of literature on fractals in quantum mechanics, relativity and quantum gravity. This shows that fractals are indeed relevant and may be too relevant to the extent that competition is not only fierce but slightly unfair to say the least.
We have not mentioned fractals in all other fields of physics. There are more publications on fractals in physics than on quantum mechanics when you consider that fractals were discovered in physics no more than 25 years ago while quantum mechanics is with us since more than 80 years. For all these reasons we think that the hard work our group had done and continues to do is more than justified and worthwhile.
In conclusion we should mention one of the most important recent papers by G.N. Ord “Quantum mechanics in two dimensional spacetime: What is a wave function”, published in Annals of Physics, 324 (2009), pp. 1211-1218. It may also be interesting to give Mohamed El Naschie’s answer to Ord’s question using his transfinite set theory. A wave function is an empty set with a topological Menger-Urysohn dimension equal minus one ( ̶ 1) which is the surface area or the neighborhood cobordism of the quantum particle which is the zero set. Two cultures divided by a common language, namely mathematics.
E-infinity Group.