Jun 19, 2011
May 15, 2011
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.
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.
Mar 24, 2011
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.
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
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.
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
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
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
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
Mar 2, 2011
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.
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
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
Jan 27, 2011
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.
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.
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.
Jan 23, 2011
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 = 82 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 82. 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.
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 = 82 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 82. 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.
Jan 22, 2011
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.
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.
Jan 20, 2011
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.
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.
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.
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.
On ‘t Hooft’s recent work and the development of Yang-Mills instanton via El Naschie’s E-infinity theory
17th January, 2011.
E-infinity communication No. 70
On ‘t Hooft’s recent work and the development of Yang-Mills instanton via El Naschie’s E-infinity theory
E-infinity theory gave considerable attention quite early on to integrating Yang-Mills theory and ‘t Hooft’s instantons into its scale invariant fractal frame work. It is probably a fact to say that Mohamed El Naschie’s relations to ‘r Hooft must have stimulated him to attempt this integration. On the other hand it must equally be a fact that the recent work of ‘t Hooft in which he introduces a dilaton field to the Lagrangian of quantum field theory was motivated primarily by the realization which he must have gained by reading El Naschie’s work on a “fractal-Cantorian” instanton.
In the present short communication we would like to direct the reader to the relevant sources from which he can come to the above very important conclusion. Dilaton is used extensively in string theory. However it would be a misconception to think that ‘t Hooft used it in the same way. That would be underestimating the importance of ‘t Hooft’s proposal which in our view is nothing less than “fractalizing” the frame work of classical quantum field theory.
El Naschie felt relatively late that Lagrangian formulation in particle physics was wrongly elevated to the status not of merely an alternative to the differential equations but to a supreme and the only right way to do theoretical particle physics. El Naschie was originally trained in the rigorous tradition of the classical theory of elasticity and Koiter’s general theory of elastic stability where a potential energy formulation goes hand in hand with differential equations and under estimated the deep seated prejudice of the community of particle physics, particularly those working in quantum field theory and that they regard a Lagrangian as sacred. We do not know exactly when he started addressing these problems but latest in 2006 he wrote a paper “Intermediate prerequisites for E-infinity theory”, CS&F, 30 (2006), pp. 622-628 in which he attempted in section 5 to clarify the issue of a Lagrangian in E-infinity theory. It is in this paper where he introduced the concept of a fuzzy Lagrangian and a fuzzy or average instanton. In the above mentioned paper as well as somewhat earlier and later papers, El Naschie made a profound discovery liking string states with instantons. The most elementary form of this discovery was stated in earlier communications and is essentially the equality of multiplying left and right movers to obtain the first massless level of Heterotic string particle-like states, namely N(0) = (504)(16) = 8064 as well as multiplying the degree of freedom of the noncompactified holographic boundary given by Klein-modular curve (336) and the instanton density of K3 Kähler manifold (24) so that we find the same value of No namely N(0) = (336)(24) = 8064. A former student of El Naschie, Dr. A. Elokaby explained the above theory in a very lucid paper “On the deep connection between instantons and string states encoded in Klein’s modular curve”, CS&F, 42 (2009), pp. 303-305. Having come that far one can easily find the exact N(0) by remembering that the exact instanton density for a fuzzy K3 Kähler is not 24 but 24 + 2 + k, which means 26 + k = 26.18033989. Similarly the 336 should be 336 + 16k = 338.8854382. Consequently, No is No = (338.8854382)(26.18033989) = 8872.135957. This is 808.1359569 states or instantons more than the classical 8064 found first by Green, Schwarz and Witten.
We could do the same for Heterotic strings by noting that 504 should be (4)( ) = 548.328157 where = 137 + ko which is the sum of all the exceptional Lie symmetry groups in the E-line (548), plus 0.328157 transfinite or fuzziness added to them. In addition the 16 should be changed to 16 + k = 16.18033989. That way one finds N(0) = (4 )(16 + k) = 8872.135956 exactly as before. El Naschie reasoned that the same procedure could be applied to ‘t Hooft’s famous instanton result, namely I(instanton) = = 78.95683521. Since the instanton density 26 + k must remain the same, all that changed is the dimensionality of the holographic boundary. In the instanton interpretation of the Heterotic strings theory leading to No = 8064 the boundary had 336 degrees of freedom (or 338.885 ≃ 339 in the compactified fuzzy exact case). For ‘t Hooft instanton our space is only 3 + 1 dimensional as is the case in classical quantum field theory. Therefore the holographic boundary must be d = (3 + 1) ̶ 1 = 3. The exact fully Yang-Mills instanton value must therefore be <> = (3)(26 + k) = 78.54101967. This may also be expressed as ( /2) + 10 = 78.54101967. Note that the classical value of ‘t Hooft is slightly larger by 0.41581554. Notice also that if we take only the integer value, namely 26 one finds <> = (3)(26) = 78. This is exactly equal to the dimension of the exceptional E6 symmetry group of the exceptional E-line. Another note worthy observation is that using the classical instanton density 24 instead of 26 one finds <> = (3)(24) = 72. This is the roots number of E6. In other words the roots number is the lowest approximation for obtaining a value for I.
Summing up we can say all the values 72, 78 as well as are equal to the volume of 16 five dimensional Ball or four dimensional spheres. The only exact value is the fuzzy value obtained by El Naschie, namely <> = (3)(26 + k) = 78.54101967. These are the elementary and obvious but deep results of Prof. Mohamed El Naschie which prompted ‘t Hooft’s courageous and mathematically elaborate and justified attempt to patch up classical quantum field theory. We, the proponents of E-infinity, wish Prof. ‘t Hooft to succeed because it is firstly good for science and secondly because it is in the interest of E-infinity’s world wide acceptance.
In conclusion we draw the attention of the reader to various important publications on the subject:
1. The relation between Yang-Mills instanton and the exceptional groups is given by El Naschie in CS&F, 38 (2008), pp. 925-927.
2. A useful dictionary translating gauge field language to fiber bundle language may be found in CS&F, 30, (2006), pp. 656-663.
3. The Weyl scaling of E-infinity is explained in CS&F, 41 (2009), pp. 869-874.
4. The relation between ‘t Hooft’s work and that of El Naschie is discussed in CS&F, 38, (2008), pp. 980-985.
5. It should be noted that Prof. El Naschie is by no means the first to consider a connection between Yang-Mills theory and fractals. In fact already in 1992 there was a paper in Physical Review Letters (23 March, 1992, Vol. 68, No. 12, pp. 1811-1813) with the title “Evidence for a Yang-Mills fractal” in which numerical evidence for fractals in a Yang-Mills system in (2 + 1) dimension was observed. Even before that T. Kawabe and S. Ohta from Japan wrote a paper on Chaos in Yang-Mills theory “Onset of chaos in time-dependent spherically symmetric SU(2) Yang-Mills theory”, Physical Review D, Vol. 41, No. 6, 15 March, 1990, pp. 1983-1988.
E-infinity communication No. 70
On ‘t Hooft’s recent work and the development of Yang-Mills instanton via El Naschie’s E-infinity theory
E-infinity theory gave considerable attention quite early on to integrating Yang-Mills theory and ‘t Hooft’s instantons into its scale invariant fractal frame work. It is probably a fact to say that Mohamed El Naschie’s relations to ‘r Hooft must have stimulated him to attempt this integration. On the other hand it must equally be a fact that the recent work of ‘t Hooft in which he introduces a dilaton field to the Lagrangian of quantum field theory was motivated primarily by the realization which he must have gained by reading El Naschie’s work on a “fractal-Cantorian” instanton.
In the present short communication we would like to direct the reader to the relevant sources from which he can come to the above very important conclusion. Dilaton is used extensively in string theory. However it would be a misconception to think that ‘t Hooft used it in the same way. That would be underestimating the importance of ‘t Hooft’s proposal which in our view is nothing less than “fractalizing” the frame work of classical quantum field theory.
El Naschie felt relatively late that Lagrangian formulation in particle physics was wrongly elevated to the status not of merely an alternative to the differential equations but to a supreme and the only right way to do theoretical particle physics. El Naschie was originally trained in the rigorous tradition of the classical theory of elasticity and Koiter’s general theory of elastic stability where a potential energy formulation goes hand in hand with differential equations and under estimated the deep seated prejudice of the community of particle physics, particularly those working in quantum field theory and that they regard a Lagrangian as sacred. We do not know exactly when he started addressing these problems but latest in 2006 he wrote a paper “Intermediate prerequisites for E-infinity theory”, CS&F, 30 (2006), pp. 622-628 in which he attempted in section 5 to clarify the issue of a Lagrangian in E-infinity theory. It is in this paper where he introduced the concept of a fuzzy Lagrangian and a fuzzy or average instanton. In the above mentioned paper as well as somewhat earlier and later papers, El Naschie made a profound discovery liking string states with instantons. The most elementary form of this discovery was stated in earlier communications and is essentially the equality of multiplying left and right movers to obtain the first massless level of Heterotic string particle-like states, namely N(0) = (504)(16) = 8064 as well as multiplying the degree of freedom of the noncompactified holographic boundary given by Klein-modular curve (336) and the instanton density of K3 Kähler manifold (24) so that we find the same value of No namely N(0) = (336)(24) = 8064. A former student of El Naschie, Dr. A. Elokaby explained the above theory in a very lucid paper “On the deep connection between instantons and string states encoded in Klein’s modular curve”, CS&F, 42 (2009), pp. 303-305. Having come that far one can easily find the exact N(0) by remembering that the exact instanton density for a fuzzy K3 Kähler is not 24 but 24 + 2 + k, which means 26 + k = 26.18033989. Similarly the 336 should be 336 + 16k = 338.8854382. Consequently, No is No = (338.8854382)(26.18033989) = 8872.135957. This is 808.1359569 states or instantons more than the classical 8064 found first by Green, Schwarz and Witten.
We could do the same for Heterotic strings by noting that 504 should be (4)( ) = 548.328157 where = 137 + ko which is the sum of all the exceptional Lie symmetry groups in the E-line (548), plus 0.328157 transfinite or fuzziness added to them. In addition the 16 should be changed to 16 + k = 16.18033989. That way one finds N(0) = (4 )(16 + k) = 8872.135956 exactly as before. El Naschie reasoned that the same procedure could be applied to ‘t Hooft’s famous instanton result, namely I(instanton) = = 78.95683521. Since the instanton density 26 + k must remain the same, all that changed is the dimensionality of the holographic boundary. In the instanton interpretation of the Heterotic strings theory leading to No = 8064 the boundary had 336 degrees of freedom (or 338.885 ≃ 339 in the compactified fuzzy exact case). For ‘t Hooft instanton our space is only 3 + 1 dimensional as is the case in classical quantum field theory. Therefore the holographic boundary must be d = (3 + 1) ̶ 1 = 3. The exact fully Yang-Mills instanton value must therefore be <> = (3)(26 + k) = 78.54101967. This may also be expressed as ( /2) + 10 = 78.54101967. Note that the classical value of ‘t Hooft is slightly larger by 0.41581554. Notice also that if we take only the integer value, namely 26 one finds <> = (3)(26) = 78. This is exactly equal to the dimension of the exceptional E6 symmetry group of the exceptional E-line. Another note worthy observation is that using the classical instanton density 24 instead of 26 one finds <> = (3)(24) = 72. This is the roots number of E6. In other words the roots number is the lowest approximation for obtaining a value for I.
Summing up we can say all the values 72, 78 as well as are equal to the volume of 16 five dimensional Ball or four dimensional spheres. The only exact value is the fuzzy value obtained by El Naschie, namely <> = (3)(26 + k) = 78.54101967. These are the elementary and obvious but deep results of Prof. Mohamed El Naschie which prompted ‘t Hooft’s courageous and mathematically elaborate and justified attempt to patch up classical quantum field theory. We, the proponents of E-infinity, wish Prof. ‘t Hooft to succeed because it is firstly good for science and secondly because it is in the interest of E-infinity’s world wide acceptance.
In conclusion we draw the attention of the reader to various important publications on the subject:
1. The relation between Yang-Mills instanton and the exceptional groups is given by El Naschie in CS&F, 38 (2008), pp. 925-927.
2. A useful dictionary translating gauge field language to fiber bundle language may be found in CS&F, 30, (2006), pp. 656-663.
3. The Weyl scaling of E-infinity is explained in CS&F, 41 (2009), pp. 869-874.
4. The relation between ‘t Hooft’s work and that of El Naschie is discussed in CS&F, 38, (2008), pp. 980-985.
5. It should be noted that Prof. El Naschie is by no means the first to consider a connection between Yang-Mills theory and fractals. In fact already in 1992 there was a paper in Physical Review Letters (23 March, 1992, Vol. 68, No. 12, pp. 1811-1813) with the title “Evidence for a Yang-Mills fractal” in which numerical evidence for fractals in a Yang-Mills system in (2 + 1) dimension was observed. Even before that T. Kawabe and S. Ohta from Japan wrote a paper on Chaos in Yang-Mills theory “Onset of chaos in time-dependent spherically symmetric SU(2) Yang-Mills theory”, Physical Review D, Vol. 41, No. 6, 15 March, 1990, pp. 1983-1988.
Physics and metaphysics in transfinite set theory and E-infinity
14th January, 2011.
E-infinity communication No. 69
Physics and metaphysics in transfinite set theory and E-infinity
Before anyone jumps to conclusions and misconstrues what we want to say here, we should better start by clarifying first two points. Metaphysics has nothing to do here with the supernatural. In fact at the time of G. Cantor for instance, metaphysics meant more or less fundamental science or stringent mathematical physics dealing with foundational issues. Naturally questions dealing with philosophical issues were included. Equally naturally, the role of religion and the existence of a higher being referred to as Deity were touched upon. In fact fundamental science used to be called natural philosophy and even until our time universities awarded degrees in all possible fields but they are all called doctor of philosophy and this could be in engineering rather than philosophy.
The second point is regarding Mohamed El Naschie who is deeply religious and counts himself as being a Moslem without any doubt but that is all. He is deeply distrustful of all forms of organized religion as well as mingling science, politics and religion in a demagogical way. Perhaps we should give two examples of his writing on this subject before we go any further. At the end of his paper “A note on quantum field theory and P-Branes in n dimensions published in CS&F, Vol. 10 (8), pp. 1413-1417 (1999) he wrote “It may seem that by reducing the contra intuitive physics of quantum mechanics to the contra intuitive geometry of the space of quantum mechanics, nothing is gained. However this is not the right way of looking at the problem. The physics of quantum mechanics has frequently opened the door to the wrong kind of mysticism which is wholly foreign to scientific thinking. By contrast all of the mind-baffling and contra intuitive results of modern geometry in hyperspaces are the end product of the most stringent rational thinking and leave no place for uncalled-for mystical beliefs what so ever. Thus we think a great deal is achieved by viewing quantum physics in what we think is the right geometrical way proposed here”. I think Prof. El Naschie made it clear where he stands. An additional quotation at the end of another paper by him may help to dispel any lingering doubts about a basically renaissance Egyptian who tolerates religious beliefs of the personal type. In CS&F, Vol. 10(2-3), pp. 567-580 (1999), at the end of a paper entitled ‘Nuclear spacetime theories, superstrings, monster groups and application’ El Naschie says “Philosophically inclined theoretical physicists have frequently wondered if, in creating the world, God had any choice in determining it. Assuming that our hypothesis about E-infinity geometry is correct, then it would seem to us that once God had created logic, he had as good as no option but to create the universe as we see it now.”
Having said all that, the situation is slightly different when we come to consider transfinite set theory, the continuum hypothesis, the axiom of choice and particularly the meaning of infinity. There is no doubt that there are two kinds of personalities among serious scientists. Those who do not believe in any form of creation or any form of non-materialistic existence and those who believe in non-materialistic existence as well as the possibility of creation. The first kind is more or less like Kronecker, the famous German mathematician. The second kind is more like G. Cantor. There are people who may be in between. In E-Infinity Mohamed El Naschie finds that the spiritual value and even religious beliefs can be an inspiration to solving extremely difficult problems. In general, one has to keep religion out of science. However there is one exception where we must be tolerant and that is when someone identifies the Almighty with the set of all sets. In fact El Naschie believes that the concept of a set of all sets leads to severe contradictions in set theory. He accepts the class of all classes but not the set of all sets.
If we search for a deep difference between Kronecker and ‘t Hooft on the one side and Cantor and El Naschie on the other side, then the line separating them scientifically is that Cantor and El Naschie believe in the infinite and yes they believe in God. This is indirectly reflected in the philosophical depth of their work where there is no more any artificial separation between physics and mathematics. In fact we recently spoke to El Naschie and he said in a recent interview that God is without doubt a mathematician and definitely not a theoretical physicist. In the same way El Naschie regards the mathematician Prof. A. Connes as the deepest physicist working among us today. In fact the history of set theory and even the mundane descriptive set theory and naturally the history of the Moscow School of Mathematics could not be understood without the role of religion. However we will talk about that in other communications.
E-infinity Group
E-infinity communication No. 69
Physics and metaphysics in transfinite set theory and E-infinity
Before anyone jumps to conclusions and misconstrues what we want to say here, we should better start by clarifying first two points. Metaphysics has nothing to do here with the supernatural. In fact at the time of G. Cantor for instance, metaphysics meant more or less fundamental science or stringent mathematical physics dealing with foundational issues. Naturally questions dealing with philosophical issues were included. Equally naturally, the role of religion and the existence of a higher being referred to as Deity were touched upon. In fact fundamental science used to be called natural philosophy and even until our time universities awarded degrees in all possible fields but they are all called doctor of philosophy and this could be in engineering rather than philosophy.
The second point is regarding Mohamed El Naschie who is deeply religious and counts himself as being a Moslem without any doubt but that is all. He is deeply distrustful of all forms of organized religion as well as mingling science, politics and religion in a demagogical way. Perhaps we should give two examples of his writing on this subject before we go any further. At the end of his paper “A note on quantum field theory and P-Branes in n dimensions published in CS&F, Vol. 10 (8), pp. 1413-1417 (1999) he wrote “It may seem that by reducing the contra intuitive physics of quantum mechanics to the contra intuitive geometry of the space of quantum mechanics, nothing is gained. However this is not the right way of looking at the problem. The physics of quantum mechanics has frequently opened the door to the wrong kind of mysticism which is wholly foreign to scientific thinking. By contrast all of the mind-baffling and contra intuitive results of modern geometry in hyperspaces are the end product of the most stringent rational thinking and leave no place for uncalled-for mystical beliefs what so ever. Thus we think a great deal is achieved by viewing quantum physics in what we think is the right geometrical way proposed here”. I think Prof. El Naschie made it clear where he stands. An additional quotation at the end of another paper by him may help to dispel any lingering doubts about a basically renaissance Egyptian who tolerates religious beliefs of the personal type. In CS&F, Vol. 10(2-3), pp. 567-580 (1999), at the end of a paper entitled ‘Nuclear spacetime theories, superstrings, monster groups and application’ El Naschie says “Philosophically inclined theoretical physicists have frequently wondered if, in creating the world, God had any choice in determining it. Assuming that our hypothesis about E-infinity geometry is correct, then it would seem to us that once God had created logic, he had as good as no option but to create the universe as we see it now.”
Having said all that, the situation is slightly different when we come to consider transfinite set theory, the continuum hypothesis, the axiom of choice and particularly the meaning of infinity. There is no doubt that there are two kinds of personalities among serious scientists. Those who do not believe in any form of creation or any form of non-materialistic existence and those who believe in non-materialistic existence as well as the possibility of creation. The first kind is more or less like Kronecker, the famous German mathematician. The second kind is more like G. Cantor. There are people who may be in between. In E-Infinity Mohamed El Naschie finds that the spiritual value and even religious beliefs can be an inspiration to solving extremely difficult problems. In general, one has to keep religion out of science. However there is one exception where we must be tolerant and that is when someone identifies the Almighty with the set of all sets. In fact El Naschie believes that the concept of a set of all sets leads to severe contradictions in set theory. He accepts the class of all classes but not the set of all sets.
If we search for a deep difference between Kronecker and ‘t Hooft on the one side and Cantor and El Naschie on the other side, then the line separating them scientifically is that Cantor and El Naschie believe in the infinite and yes they believe in God. This is indirectly reflected in the philosophical depth of their work where there is no more any artificial separation between physics and mathematics. In fact we recently spoke to El Naschie and he said in a recent interview that God is without doubt a mathematician and definitely not a theoretical physicist. In the same way El Naschie regards the mathematician Prof. A. Connes as the deepest physicist working among us today. In fact the history of set theory and even the mundane descriptive set theory and naturally the history of the Moscow School of Mathematics could not be understood without the role of religion. However we will talk about that in other communications.
E-infinity Group
Jan 19, 2011
The dilaton in Mohamed El Naschie’s early papers and the very recent paper by G. ‘t Hooft incorporating the dilaton in quantum field theory
22nd December, 2010.
E-infinity communication No. 68
The dilaton in Mohamed El Naschie’s early papers and the very recent paper by G. ‘t Hooft incorporating the dilaton in quantum field theory
We already very briefly discussed the recent proposal of Nobel Laureate in Physics (1999) Gerard ‘t Hooft regarding the introduction of a dilaton scalar field into his version of deterministic quantum field theory. In the present short note we return to this subject and comment on it again, this time in the light of El Naschie’s early papers about the role of dilaton in E-infinity theory.
The idea of a scalar dilaton field and the hypothetical particle dilaton goes back most probably to the work of Kaluza and may be earlier on in connection with the work of the Finish theoretical physicist and colleague of Kaluza, Gunar Nordstrom. The writers remember seeing the television science program of Prof. El Naschie during which he told an engaging story about the man who invented the fifth dimension. This man is not T. Kaluza but it is G. Nordstrom. However Nordstrom space is flat which is extremely important for ‘t Hooft’s work as pointed out by El Naschie. In his first paper dedicated to the subject in 2008 El Naschie considered certain dualities pertinent to the Nordstrom-Kaluza-Klein theories which were obviously important for quantum field theory. The paper in question is “On dualities between Nordstrom-Kaluza-Klein, Newtonian and quantum gravity”, CS&F, 36 (2008), pp. 808-810. A second paper followed where the need for a coupling constant and consequently coupling “particle” between gravity and electromagnetism was discussed. This turned out to probably be a pseudo Goldstone boson. This paper was entitled “Kaluza-Klein unification – some possible extensions”, CS&F, 37 (2008), pp. 16-22. El Naschie calculated the number of elements of the matrix and found that 14 of them are due to Einstein + Maxwell. The additional pseudo Goldstone boson resulting from fusing both theories is what makes the 14 + 1 = 15. Subsequently El Naschie showed in this paper that the K-K theory can be generalized to Witten’s five Brane in eleven dimensions theory and calculates the well known 528 states.
The preceding discussion may be considered the beginning of the motivation for a dilaton field and particle. The direct predecessor was thus the Jordan, Brans and Dicke scalar. The scalar appears in the form of a dilaton in all perturbative string theories. The exception is M theory. Dilaton is important for conformal invariance. That means dilaton is important for scale invariance. The introduction of the dilaton field in quantum field theory is almost equivalent to the introduction of L. Nottale’s scale relativity into it or making quantum field theory a little more “fractal field-like”. Using the dilaton field one gains two things indirectly. First a fractal-like scale invariance and second an empty set-like vacuum with an expectation value and negative dimension. This is of course a very loose non-mathematical description of the hidden idea behind the introduction of the dilaton. The dilaton featured in El Naschie’s older publication at many locations to show the connections between his Cantorian spacetime approach and the dilaton for which there is no urgent need what so ever in E-infinity theory because Cantorian set theoretical formulation has a natural inbuilt dilaton in it so to speak. The first substantial discussion of the dilaton in E-infinity formalism was given by Prof. El Naschie ten years ago in his first outline for a general transfinite spacetime theory in a paper entitled “A general theory for the topology of transfinite Heterotic strings and quantum gravity” published in CS&F, 12 (2001), pp. 969. In section 7.3 on page 975 under the heading “Relation to 10 dimensional Newton’s constant and dilaton” El Naschie gave the exact transfinite value of g(dilaton) to be equal 0.723606797. That means 10 copies of g gives the dimension of a Milnor fractal sphere in seven fractal dimensions. Subsequently he calculates the real part of the dilaton S(D) namely S(R) and finds S(R) = 26.18033989 = 26 + k where k is a function of the golden mean to the power 3, namely k = 0.18033989. Incidentally in this paper El Naschie considers at some length the important work of J. Ambjorn of dynamical triangulation and introduces fractal fuzziness to it, something which Ambjorn et al incorporated in their excellent paper published in Scientific American in 2008 with Dr. R. Loll. In a second, mainly survey paper entitled “On a class of general theories for high energy particle physics”, CS&F, 14 (2002), pp. 649-669, El Naschie consider the dilaton again. For instance in Table 5 on page 657 he reports on the values of S® and T® estimated in a paper by F. Quevedo. These were found to be S (R) 25 and T(R) 1. These are quite near the exact transfinite values found by El Naschie, namely S(R) = 26 + k. It is tempting at this point to speculate that S(R) + T(R) = 26 + k. This is supported by the numeric (S(R) 25) + (T(R) 1) = 26 but the more important fact is that the frequent appearance of 26 in E-infinity theory must have some hidden reason. We recall the 26 + k = 26.18033989 was found to be the value for all of the following. It is the value for the bosonic dimension of Heterotic string theory, the inverse super symmetric unification of all fundamental forces, the curvature of the core of E-infinity spacetime, the Euler characteristic of the fuzzy K3-Kähler manifold of E-infinity theory as well as the instanton density of the same. Now that even the real part of the dilaton turned out to be 26, we think the reason is the topology and dynamics of the holographic boundary of E-infinity. This boundary is a compactified Klein modular curve with 336 + 16k degrees of freedom equal nearly to 336 + 3 = 339. Two of the most important orbits in this modular curve are the 42 and the 26 orbit.
In conclusion we think incorporating the dilaton in quantum field theory ‘t Hooft is essentially realizing very deeply that his deterministic quantum mechanics is a homomorphism of E-infinity indetermanistic, chaotic and fractal mechanics.
E-infinity Group
E-infinity communication No. 68
The dilaton in Mohamed El Naschie’s early papers and the very recent paper by G. ‘t Hooft incorporating the dilaton in quantum field theory
We already very briefly discussed the recent proposal of Nobel Laureate in Physics (1999) Gerard ‘t Hooft regarding the introduction of a dilaton scalar field into his version of deterministic quantum field theory. In the present short note we return to this subject and comment on it again, this time in the light of El Naschie’s early papers about the role of dilaton in E-infinity theory.
The idea of a scalar dilaton field and the hypothetical particle dilaton goes back most probably to the work of Kaluza and may be earlier on in connection with the work of the Finish theoretical physicist and colleague of Kaluza, Gunar Nordstrom. The writers remember seeing the television science program of Prof. El Naschie during which he told an engaging story about the man who invented the fifth dimension. This man is not T. Kaluza but it is G. Nordstrom. However Nordstrom space is flat which is extremely important for ‘t Hooft’s work as pointed out by El Naschie. In his first paper dedicated to the subject in 2008 El Naschie considered certain dualities pertinent to the Nordstrom-Kaluza-Klein theories which were obviously important for quantum field theory. The paper in question is “On dualities between Nordstrom-Kaluza-Klein, Newtonian and quantum gravity”, CS&F, 36 (2008), pp. 808-810. A second paper followed where the need for a coupling constant and consequently coupling “particle” between gravity and electromagnetism was discussed. This turned out to probably be a pseudo Goldstone boson. This paper was entitled “Kaluza-Klein unification – some possible extensions”, CS&F, 37 (2008), pp. 16-22. El Naschie calculated the number of elements of the matrix and found that 14 of them are due to Einstein + Maxwell. The additional pseudo Goldstone boson resulting from fusing both theories is what makes the 14 + 1 = 15. Subsequently El Naschie showed in this paper that the K-K theory can be generalized to Witten’s five Brane in eleven dimensions theory and calculates the well known 528 states.
The preceding discussion may be considered the beginning of the motivation for a dilaton field and particle. The direct predecessor was thus the Jordan, Brans and Dicke scalar. The scalar appears in the form of a dilaton in all perturbative string theories. The exception is M theory. Dilaton is important for conformal invariance. That means dilaton is important for scale invariance. The introduction of the dilaton field in quantum field theory is almost equivalent to the introduction of L. Nottale’s scale relativity into it or making quantum field theory a little more “fractal field-like”. Using the dilaton field one gains two things indirectly. First a fractal-like scale invariance and second an empty set-like vacuum with an expectation value and negative dimension. This is of course a very loose non-mathematical description of the hidden idea behind the introduction of the dilaton. The dilaton featured in El Naschie’s older publication at many locations to show the connections between his Cantorian spacetime approach and the dilaton for which there is no urgent need what so ever in E-infinity theory because Cantorian set theoretical formulation has a natural inbuilt dilaton in it so to speak. The first substantial discussion of the dilaton in E-infinity formalism was given by Prof. El Naschie ten years ago in his first outline for a general transfinite spacetime theory in a paper entitled “A general theory for the topology of transfinite Heterotic strings and quantum gravity” published in CS&F, 12 (2001), pp. 969. In section 7.3 on page 975 under the heading “Relation to 10 dimensional Newton’s constant and dilaton” El Naschie gave the exact transfinite value of g(dilaton) to be equal 0.723606797. That means 10 copies of g gives the dimension of a Milnor fractal sphere in seven fractal dimensions. Subsequently he calculates the real part of the dilaton S(D) namely S(R) and finds S(R) = 26.18033989 = 26 + k where k is a function of the golden mean to the power 3, namely k = 0.18033989. Incidentally in this paper El Naschie considers at some length the important work of J. Ambjorn of dynamical triangulation and introduces fractal fuzziness to it, something which Ambjorn et al incorporated in their excellent paper published in Scientific American in 2008 with Dr. R. Loll. In a second, mainly survey paper entitled “On a class of general theories for high energy particle physics”, CS&F, 14 (2002), pp. 649-669, El Naschie consider the dilaton again. For instance in Table 5 on page 657 he reports on the values of S® and T® estimated in a paper by F. Quevedo. These were found to be S (R) 25 and T(R) 1. These are quite near the exact transfinite values found by El Naschie, namely S(R) = 26 + k. It is tempting at this point to speculate that S(R) + T(R) = 26 + k. This is supported by the numeric (S(R) 25) + (T(R) 1) = 26 but the more important fact is that the frequent appearance of 26 in E-infinity theory must have some hidden reason. We recall the 26 + k = 26.18033989 was found to be the value for all of the following. It is the value for the bosonic dimension of Heterotic string theory, the inverse super symmetric unification of all fundamental forces, the curvature of the core of E-infinity spacetime, the Euler characteristic of the fuzzy K3-Kähler manifold of E-infinity theory as well as the instanton density of the same. Now that even the real part of the dilaton turned out to be 26, we think the reason is the topology and dynamics of the holographic boundary of E-infinity. This boundary is a compactified Klein modular curve with 336 + 16k degrees of freedom equal nearly to 336 + 3 = 339. Two of the most important orbits in this modular curve are the 42 and the 26 orbit.
In conclusion we think incorporating the dilaton in quantum field theory ‘t Hooft is essentially realizing very deeply that his deterministic quantum mechanics is a homomorphism of E-infinity indetermanistic, chaotic and fractal mechanics.
E-infinity Group
Foreward
21st December, 2010.
E-infinity communication No. 67
Foreward
In around 1930 Einstein wrote in an article published in the Journal of the Franklin Institute:
“ . . . it has been pointed out that the introduction of a space-time continuum may be considered
as contrary to nature in view of the molecular structure of everything which happens on a small
scale. It is maintained that perhaps the success of the Heisenberg method points to a purely
algebraic method of description of nature that is to the elimination of continuous functions from
physics. Then however, we must also give up, by principle the space-time continuum. It is not
unimaginable that human ingenuity will some day find methods which will make it possible to
proceed along such a path. At present time however, such a program looks like an attempt to
breath in empty space”.
I must say, seeing the present collection of articles on strings and quantum field theory, and
reading the preceding lines almost seventy years after Einstein first wrote them, it is becoming
evident to me that we have came a long way since then. We can no longer hide that we have
“almost” given up the continuum. Nevertheless we are not breathing in vacuum thanks to the
pioneering work of physicists like D. Bohm, A. Wheeler, R. Feynman, D. Finkelstein, R.
Penrose, E. Witten, L. Nottale and many others. These scientists have virtually revolutionized
our conventional notion of spacetime geometry. The task could not have been successful in
revolutionizing quantum physics if it would not have been for an equally vigorous and not less
revolutionary development on the “pure” side by mathematicians like G. Cantor, K. Menger, F.
Hausdorff, J. von Neumann, M. Atiyah, W. Thurston, V. Jones, S. Donaldson, M. Freedman
and A, Connes.
Finally, the unbelievable advances in the power of our electronic computational and experimental
capabilities have enabled us to make progress in fields where the mere mention of a real
experimental verification or computer simulation would have triggered the laughter of disbelieve
only three or four decades ago.
So it is not disrespectful that we are trying and have partially proven many of the giants of
quantum physics to be wrong on various occasions. Schrodinger was wrong to think that we
could not experiment with a single quantum particle. Heisenberg was wrong to think that
quantum particles have at best a reduced reality, and at worse there are no real particles at all.
We know now that elementary particles are different from classical objects, but they are very real
and can be captured and tamed. Finally even Feynman and other well-known names of theoretical
physics were wrong about superstrings. There are already reasonably realistic proposals for
indirect experimental verification of strings which could be carried out in the foreseeable future.
In the present volume most of the contributions are basically challenging many of Einstein’s
ideas about spacetime being a continuum something which was also taken for granted in classical
quantum physics. This is however the heart of Einsteinian thinking because he has always had
the courage and the foresight to challenge any conventional wisdom if he thought that there
ought to be a better way to describe nature. Famous names never worried Einstein. Following
this example I would like to gather my courage and state clearly that I believe that it is wrong to
think that micro spacetime has a fixed finite dimension.
In this sense the present volume is in the best tradition of Einstein’s thinking in more than one
sense and I would like to extend our thanks to all those who have contributed to this volume
and give our highest esteem to Prof. Carlos Castro on behalf of the Editorial Board for his
excellent effort in producing this issue. Particular words of thanks are also due to Prof. Sir R.
Penrose for contributing to the present volume, even though he was travelling abroad and under
enormous time pressure.
M.S. El Naschie
Cambridge 1998
E-infinity communication No. 67
Foreward
In around 1930 Einstein wrote in an article published in the Journal of the Franklin Institute:
“ . . . it has been pointed out that the introduction of a space-time continuum may be considered
as contrary to nature in view of the molecular structure of everything which happens on a small
scale. It is maintained that perhaps the success of the Heisenberg method points to a purely
algebraic method of description of nature that is to the elimination of continuous functions from
physics. Then however, we must also give up, by principle the space-time continuum. It is not
unimaginable that human ingenuity will some day find methods which will make it possible to
proceed along such a path. At present time however, such a program looks like an attempt to
breath in empty space”.
I must say, seeing the present collection of articles on strings and quantum field theory, and
reading the preceding lines almost seventy years after Einstein first wrote them, it is becoming
evident to me that we have came a long way since then. We can no longer hide that we have
“almost” given up the continuum. Nevertheless we are not breathing in vacuum thanks to the
pioneering work of physicists like D. Bohm, A. Wheeler, R. Feynman, D. Finkelstein, R.
Penrose, E. Witten, L. Nottale and many others. These scientists have virtually revolutionized
our conventional notion of spacetime geometry. The task could not have been successful in
revolutionizing quantum physics if it would not have been for an equally vigorous and not less
revolutionary development on the “pure” side by mathematicians like G. Cantor, K. Menger, F.
Hausdorff, J. von Neumann, M. Atiyah, W. Thurston, V. Jones, S. Donaldson, M. Freedman
and A, Connes.
Finally, the unbelievable advances in the power of our electronic computational and experimental
capabilities have enabled us to make progress in fields where the mere mention of a real
experimental verification or computer simulation would have triggered the laughter of disbelieve
only three or four decades ago.
So it is not disrespectful that we are trying and have partially proven many of the giants of
quantum physics to be wrong on various occasions. Schrodinger was wrong to think that we
could not experiment with a single quantum particle. Heisenberg was wrong to think that
quantum particles have at best a reduced reality, and at worse there are no real particles at all.
We know now that elementary particles are different from classical objects, but they are very real
and can be captured and tamed. Finally even Feynman and other well-known names of theoretical
physics were wrong about superstrings. There are already reasonably realistic proposals for
indirect experimental verification of strings which could be carried out in the foreseeable future.
In the present volume most of the contributions are basically challenging many of Einstein’s
ideas about spacetime being a continuum something which was also taken for granted in classical
quantum physics. This is however the heart of Einsteinian thinking because he has always had
the courage and the foresight to challenge any conventional wisdom if he thought that there
ought to be a better way to describe nature. Famous names never worried Einstein. Following
this example I would like to gather my courage and state clearly that I believe that it is wrong to
think that micro spacetime has a fixed finite dimension.
In this sense the present volume is in the best tradition of Einstein’s thinking in more than one
sense and I would like to extend our thanks to all those who have contributed to this volume
and give our highest esteem to Prof. Carlos Castro on behalf of the Editorial Board for his
excellent effort in producing this issue. Particular words of thanks are also due to Prof. Sir R.
Penrose for contributing to the present volume, even though he was travelling abroad and under
enormous time pressure.
M.S. El Naschie
Cambridge 1998
Some of the most important results of M.S. El Naschie’s research in E-infinity theoretical quantum physics
21st December, 2011.
E-infinity communication No. 66
Some of the most important results of M.S. El Naschie’s research in E-infinity theoretical quantum physics
1. E-Infinity theory (E (∞)) has clearly shown that random Cantor sets are the basic building blocks or atoms of quantum spacetime. The intrinsic topological dimension of these building blocks is zero, its Hausdorff fractal dimension is the golden ratio (0.618033989…) and its embedding dimension is unity.
2. Quantum spacetime is described fully by not one but three dimensions. First a topological (Menger-Urysohn) dimension equal to exactly 4. Second a Hausdorff dimension equal to 4 plus the golden ratio to the power of 3 which means 4.236067977… This is effectively a 4D cube inside a 4D cube and so on ad infinitum. Third a formal dimension equal to infinity. In other words our quantum spacetime is an infinite dimensional but hierarchal Cantor set of measure zero.
3. Because orthodox quantum mechanics is totally insensitive to fractals and does not consider the Cantorian nature of quantum spacetime, many paradoxes follow.
4. The mass spectrum of all known elementary particles of the standard model as well as some composite particles was predicted with astonishing accuracy compared to the known experimental value. Because of KAM theorem they are always a function of the golden ratio as confirmed experimentally by the quantization and the experiment at the Helmholtz Centre.
5. Many of the fundamental constants of nature were derived from first principles. This includes the coupling constant of quantum gravity as well as the electromagnetic fine structure constant and Newton’s gravity constant. In particular a fundamental equation was established relating the Bulk (E8E8) with the holographic boundary and gravity from which = 137 was derived. In addition the inverse coupling of unification of all fundamental forces were found to be = 26 + k ≃ 26.
6. An exact renormalization equation was constructed which derived quarks confinement as an exact result. Confinement is a theorem of the golden quantum field theory.
7. The relationship between Lie symmetry groups as well as two and three Stein spaces and high energy physics was outlined. In particular the role played by E8 in this respect was analyzed. The total number of elementary particles in an extended standard model was shown to be 137 particles. This includes 10 spacetime quasi dimensions. E-infinity P-Adic reasoning was also employed.
8. The theory predicted the existence of 8 dimensional Higgs field with at least one Higgs particle or five Higgs particles becoming manifest. The Higgs mass was determined to be approximately 169 Gev.
9. The Euler characteristic as well as the curvature of the quantum spacetime was shown to be equal to 26 + k 26. In fact even the real part of the dilaton is given by S® = 26. In addition arguments were given to show that our universe is likely to be compact. An alternative theory using instanton density to calculate the first massless particle-like states corrected the classical well known solution of Heterotic string theory, namely 8064 to the exact solution 8872.
10. The stationary state of quantum mechanics was shown to be that of the VAK, i.e. the vague attractor of Kolmogorov.
11. A quantum particle may be modelled as a fractal point represented by the two dimensions of a zero set dim P (0,ϕ) where . The quantum wave is the surface of a fractal point, i.e. a fractal surface and may be represented by the two dimensions of an empty set dim W . The empty set accounts for the paradoxal outcome of the two-slit experiment and the disappearance of the interference fringes.
12. It is reasoned that in a totally disjointed infinite dimensional but hierarchal Cantor-like space time manifold, calculus must be replaced by Weyl-like golden mean scaling which effectively represents a new version of quantized calculus. The set theoretical analogue of this is Suslin scaling of descriptive set theory.
13. El Naschie made several suggestions regarding a Banach-Tarski-like big bang theory based on paradoxal decomposition of spheres.
14. Mohamed El Naschie and his group proved the equivalence of his basic equations of E-infinity theory and the dimensional function of von Neumann’s continuous geometry and A. Connes’ noncommutative geometry.
15. Very recently El Naschie discovered several facts connecting his E-infinity theory to K-theory and the mathematical E-infinity theory of highly structured ring spectrum and the E4 of operad of 4 dimensional cubes.
16. It is conjectured that E-infinity building blocks represent at a minimum a O-category but could also be n-category for n → ∞.
17. The symmetry group of E-infinity is fuzzy and is a fractal symmetry given by the sum of all the dimensions of the E-line exceptional Lie groups 584 and not 496 as well as the sum of all Stein spaces with dimension equal 685. The integer value of all symmetry groups including Stein spaces of the first kind is 3689. When adding the super strings 10 dimensions then one finds (27)(137) = 3699.
18. E-infinity is shown to combine numbers and number theoretical reasoning with topology exactly as in K-theory.
19. Ervin Goldfain demonstrated that measurement of cosmic radiation confirms the correctness of E-infinity theory. El Naschie showed the consistency of the COBE measurement with E-infinity predictions. Malcolm H. MacGregor demonstrated the consistency of the experimental value of the mass spectrum with his alpha quantization of the masses (book published by World Scientific 2007). Alpha quantization is coarse graining of the golden mean quantization. The dependency on the golden mean was demonstrated for the first time experimentally in the Helmholtz-University of Oxford joint work announced earlier this year.
20. Recent work by Ali Chamseddine and Alain Connes (arXiv paper 2010) reinforces some fundamental results of E-infinity theory.
21. The recent results published by Ambjorn and R. Loll are particularly appealing confirmation of E-infinity theory. The spectral dimension of their work, namely 4.02 is a direct Weyl-Suslin scaling in exactly 16 steps from the first massless level of Heterotic strings (8872 integer value) which corresponds to the classical Green-Schwartz and Witten value (8064). The difference comes from the fact that E-infinity uses the entire 8 exceptional Lie group line which adds to 548 while the classical value (8064) considers only a subset of the exceptional Lie groups leading to only 504.
E-infinity communication No. 66
Some of the most important results of M.S. El Naschie’s research in E-infinity theoretical quantum physics
1. E-Infinity theory (E (∞)) has clearly shown that random Cantor sets are the basic building blocks or atoms of quantum spacetime. The intrinsic topological dimension of these building blocks is zero, its Hausdorff fractal dimension is the golden ratio (0.618033989…) and its embedding dimension is unity.
2. Quantum spacetime is described fully by not one but three dimensions. First a topological (Menger-Urysohn) dimension equal to exactly 4. Second a Hausdorff dimension equal to 4 plus the golden ratio to the power of 3 which means 4.236067977… This is effectively a 4D cube inside a 4D cube and so on ad infinitum. Third a formal dimension equal to infinity. In other words our quantum spacetime is an infinite dimensional but hierarchal Cantor set of measure zero.
3. Because orthodox quantum mechanics is totally insensitive to fractals and does not consider the Cantorian nature of quantum spacetime, many paradoxes follow.
4. The mass spectrum of all known elementary particles of the standard model as well as some composite particles was predicted with astonishing accuracy compared to the known experimental value. Because of KAM theorem they are always a function of the golden ratio as confirmed experimentally by the quantization and the experiment at the Helmholtz Centre.
5. Many of the fundamental constants of nature were derived from first principles. This includes the coupling constant of quantum gravity as well as the electromagnetic fine structure constant and Newton’s gravity constant. In particular a fundamental equation was established relating the Bulk (E8E8) with the holographic boundary and gravity from which = 137 was derived. In addition the inverse coupling of unification of all fundamental forces were found to be = 26 + k ≃ 26.
6. An exact renormalization equation was constructed which derived quarks confinement as an exact result. Confinement is a theorem of the golden quantum field theory.
7. The relationship between Lie symmetry groups as well as two and three Stein spaces and high energy physics was outlined. In particular the role played by E8 in this respect was analyzed. The total number of elementary particles in an extended standard model was shown to be 137 particles. This includes 10 spacetime quasi dimensions. E-infinity P-Adic reasoning was also employed.
8. The theory predicted the existence of 8 dimensional Higgs field with at least one Higgs particle or five Higgs particles becoming manifest. The Higgs mass was determined to be approximately 169 Gev.
9. The Euler characteristic as well as the curvature of the quantum spacetime was shown to be equal to 26 + k 26. In fact even the real part of the dilaton is given by S® = 26. In addition arguments were given to show that our universe is likely to be compact. An alternative theory using instanton density to calculate the first massless particle-like states corrected the classical well known solution of Heterotic string theory, namely 8064 to the exact solution 8872.
10. The stationary state of quantum mechanics was shown to be that of the VAK, i.e. the vague attractor of Kolmogorov.
11. A quantum particle may be modelled as a fractal point represented by the two dimensions of a zero set dim P (0,ϕ) where . The quantum wave is the surface of a fractal point, i.e. a fractal surface and may be represented by the two dimensions of an empty set dim W . The empty set accounts for the paradoxal outcome of the two-slit experiment and the disappearance of the interference fringes.
12. It is reasoned that in a totally disjointed infinite dimensional but hierarchal Cantor-like space time manifold, calculus must be replaced by Weyl-like golden mean scaling which effectively represents a new version of quantized calculus. The set theoretical analogue of this is Suslin scaling of descriptive set theory.
13. El Naschie made several suggestions regarding a Banach-Tarski-like big bang theory based on paradoxal decomposition of spheres.
14. Mohamed El Naschie and his group proved the equivalence of his basic equations of E-infinity theory and the dimensional function of von Neumann’s continuous geometry and A. Connes’ noncommutative geometry.
15. Very recently El Naschie discovered several facts connecting his E-infinity theory to K-theory and the mathematical E-infinity theory of highly structured ring spectrum and the E4 of operad of 4 dimensional cubes.
16. It is conjectured that E-infinity building blocks represent at a minimum a O-category but could also be n-category for n → ∞.
17. The symmetry group of E-infinity is fuzzy and is a fractal symmetry given by the sum of all the dimensions of the E-line exceptional Lie groups 584 and not 496 as well as the sum of all Stein spaces with dimension equal 685. The integer value of all symmetry groups including Stein spaces of the first kind is 3689. When adding the super strings 10 dimensions then one finds (27)(137) = 3699.
18. E-infinity is shown to combine numbers and number theoretical reasoning with topology exactly as in K-theory.
19. Ervin Goldfain demonstrated that measurement of cosmic radiation confirms the correctness of E-infinity theory. El Naschie showed the consistency of the COBE measurement with E-infinity predictions. Malcolm H. MacGregor demonstrated the consistency of the experimental value of the mass spectrum with his alpha quantization of the masses (book published by World Scientific 2007). Alpha quantization is coarse graining of the golden mean quantization. The dependency on the golden mean was demonstrated for the first time experimentally in the Helmholtz-University of Oxford joint work announced earlier this year.
20. Recent work by Ali Chamseddine and Alain Connes (arXiv paper 2010) reinforces some fundamental results of E-infinity theory.
21. The recent results published by Ambjorn and R. Loll are particularly appealing confirmation of E-infinity theory. The spectral dimension of their work, namely 4.02 is a direct Weyl-Suslin scaling in exactly 16 steps from the first massless level of Heterotic strings (8872 integer value) which corresponds to the classical Green-Schwartz and Witten value (8064). The difference comes from the fact that E-infinity uses the entire 8 exceptional Lie group line which adds to 548 while the classical value (8064) considers only a subset of the exceptional Lie groups leading to only 504.
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