Spacetime May Have Fractal Properties on a Quantum Scale

Mar 25, 2009 By Lisa Zyga feature
As scale decreases, the number of dimensions of k-Minkowski spacetime (red line), which is an example of a space with quantum group symmetry, decreases from four to three. In contrast, classical Minkowski spacetime (blue line) is four-dimensional on all scales. This finding suggests that quantum groups are a valid candidate for the description of a quantum spacetime, and may have connections with a theory of quantum gravity. Image credit: Dario Benedetti.

(PhysOrg.com) -- Usually, we think of spacetime as being four-dimensional, with three dimensions of space and one dimension of time. However, this Euclidean perspective is just one of many possible multi-dimensional varieties of spacetime. For instance, string theory predicts the existence of extra dimensions - six, seven, even 20 or more. As physicists often explain, it’s impossible to visualize these extra dimensions; they exist primarily to satisfy mathematical equations.

As if extra dimensions weren’t strange enough, new research has probed an even more mind-bending possibility: that spacetime has dimensions that change depending on the scale, and the dimensions could have on small scales. In a recent study, Dario Benedetti, a physicist at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, has investigated two possible examples of spacetime with scale-dependent dimensions deviating from classical values at short scales. More than being just an interesting idea, this phenomenon might provide insight into a quantum , which also has been suggested to have scale-dependent dimensions. Benedetti’s study is published in a recent issue of .

“It is an old idea in that at short scales spacetime might appear foamy, fuzzy, fractal or similar,” Benedetti told PhysOrg.com. “In my work, I suggest that quantum groups are a valid candidate for the description of such a quantum spacetime. Furthermore, computing the spectral dimension, I provide for the first time a link between quantum groups/noncommutative geometries and apparently unrelated approaches to quantum gravity, such as Causal Dynamical Triangulations and Exact Renormalization Group. And establishing links between different topics is often one of the best ways we have to understand such topics.”

In his study, Benedetti explains that a spacetime with quantum group symmetry has in general a scale-dependent dimension. Unlike classical groups, which act on commutative spaces, quantum groups act on nocommutative spaces (e.g. where xy doesn’t equal yx), which emerges through their unique curvature and quantum uncertainty. Here, Benedetti considers two types of spacetime with quantum group symmetry - a and k-Minkowski spacetime - and calculates their dimensions. In both spaces, the dimensions have fractal properties at small scales, and only reach classical values at large scales.

“In simple words, the relation between quantum groups and noncommutative geometry is as follows,” he explained. “Classically, we know that certain spaces are invariant under the action of some classical groups; for example, Euclidean space is invariant under rotations and translations. A quantum group is a deformation of a given classical group, and is such that no classical space can have it as a symmetry group. The invariant space has to be as well a deformation of a classical space, a deformation that makes it noncommutative. No relation of all this to fractals is known, but in my work I've found that they do have a common property, that is, a non-integer dimension (at some scale).”

Compared to a Euclidean sphere, a quantum sphere’s curvature and uncertainty make it a noncommutative space. When calculating the spectral dimension of the quantum sphere, Benedetti found that it closely resembles a standard sphere on large scales; however, as the scale decreases, the dimensions of the quantum sphere deviate and go down to zero. He describes this phenomenon as a signature of the fuzziness, or uncertainty, of the quantum sphere, and also as resulting from fractal behavior at small scales.

In the second kind of space, k-Minkowski spacetime, the dimensions also deviate from the constant behavior of classical Minkowski spacetime. While the latter always has four dimensions, independent of the scale, the number of dimensions in the quantum version decreases to three as a function of the scale. In both k-Minkowski spacetime and the quantum sphere, the dimensionality becomes non-integral, which is a typical signature of fractal geometry.

Benedetti’s results match previous approaches to quantum gravity, which also point to the emergence of a ground-scale spacetime with fractal properties. Together, these studies may help scientists understand the unique Planck scale properties of spacetime, and possibly tie in to a of gravity. For instance, as Benedetti explains, the fractal nature of quantum spacetime might enable gravity to cure its own ultraviolet behavior by dimensional reduction.

“The main problem with gravity is that apparently it cannot be quantized as other field theories; in jargon it is said to be non-renormalizable,” he said. “This problem is specific to four-dimensional spacetime. If spacetime had only two dimensions, then quantum gravity would be much simpler and treatable. The problem with a two-dimensional theory is that it is unphysical, as we see four dimensions at our scales. Things can be solved combining four and two dimensions at different scales. That is, if gravity itself provides a mechanism by which the dimension of spacetime depends on the scale at which we probe it (four at our and larger scales and two at very short scales), then we could have a physical theory (compatible with observations) that is free of quantum (short scale) troubles.”

More information: Benedetti, Dario. “Fractal Properties of Quantum Spacetime.” Physical Review Letters 102, 111303 (2009).

Copyright 2009 PhysOrg.com.
All rights reserved. This material may not be published, broadcast, rewritten or redistributed in whole or part without the express written permission of PhysOrg.com.

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googleplex
3 / 5 (6) Mar 25, 2009
Clearly the universe is scale dependent. I have often wondered how this could be understood in mathematical terms.
I am also convinced that there are extra dimensions. One of these extra dimensions explain the curvature of space time.
earls
3.8 / 5 (5) Mar 25, 2009
Variable measurable dimensions on a sliding scale. Hrm... googleplexing.

The question is then, which at "scale" is the "baseline" that which all the dimensions can be measured at once? If one exists at all...

Perhaps such would be relegated to a "virtual" scale independent equation.

Identifying all of the variables on the different scales seems like quite a insurmountable task. How would we know if we even quantified them all... I suppose "when the equation breaks for the known set of data."

Definitely one of the more interesting and thought provoking articles on Physorg (physics) as of late.
Modernmystic
3.3 / 5 (3) Mar 25, 2009
I think they're really on to something. When we get quantum gravity it will be something like this I think. Something that kind of makes you say "Well yeah that makes perfect sense, but I never would have thought of it".
yyz
3 / 5 (4) Mar 25, 2009
This study touches on two important questions in physics today 1) What is the nature and structure of spacetime 'foam' at Planck scale? and 2)How can singularities be dealt with in quantum mechanics (QM)? Will this research lead to the long-sought quantum description of gravity?
Nevertheless
2 / 5 (8) Mar 25, 2009
My principal interest in all of this is pragmatic. Neuroscientists will have an easier transition to the idea that consciousness is external to the brain if there is a theory that helps explain how this is possible, see, e.g.,

http://www.quantu...nts.html

As the Singulaity approaches we will be very interested in whether machines can have conscious minds. A theory concerning how this can happen should help things along.
Alizee
Mar 26, 2009
This comment has been removed by a moderator.
Alizee
Mar 26, 2009
This comment has been removed by a moderator.
magpies
1.5 / 5 (8) Mar 26, 2009
Yeah and perhaps if you go to a short enoth scale you will find elfs and hobogoblins!

Get real...
Suzu
5 / 5 (6) Mar 26, 2009
Yeah and perhaps if you go to a short enoth scale you will find elfs and hobogoblins!



Get real...


Elves* you numbnut. You are also welcome to bring us something that proves you enlightened point of view.
magpies
1.1 / 5 (8) Mar 26, 2009
Look the reason you get something that looks fairytailish when you look at a smaller and smaller scale... Is simple. Everything works the way it works... When you try to define the way something works your perspective itself shapes how you see it work. As a race of beings we cant understand something we are not ready to understand. Yet we will be trying to understand everything as beings. If you could remove yourself from your own perspective you could understand something elses own perspective possibly but untill then no. This is part of what makes us human and also part of what that makes our science wrong.
phystic
4 / 5 (1) Mar 26, 2009
Alizee, might you be meaning and "entropy-like" stuff when you mention "random stuff"?
mattytheory
2.4 / 5 (5) Mar 26, 2009
I hate to say it but both magpies and alizee have a point. At least to me they do... but I am drunk. So, what do I know? I'll probably read this tomorrow and wish I could delete my comment. I hope not...
Alexa
1.6 / 5 (7) Mar 26, 2009
..you be meaning and "entropy-like" stuff ..

Of course, and ideal gas is the simplest representation of every entropic system. We shouldn't replace natural phenomena by abstract concepts, because lost of general insight, connections and information. If we are saying, "quantum mechanics is of entropic character", we can deduce another atributes of quantum mechanic systems, because these systems share many common aspects of their behavior, not just probability or entropy distribution. Density fluctuations of gas have shape of strings and membranes and many other atributes, which can be predicted/computed by this model.
Alexa
1.2 / 5 (6) Mar 26, 2009
..problem with a two-dimensional theory is that it is unphysical, as we see four dimensions at our scales...
This is the reason, AWT doesn't assume some particular number of dimensions, it considers virtually infinite number of dimensions and lets the probability and geometric constrains to limits the number of dimensions.

For example, the density fluctuations inside of gas aren't always flat. If the gas is sparse, they have rather spherical character. After then we can model fluctuations as a system of kissing hypersheres, where just the 3D hypersheres allows the most compact distribution. This determines the number of spatial dimensions of our space-time. It appears as so large, because it's the most compact one possible.

Inside of more dense gas, though, a less compact density fluctuations appears: membranes and strings. The lower number of space-dimensions is compensates by increasing number of time dimensions and the system becomes chaotic from our 3D perspective. Therefore our view of reality doesn't differ from perspective, which "would see" some 3D fluctuation of gas, when interacting with other fluctuations (strings, branes or hyperspheres).
Alexa
1.2 / 5 (6) Mar 26, 2009
In general, we should decide question, if we want to adhere to "4D-only" space-time and "four fundamental forces" model to handle all violations of inverse square law like "natural nonlinearities", "gravity lensing" or "dispersion phenomena" - while still looking for "hidden dimensions" and "Lorentz symmetry violations". Because every nonlinearity from ISL IS a manifestation of hidden dimensions, in fact.

Or whether we would interpret things like dispersion, refraction and nonlinear energy spreading like manifestation of supergravity, hidden dimensions and Lorentz symmetry violation phenomena.

This is radical new approach to reality, but I'm convinced, we cannot consider both variants at the same moment from consistency reasons. For example, we shouldn't expect/look for hidden dimensions and Lorentz invariance at the same moment.

http://aetherwave...ory.html

From practical reasons it's always possible to handle exceptions separatelly, but when spending money for basic research in physics, we should make clear for yourself, which model we want to consider for future. The laymans can retain their plural/fragmented view of reality, but theorists should be perfectly aware of what they're looking for.
dev2000
5 / 5 (2) Mar 26, 2009
"In simple words..."

Too funny.
benhanson
2.7 / 5 (3) Mar 27, 2009
Have we disproved gravity's action at a distance? Doesn't gravity's "faster than light" interaction imply that it is a property of space as a medium rather than an object within that medium? By medium, I'm simply talking about the laws of nature/physics, the framework. It always seems to me that the interesting question that no one seems to have the answer to, is how does matter interact with and thus alter the medium? How can mass warp space? The idea of a fractal dimension is interesting in that it could explain the variability of the medium given the influence of matter, if someone could explain how energy influences a variable in the fractal equation.
earls
5 / 5 (2) Mar 27, 2009
It's been claimed that a successful experiment has measured the speed of gravity at the speed of light. There is no "faster than light" interaction.

Regarding how mass can warp space(time)...

Imagine a rubber sheet stretched over a frame - like a drum. If you compact an area of the sheet, it becomes denser - that's your mass. The area of the sheet surrounding the mass is your "warping" of spacetime - gravity.

When this analogy is usually delivered, it is said "the mass is sitting on top of the rubber sheet..."

This is a mistake... Mass and spacetime are not separate entities, they are one in the same, but simply density fluctuations of the "medium."

If you imagine this concept in three dimensions, you'll have a much more accurate picture of reality.
Alizee
Mar 27, 2009
This comment has been removed by a moderator.
Alizee
Mar 29, 2009
This comment has been removed by a moderator.
Alizee
Mar 29, 2009
This comment has been removed by a moderator.
yyz
1 / 5 (1) Mar 29, 2009
Check out this post at the BAUT forum and draw your own conclusions about El Naschie: http://www.bautfo...579.html .
Alizee
Mar 29, 2009
This comment has been removed by a moderator.
Alizee
Mar 29, 2009
This comment has been removed by a moderator.
Alexa
1 / 5 (3) Mar 30, 2009
..Spacetime May Have Fractal Properties on a Quantum Scale..
Strictly speaking, from lower dimensionality of space-time at shorter scale doesn't follow, space-time has a fractal nature at all. The same observation can be made at water surface, where for shorter wavelengths we can observe a distortion of waves due the Brownian motion of water molecules.

In this way, such observation is rather evidence of particle Aether model, then the fractal nature of it and every notion of fractals is irrelevant here. BTW every decreasing of dimensionality is manifestation of ISL violation for light and gravity and Lorentz symmetry violation as well (therefore the violation of string theory, which is based on special relativity and it assumes the existence of additional dimensions instead of reduction of their number), etc...
remoran
4 / 5 (2) Mar 30, 2009
I am not a math person at all so if I show profound ignorance of same, bear with me. Question, wouldn't space have inherent granularity to allow for the interaction of universes within the multiverse, thus giving rise to the bizarre effects of the quantum as seen, for example, by the elegant Thomas Young experiment showing the duality of light?

To me, the quantum foam (as interface) operating at Planck scale enables the many worlds of Hugh Everett to emerge and perhaps, this fractal theory could, in conjunction with quantum gravity, could explain how this amazing process happens.

Just a thought.
Modernmystic
3 / 5 (2) Mar 30, 2009
Question, wouldn't space have inherent granularity to allow for the interaction of universes within the multiverse,


Well if we find it does then we've learned something profoud about the multiverse. If we can interact with them, then they must at some level be operating on the same laws and princiles ours does...or at least some of them do...
Alizee
Mar 30, 2009
This comment has been removed by a moderator.
Alizee
Mar 30, 2009
This comment has been removed by a moderator.
Modernmystic
3.7 / 5 (3) Mar 30, 2009
Shouldn't that have read:

"By AWT Youre all naive religious tribesmen, face it."?

Or did you just forget the cool aid/brainwashing for a second?
Alizee
Mar 30, 2009
This comment has been removed by a moderator.
Alizee
Mar 31, 2009
This comment has been removed by a moderator.
spice_guru
3.7 / 5 (3) Apr 02, 2009
A genuinely interesting article, but that headline will act as a beacon for every dope-smoking alt.physics troll on the internet!
Alexa
1 / 5 (4) Apr 03, 2009
I don't understand, why you calling people a trolls, just because they're propose a more transparent explanations of the same things. You can be either religious layman and after than you should save your arrogance, which manifests only your personal frustrations.

Or you're one of mainstream scientists - and after then a generally better line of reasoning is expected. Natural competetion of ideas should be usefull even for science, don't you think? Nobody has a patent for definite knowledge - until you're not a Pope, indeed.
yyz
1 / 5 (1) Apr 05, 2009
%u2018Many cosmologists find fault with their analysis, largely because a fractal matter distribution out to such huge scales undermines the standard model of cosmology. According to the accepted story of cosmic evolution, there simply hasn%u2019t been enough time since the big bang nearly 14 billion years ago for gravity to build up such large structures.%u2019 And: %u2018Modelling a fractal universe with general relativity is possible in theory, but in reality it would be devilishly complicated. That would leave cosmologists without a working model, like acrobats without a net.%u2019 And according to Neil Turok: %u2018The standard picture of a homogeneous universe on large scales is holding up very well when tested with very large-scale observations like those mapping the cosmic background radiation, X-rays and radio galaxies,%u2019 says physicist Neil Turok of Cambridge University in the UK.

%u2018If the observations of galaxies in optical surveys don%u2019t agree, there may be a number of possible explanations, without resorting to an extremely inhomogeneous, fractal universe,%u2019 he told New Scientist. These are direct quotes from the New Scientist link here http://www.newsci...page=1.I remain dubious of the claims presented in their paper & await further study of the subject.
yyz
1 / 5 (1) Apr 05, 2009
The article in question at the New Scientist site is entitled 'Galaxy map hints at fractal universe' by Amanda Gefter June 25, 2008. Sorry for the broken link above.
Alexus
1 / 5 (2) Apr 05, 2009
I am only interested in the scientific content of the work of Prof. Mohamed Elnaschie. His name, religion and nationality or sexual preferences are of no concern to me whatsoever. It is a matter of irrelevance as far as I am concerned where he publishes his work or how many papers he has written. As a scientist I am only concerned with the scientific content of what he is writing. I think the best theoretical base of his work was laid down in a four-page paper entitled: On Certain %u201CEmpty%u201D Cantor Sets and their Dimensions. A global view of his theory with an interesting connection to Godel Theorem maybe found in his 1993 paper titled: Statistical Mechanics of Multi-dimensional Cantor Sets, Godel Theorem and Quantum Spacetime. This paper was published in the Journal of the Franklin Institute, vol. 330, No. 1, pp.199-211. His other mathematical paper on the empty sets was published in Chaos, Solitons & Fractals, vol. 4, No. 2, pp 293-296, 1994. The basic idea of his work as far as I can see it is the existence of a VAK. This is a shorthand writing for the Vague Attractor of Kolmogorov. It was Rene Thom the renowned French topologist who conjectured in his acclaimed book: Structure, Stability and Morphogenesis that the VAK represents the stationary state of quantum mechanics. Elnaschie has extended the notion to mean the limit set of a dynamical fractal spacetime. His famous paper on this subject was the VAK of vacuum fluctuation, self-organization and the math spectrum in high energy physics. Following Menger and Urysohn, Elnaschie set the dimensions of the empty sets to be minus one. He then proved that a corresponding Hausdorff dimension will be Phi square where Phi stands for the Golden Mean which is roughly 0.618033989. From these simple well-established mathematical facts he was able to develop over a period of about twenty years a very comprehensive spacetime theory for high energy physics. The theory is called frequently E-Infinity Theory or Cantorian Fractal Spacetime Theory. The corresponding quantum field theory is sometimes referred to as Quantum Golden Field Theory. Elnaschie makes extensive use of all the concepts of nonlinear dynamics and the general mathematical theory of dimensions. He was able to prove that nothingness has a Hausdorff dimension equal to zero and a corresponding Menger-Urysohn dimension equal to minus infinity. His theory must have been shocking to traditional physicists who know almost nothing about the two pillars upon which his theory is based namely nonlinear dynamics and the mathematical theory of dimension developed by Poincare, Brouwer, Karl Menger and Paul Urysohn. However meantime there are many people who are using his theory. A very encouraging sign is a recent profound paper by a well-known British scientist, T.N. Palmer who is a fellow of the Royal Society. Palmers paper has been recently reviewed by Buchanan in the Newscientist. It will appear shortly in the Proceedings of the Royal Society. Palmers paper is an independent confirmation of all the fundamental results of Elnaschies theory and seems to have been developed largely independent of Elnaschies work. I hope I can answer in the future in some more detail any question which the reader of this site may pose.
Andie
1 / 5 (1) Apr 06, 2009
My attention has been drawn to the comment by Alizee of March 29, 2009 a few moments ago. This is a direct response to his comment reflecting a fundamentally correct attitude. However in this particular case, and I know without a shred of a doubt that Prof. Mohamed Elnaschie is a victim other than the villain who some people for reasons better known to themselves are trying to portray. The history of Mohamed Elnaschie is well known. He was from the age of 30 a highly successful person in each conceivable endeavor and still is until today. The campaign against him started because of something completely different from what the superficial reader thinks. It is ridiculous to even suggest that someone would publish in his own journal so many papers without anyone noticing it. John Baez once he achieved the scandal he wanted, closed his site. Is not that an admission that he feared the long arm of the law. If you have such fear is not that an admission that you know you are wrong. Did not Nature take its allegations out of Nature online? Is that not an admission that these allegations must have been wrong? If there is a single person who never benefited financially from publishing papers in Physics, then this person must be Prof. Mohamed Elnaschie. John Baez is a close friend of Renate Loll and he is the one who drafted Schermeier of Nature and Christopher Drosser from Die Zeit. Baez, Loll and Dawker are all friends and closely connected with you know how. The whole issue is very distasteful I can assure you and no matter how long it takes, the truth will come out and someone will be telling you I told you so. But there is something more important to what Alizee is saying. To many people Werner Heisenberg is the greatest physicist of all time. For many others he is a Nazi collaborator who wanted to help Hitler get the bomb. The truth is that he is a German patriot who thought he was doing what he had to do for his country which was at war. His science and his Uncertainty relations have nothing to do with Nazism or Communism or any other isms. There is something fascistic about the internet. You sit there safe and sound behind a pseudo name and a pseudo address and a computer screen. Suddenly you appoint yourself judge and jury. In the Internet you are guilty until proven innocent. Anyone could defame anyone at will. The fact that these are real people with families, children, wives and parents is totally irrelevant. We should all think about that before making definite unfounded allegations. If we don%u2019t do that we either will have to introduce unheard of restrictions on the internet or we will end in total anarchies. I think you all will agree that self control is the best and most efficient way to control.
Alexus
1 / 5 (1) Apr 08, 2009

The paper by Dario Benedetti in Physical Review Letters is undoubtedly an important paper. It comes meantime as no surprise however that Mohamed Elnaschie was there first. Please see the paper by him titled: Quantum Groups and Hamiltonian Sets on a Nuclear Spacetime Cantorian Manifold, in Chaos, Solitons & Fractals, vol. 10, pp 1251-1256. This paper is 1999 almost ten years ago. In table 1 Elnaschie calculates what seems to me to be spectral dimensions and Hausdorff dimensions. Two other papers from the same author in the same year and same Journal seem to discuss various related topics in more details. The first is Jones%u2019 Invariant, Cantorian Geometry and Quantum Spacetime, vol. 10, No 7, pp 1241-1251. The second is The Golden Mean in Quantum Geometry, Knot Theory and Related Topics, Vol. 10 No 8, pp 1203-1307. I have started to form my own opinion about certain things. In all events the claim of priority reported in the article seems to be inaccurate. This is not a calamity but as I said it seems to explain to me certain, how should I say, anomalies.

Andy Somer
chaosnet3
not rated yet Apr 23, 2009
Fascinating stuff. I mean what comes out as a conclusion, what is written in the title %u201CSpacetime Fractal Properties on the Quantum Scale%u201D. By not being a physicist myself, I would skip the parts that I do not have a secure knowledge of , but without dismissing their significance in making it possible, for the individuals to formulate their theories. Myself , armed with a few concepts at my disposal, driven by an impulse to make them meaningful for me, I have toyed with these thoughts, and I have posted them in the web. In successive, bursts of thoughts I would say, these are:



Fractal universe, fractal reality?



Fractal corridors? Existence, reality, universe is built by?



As space expands outwards since the big bang, it expands inwards too?



Observer to Planck length distance, fractal? Why we perceive continuity, out of the energy packets (quanta), the physical world is made out of?



Fractal dimensions harbour parallel worlds?



Sub-atomic particles having zero rest mass. Universe, an infinitesimally thin sheet?



Space, a result of forces? Interaction of forces? It is bound to be.



Arrived to thoughts that looks at the world, the universe as fractal, by following a path where chaos, is seen as the generator of the reality, we live in. Chaos, being ubiquitous and universal, becomes a tool to provide the explanations human individuals are seeking out.



Individuals experience the world. What experience is based upon. Would it be diferent if:



Human experience. What if, our human resolving time was on a par with the resolving time of a fast electronic device?



and how do we amass knowledge? Comes out of agreement or compromise,



Convention? What kind of convention is that? Agreement or compromise?



propellerdiver
not rated yet Jul 19, 2009
I did not know this was an existing branch of physics. I've been calling this the "Horton/Who" universe, and writing on it as I try to justify my sense of a scientifically justifiable God and some form of non-chaotic Universe. The thought that the "the universe" could be generally spherical and have a boundary didn't make sense- a fractal structure does, and the fractal structure could exist in an "up and down" and "sidewise" direction, too. I wonder if Dr. Seuss was not a stealth physicist, who was trying to prepare young brains for understanding fractals later in life.
Rexy
not rated yet Sep 04, 2009
I agree with Andy Somer but think that the most important paper which demonstrates what he said about Mohamed El Naschie work is the following paper. Semiclassical dimensions, quantum groups and the Cantorian manifold. Chaos, Solitons and Fractals 11(2000)pages 1137 to 1144. In this paper El Naschie introduced an amazing mathematical device which he calls semiclassical dimensions. He uses the theory of quantum groups and cantorian spacetime to drive this dimension. If he sets the Hausdorff dimension of his fundamental cantor set equal to 1, he gets the space dimension equal 3. Setting for the same set the Hausdorff dimension two third he obtains the space time dimension 4. For the Hausdorff dimension equal half one finds the fermionic dimension 5. Finally for a Hausdorff dimension equal the golden mean which means a random triadic cantor set, he obtains the dimension of a four dimensional Hilbert cube which is 4 plus phi to the power of 3. This means 4.236067977 as demonstrated very recently by Ji Huan-He in Shanghai, China. I think the reason for the success of T. N. Palmer's approach and Mohamed El Naschie's analysis lies in the fact that they stopped patching work. Rather than to go on adding little bits and pieces here and there to an old fashioned building, they started from scratch. You can go patching quantum field theory as much as you want and you will always get at best an aberration of what should have been an Apollo. I find it also mildly disturbing that someone in the world renown Perimeter Institute is not aware of fundamental contributions of T. N. Palmer and his papers which are published in the proceedings of the Royal Society as well the fundamental contribution of Mohamed El Naschie who used the aforementioned semiclassical dimension using quantum group theory.

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