# Mohamed El Naschie answers a few questions about this month's new hot paper in the field of Engineering. *September 2006*

*September 2006*

### From www.esi-topics.com follow link to view original article

### Field: Engineering Article

# Title: On a fuzzy Kahler-like manifold which is consistent with the two slit experiment

### Authors: El Naschie, MS

### Journal: INT J NONLINEAR SCI NUMER SIM

### Volume: 6

### Issue: 2

### Page: 95-98

### Year: 2005

### * POB 272, Surrey KT11 2FQ, England.

### * Univ Alexandria, Fac Sci, Dept Phys, Alexandria, Egypt.

# Why do you think your paper is highly cited?

This paper was an invited paper on the occasion of the world year of Physics which was dedicated to Albert Einstein’s legacy. The paper may have triggered interest because it links two quite different aspects of science and technology, namely a fundamental question about the possible transfinite discreteness of space and time and a rather down-to-earth but equally fundamental experiment with technological engineering applications, the famous two-slit experiment with quantum particles which encapsulates the quintessence of all the weirdness of the quantum world, such as the wave-particle duality, quantum entanglement, and the situation of non-locality.

Thus, although the paper was a contribution towards the resolution of some fundamental questions linking spacetime geometry and topology to high-energy particle physics, it nevertheless approached this highly mathematical subject from a rather experimentalist viewpoint.

Perhaps this is due to the fact that, despite having previously been a professor of engineering and formally trained in the university during my under- and postgraduate years as such, I came at a very tender age "under the spell of physics"—to quote the words of a dear friend who happens to be a distinguished theoretical physicist and who has deeply relished the same beautiful "curse" from the very beginning until this very moment.

In fact, I have never recognized the traditional lines of demarcation between the sciences, not even between theoretical physics and engineering, let alone pure mathematics and applied physics. Thus, the melting of math, physics, and experimental realism may have appealed to similarly-inclined researchers and thus led to the high citation rate of this particular paper.

However, in any event, one should not forget that my approach in this paper, namely geometrizing physics, is in a direction where the majority of theoretical physicists working on the Minkowski-Einstein program are involved, and that the two-slit experiment which I attempt to resolve in the same paper is arguably the most famous and most difficult problem in quantum mechanics. There are also possible applications, as yet undreamed, for this experiment in nano and quantum technology. This may also have contributed to the high citation rate.

# Does it describe a new discovery, methodology, or synthesis of knowledge?

There is an element of all three points, essentially a new discovery about the nature of spacetime using a new methodology—namely transfinite calculations and scaling, which is a synthesis of various mathematical results from nonlinear dynamics, deterministic chaos and fractals, as well as insights gained from fuzzy sets and complex manifolds which have been added.

In the last few years, I have been attempting to reformulate E-infinity Cantorian spacetime theory—incidentally also featured in ESI—in terms of a more conventional model and, if at all possible, with a direct reference to some experiments.

It took a while for me to realize that a four-dimensional Kähler manifold used frequently in the M. Green, J. Schwarz, and D. Gross string theory called K3 could model E-infinity spacetime provided we introduce "fractal" fuzziness into it. In super string theory, K3 is used to compactify the extra six dimensions, leaving only the classical four dimensions visible. In our case the fuzzy K3, which mimics E-infinity, is postulated to be our real quantum spacetime.

The result was a set of brand-new ideas which have recently been extended to include Gerard ‘t Hooft’s holographic principle and the anti de Sitter spacetime model discussed by E. Witten. However, this particular paper, published in Int. J. Nonlinear Sci. & Numerical Simulation in 2005, was one of the first of this series of articles and apparently the most cited one which, I must admit, I did not anticipate, and which has come as a very nice surprise to me.

# Could you summarize the significance of your paper in layman’s terms?

The wave nature of light was established beyond any doubt by the famous interference experiment of T. Young. Take a torch and put a large piece of dark cardboard with two adjacent tiny holes in front of it. Go in a dark room and project the result on the wall. With some luck, and after manipulating the various distances between the torch and the cardboard, as well as the cardboard and the wall, one will notice concentric rings of dark shadows and light on the wall.

In simplistic terms, light plus light does not result always in a more intense light but could lead to darkness. In scientific terms, lights must have a wave form and can annihilate each other when out of phase, just as opposing water waves, out of phase, annihilate each other.

However, ensuing technical developments which enabled the experimentalist to emit a single light packet, or what Newton called "corpuscles," coupled with the advent of quantum mechanics, showed, without any ambiguity, that light behaves also as particles. In fact, light seems to be a particle when emitted, as well as when it arrives at the detection screen, although it seems to propagate as a wave.

This is what Louis de Broglie—the winner of the Nobel Prize in Physics in 1929—formulated mathematically, and called "material waves," and is known as wave-particle duality. However, material waves defy, not only classical Newtonian mechanics, but also common sense. The paradoxical nature of the two-slit experiment is explained in numerous popular scientific writings and books and may be summarized in the inescapable conclusion that a photon or an electron—or, for that matter, any quantum particle, including a Buckyball molecule (C60)—could be said to have passed through both slits in the screen simultaneously without splitting in two.

A fictitious macroscopic analogue of this experiment would be equivalent to a skier sliding on both sides of a tree simultaneously without hitting the tree or injuring himself. A parody on this situation is shown in Fig. (1).

Through my work on E-infinity theory, I realized that, in a spacetime manifold which is infinite-dimensional, a dimensional fractal, such as a classically impossible skiing trick, is possible in fractal land. The extra dimensions are the logical loop holes. For instance, in two dimensions, by putting both our hands on a table, it is impossible, no matter how hard we try, to bring our left and right hands to be congruent.

However, by turning one hand in the extra third dimension, we can rotate it and bring it to exactly cover the other hand on the table.

This magic can be continued in E-infinity, in a manner of speaking, indefinitely, so that in E-infinity spacetime, as in its fuzzy Kähler model, we can do infinitely many more things that we cannot do in the 3+1 Euclidean spacetime of our daily experience. Thus, I started constructing a space based on the two-slit experiment, which is infinite-dimensional in the fractal self-similar hierarchal sense, when observed with quantum mechanical high resolution. However, at our low resolution, low-energy scale of classical mechanics, the very same spacetime manifold looks like an ordinary 3+1=4 dimensional spacetime.

Proceeding in this way, we found that the space of E-infinity theory which is an infinite dimensional but hierarchal fractal called a Cantor set, may be modeled by a classical geometrical structure called K3 manifold, provided this manifold is made fuzzy. The mathematical theory of fuzzy sets is highly developed and used extensively in many practical and engineering problems.

Fractal geometry is, by its very nature, fuzzy, and that is how we were able to give K3—which is used in string theory for other purposes—a fuzzy outlook. Proceeding in this way, we did not only give a geometrical topological rational explanation for the two-slit experiment, but were also able to determine the particle content of this spacetime manifold. This led to the startling conclusion that the standard model should have, besides the 60 particles (or degrees of freedom) believed to have already been discovered experimentally, a maximum of additional particles equal to 9.

These 9 particles are thought to be 1 graviton and 8 Higgs degrees of freedom, of which, either 1 or 5 should manifest themselves as particles. Thus, starting from a fundamental quantum experiment, we were able to make predictions involving all the fundamental interactions, including gravity.

# How did you become involved in this research and were there any obstacles along the way?

My work is mainly in the theory of unification. This is a theory which believes that all forces in nature must have the same origin. Just as Achnaton of ancient Egypt believed in only one God, most of theoretical physicists believe in a theory of unification.

There are several examples of partial unification. The most successful unification is that of Maxwell equations unifying electricity and magnetism and giving us our modern present civilization. Similarly, the theory of the electroweak unified the electromagnetism with the weak force which is responsible for radioactive decay. My work in unification, however, used Einstein and his general relativity theory as a role model.

If we believe in unification and if we accept that gravity is a manifestation of the curvature of spacetime, or, more generally, the geometry and topology of our large-scale spacetime manifold, then it must be that all other forces are intimately linked with the geometry and topology of spacetime in the smaller, i.e., micro or quantum spacetime. That is the way that E-infinity theory was developed. However, I wanted something more, namely a derivation using mainstream mathematics as well as a strong relation to engineering and the experimental world. That is how I came to write the paper in question.

# Are there any social or political implications for your research?

My work is derived from complexity theory, nonlinear dynamics, and fractals, and I applied it to high-energy physics in the first place. However, as more researchers became aware of this theory, it has been applied to many other disciplines, such as biology and also brain research.

Indeed, there are many other applications far away from the so-called hardcore science—such as philosophy and political economy as applied to conflict situations. This is not strange when considering that my theory is embedded in nonlinear dynamics and deterministic chaos, which is in turn widely used in the fields of the social sciences and humanities.

### Prof. Dr. Mohamed El Naschie Distinguished Fellow of the Frankfurt Association for the Advancement of Fundamental Research in Physics Institute of Physics, University of Frankfurt Frankfurt, Germany and Principal Adviser for Science & Technology King Abdul Aziz City of Science & Technology Riyadh, Saudi Arabia