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Einstein’s dream and fractal geometry
Abstract
A comparison of the relativity with Cantorian fractals to highlight the achievements and shortcomings of Einstein's alleged dream of geometrizing physics was presented. The eight-dimensional spacetime of supersymmetry where the distinction between matter and force field was dissolved was studied. The 3-step symmetry breaking in which shadow matter as well as chiral and super symmetry were eliminated was applied. The Einstein's tensor, super gravity and the number of elementary particles in the standard model were also analyzed.
... Moreover, recent evolution of the physical theory, centered on nonlinearity, non-extensivity and fractality, shows that Prigogine's point of view was not as extreme as it was considered at the beginning. After all, Tsallis extension of statistics [2] and the fractal extension for dynamics of complex systems as it was developed by Zaslavsky [24], Castro [12], Tarasov [25,26], El Naschie [27], Milovanov and Rasmussen [28], El-Nabulsi [29], Nottale [13], Goldfain [30] and many others scientists, are the double face of a unified novel theoretical framework. In this way they constitute the appropriate base for the modern study of nonequilibrium dynamics, since the q-statistics is related, at its foundation, to the underlying fractal dynamics of the non-equilibrium states. ...
... Also, the fractional extension of dynamics includes the non-Gaussian scale invariance, related to the multiscale coupling and non-equilibrium extension of the renormalization group theory [24]. Moreover, Tarasov [25,26], Goldfain [30], Cresson and Greff [40], El-Nabulsi [29] and other scientists generalized the classical or quantum dynamics in a continuation of the original break through by Ord [76], El-Naschie [27], Nottale [13], Castro [12] and others concerning the fractal generalization of physical theory. ...
... This leads to the utilization of fractal geometry into the extended unification of physical theories, since the fractal geometry includes properties of stochastic multiscale and selfsimilar character. Scientists like Ord [76], El Naschie [27], Nottale [13] and others, introduced the idea of fractality into the geometry of space-time itself negating the notion of differentiability of physical variables. Also the fractional geometry is connected to non-commutative geometry since for fractional objects the principle of self similarity negates the Eycleidian notion of the simple geometrical point (as that which has no-parts), just like it negates the idea of differentiability and determinism. ...
In this study, we present the highlights of complexity theory (Part I) and significant experimental verifications (Part II) and we try to give a synoptic description of complexity theory both at the microscopic and at the macroscopic level of the physical reality. Also, we propose that the self-organization observed macroscopically is a phenomenon that reveals the strong unifying character of the complex dynamics which includes thermodynamical and dynamical characteristics in all levels of the physical reality. From this point of view, macroscopical deterministic and stochastic processes are closely related to the microscopical chaos and self-organization. The scientific work of scientists such as Wilson, Nicolis, Prigogine, Hooft, Nottale, El Naschie, Castro, Tsallis, Chang and others is used for the development of a unified physical comprehension of complex dynamics from the microscopic to the macroscopic level. Finally, we provide a comprehensive description of the novel concepts included in the complexity theory from microscopic to macroscopic level. Some of the modern concepts that can be used for a unified description of complex systems and for the understanding of modern complexity theory, as it is manifested at the macroscopic and the microscopic level, are the fractal geometry and fractal space-time, scale invariance and scale relativity, phase transition and self-organization, path integral amplitudes, renormalization group theory, stochastic and chaotic quantization and E-infinite theory, etc.
... The physical characteristics of complexity theory can be manifest as a physical system driven in states that are far from equilibrium (FFE). FFE statistics and dynamics can be unified via the non-extensive statistics included in Tsallis q-entropy theory [1] and the fractal generalization of dynamics included in theories developed by many scientists [2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The characteristics of nonequilibrium physical theory for space plasmas and other physical systems have been verified for many cases [16][17][18][19][20][21]. ...
... Moreover, the development of complexity theory led to extension of the probabilistic character of dynamics from a microscopic (quantum theory) to a macroscopic (classical theory) level. This showed the deeper meaning of the Boltzmann entropy principle through fractal extension of dynamics, as well as q-extension of statistics according to Tsallis nonextensive entropy theory [3][4][5]9,11,14,25,10]. ...
... Thus, fractal geometry can be used for extended unification of physical theories because it includes stochastic multi-scale and self-similar properties. Scientists then introduced the idea of fractality in the geometry of space-time, negating the notion of the differentiability of physical variables [2][3][4]. Fractal geometry is also linked to non-commutative geometry since the principle of self-similarity for fractal objects negates the Euclidian notion of a simple geometric point (one that has no parts), just as it negates the idea of differentiability and determinism. Therefore, the fractal geometry of space-time leads to the fractal extension of dynamics exploiting fractal calculus (fractal integrals and derivatives) [9,13,14,40,10]. ...
... If we switch this light source on, then light will propagate in an ever increasing circle, around the source with a velocity C which is that of light. Now, if we blot the elapsing time needed for the propagation of light on a vertical axis perpendicular to our surface with its zero point exactly at the light source point, then we obtain the light cone which plays a fundamental role in relativity theory [9]. Here we will use an E-infinity fuzzy light cone [9]. ...
... Now, if we blot the elapsing time needed for the propagation of light on a vertical axis perpendicular to our surface with its zero point exactly at the light source point, then we obtain the light cone which plays a fundamental role in relativity theory [9]. Here we will use an E-infinity fuzzy light cone [9]. The light cone gives us the means to differentiate between three types of geodesics. ...
... Special relativity states that all the matter must move at speed of light. This means that all bodies must move on time like the null geodesics, i.e., unless acted upon by forces other than gravity [9]. ...
In this paper, we will introduce a new type of geodesics of chaotic flat space, it is a retraction of chaotic flat space from the view point of Einstein’s dream and fractal geometry, as presented by El-Naschie. The relation between the limit of folding of chaotic flat space and its retraction is deduced.
... With the arrival of the special theory of relativity it became advantageous to formulate our dynamical laws in a flat four-dimensional Minkowski space-time manifold. Subsequently, it was shown in the general theory of relativity that gravity could be given an elegant geometrical interpretation if we imagined space-time to be a curved fourdimensional manifold [6][7][8]. ...
... It may seem odd that a particle should travel along each and every path simultaneously in order to restore a spacetime trajectory in quantum mechanics, but this is what the path integral formalism implies. The present author has shown sometime ago, following earlier work by Abbott and Wise as well as Ord and Nottale that the path integral could be taken more physically to mean that quantum space-time is a Cantor-set-like fractal [5][6][7][8]. This interpretation was not developed to a mathematical theory in one go, but was developed rather slowly over a number of years. ...
... Because the experimental fact implies that a quantum particle could pass through two holes simultaneously we are faced with two simple logical possibilities. Either the particle does indeed move along many paths simultaneously which is classically absurd or this space is highly complex and topologically multiply connected space-time with a high dimensionality [5][6][7][8]. ...
An idealized two-slit experiment is envisaged in which the hypothetical experimental set-up is constructed in such a way as to resemble a toy model giving information about the structure of quantum space–time itself. Thus starting from a very simple equation which may be interpreted as a physical realization of Gödel’s undecidability theorem, we proceed to show that space–time is very likely to be akin to a fuzzy Kähler-like manifold on the quantum level. This remarkable manifold transforms gradually into a classical space–time as we decrease the resolution in a way reversibly analogous to the processes of recovering classical space–time from the Riemannian space of general relativity.
... fold symmetry when mirror-image transformation is permitted. Let us leave this 336-fold symmetric pattern and consider the famous Riemannian tensor used in EinsteinÕs theory of relativity [13]. ...
... Let us look at the problem from another side, namely that of anomaly cancellation of superstring theory. This may be written in the necessary form [9][10][11]13,14] n H þ 29n t À n r ¼ 273 ...
... There we find a theorem which says that given the dimension of a metric space, then we can immediately give the maximal number possible for the isometrics of this space which is equal to the number of independent killing vector fields [9][10][11]13,14]: This is about 20 states less than the corrected 504 namely 548.3281. However, breaking the symmetry of the 528 leads to ...
The role of elementary number theory in high energy physics is discussed in some details.
... Connes' noncommutative geometry [15][16][17][18][19] which is based on von Neumann's pointless continuous geometry [24,32] and where the Menger-Urysohn dimensional theory also plays a pivotal role. We added to that the bijection formula [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] to replace and extend Connes' dimensional function and made extensive use of the Menger-Urysohn deductive theory of dimensions [9-14, 26, 36] which resulted in a giant step forward in our understanding of the essential differences between zero, empty and nothing and by realizing that way that the empty set must have negative topological dimension starting at minus one [38][39][40][41][42]. We did not stop there but extended the notion of empty set using the concept of the degree of emptiness of an empty set introduced some time ago by the late B. Mandelbrot [39] and that way made a profound conclusion preventing us from confusing zero with empty and empty with nothingness so that in this respect, we can claim to have the highest rigour possible not only in physics but also in pure mathematics [39]. ...
... The bijection formula of E-infinity is central to our present theory which as we will reason here shortly, is not only more convenient to use than Connes' dimensional function of Penrose universe [31][32][33][34][35][36][37][38] but is also much more general and much less restricted or confined to a model [38]. In all events the important point behind both mathematical formulas is to find the Hausdorff dimension corresponding to a certain topological dimension. ...
We start from a set theoretical foundation of quantum mechanics and go all the way to giving a soap bubble minimal surface interpretation of quantum wave collapse. We outline first the quintessence of a fairly new quantum theory based on transfinite set theory representing a synthesis of the work of J. von Neumann's pointless continuous geometry, A. Connes' noncommutative quantum geometry, Menger-Urysohn dimensional theory, Sir R. Penrose's fractal tiling universe, 'tHooft's renormalon as well as D. Shectman quasi crystals and Einstein-Kaluza-Klein space-time theories. We conclude that the Aether is a real non-materialistic material and that is essentially what we commonly call space-time which can be accurately described as the empty five dimensional set in a Penrose quasi crystal universe. Seen in the light of the above, empty space-time can be the source of infinite clean energy and the realization of NASA's EM drive.
... We stress once more that the sources and elements of the present synthesis is an amalgamation of the thinking and results of D. Shechtman's quasi crystals [29,31,52], its geometrical topological description by Penrose fractal tiling universe [31] and its algebraic analytical description by A. Connes' noncommutative geometry [15][16][17][18][19] which is based on von Neumann's pointless continuous geometry [24,32] and where the Menger-Urysohn dimensional theory also plays a pivotal role. We added to that the bijection formula [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] to replace and extend Connes' dimensional function and made extensive use of the Menger-Urysohn deductive theory of dimensions [9][10][11][12][13][14] [ 26,36] which resulted in a giant step forward in our understanding of the essential differences between zero, empty and nothing and by realizing that way that the empty set must be negative topological dimension starting at minus one [38][39][40][41][42]. We did not stop there but extended the notion of empty set using the concept of the degree of emptiness of an empty set introduced some time ago by the late B. Mandelbrot [39] and that way made a profound conclusion preventing us from confusing zero with empty and empty with nothingness so that in this respect, we can claim to have the highest rigour possible not only in physics but also in pure mathematics [39]. ...
... The bijection formula of E-infinity is central to our present theory which as we will reason here shortly, is not only more convenient to use than Connes' dimensional function of Penrose universe [31][32][33][34][35][36][37][38] but is also much more general and much less restricted or confined to a model [38]. In all events the important point behind both mathematical formulas is to find the Hausdorff dimension corresponding to a certain topological dimension. ...
... In the present work, we introduce E-infinity theory in which we can, in a manner of speech, "differentiate" and "integrate" the non-differentiable and non-integrible, namely geometrical structures made up entirely from transfinite point set of higher dimensional Cantor sets [12]- [16]. At a minimum, this is shockingly unconventional at least in the first few moments. ...
... This is the inverse Sommerfeld fine structure constant. The integer part is the famous prime number 137 [12]- [20]. In addition we have 0.036 ∆ ≅ leading to the experimentally well founded value 137.036 o α ≅ . ...
Starting from Witten’s eleven dimensional M-theory, the present work develops in an analogous way a corresponding dimensional fractal version where . Subsequently, the new fractal formalism is utilized to determine the measured ordinary energy density of the cosmos which turns out to be intimately linked to the new theory’s fractal dimension via non-integer irrational Lorentzian-like factor: where is Hardy’s probability of quantum entanglement. Consequently, the energy density is found from a limiting classical kinetic energy to be Here, is ‘tHooft’s renormalon of dimensional regularization. The immediate logical, mathematical and physical implication of this result is that the dark energy density of the cosmos must be in astounding agreement with cosmic measurements and observations.
... Moreover, far from equilibrium, statistics and dynamics can be unified through the Tsallis nonextensive statistics included in Tsallis q-entropy theory [1] and the fractal generalization of dynamics included in theories developed by Ord [2], El-Naschie [3], Nottale [4], Castro [5], Zaslavsky [6], Shlesinger [7], Kroger [8], Tarasov [9,76], El-Nabulsi [10,77], Cresson [11], Goldfain [12,78] and others. Also, the new characteristics of non-equilibrium physical theory for space plasmas and other physical systems were verified until now by Pavlos [13,79,80], Iliopoulos [22] and Karakatsanis [14,81] in many cases. ...
... Also, the fractal extension of dynamics includes an extension of non-Gaussian scale invariance, related to the multiscale coupling and non-equilibrium extension of the renormalization group theory [6]. Moreover Tarasov [9], Coldfain [12], Cresson [11], El-Nabulsi [10] and other scientists generalized the classical or quantum dynamics in a continuation of the original break through of El-Naschie [3], Nottale [4], Castro [5] and others concerning the fractal generalization of physical theory. ...
In this study the general theoretical framework of Complexity theory is presented, viewed through non-extensive statistical theory introduced by Constantino Tsallis in 1988. Nonlinear dynamics, fractional dynamics, thermodynamics and statistical physics constitute the basic elements of Complexity theory which can be applied to processes near, as well as, far from equilibrium. Moreover, the nonextensive statistics can be the fundamental linchpin of Complexity theory from microscopical to macroscopical level. Finally, a brief review of results concerning nonlinear analysis of time series corresponding to different complex systems is given.
... . Starting more or less from there it became the Author's lifelong work and even magical fascination to incorporate the basic structure of quantum mechanics into the very topology and geometry of space and time [7]- [428]. To do this, the author followed a path inspired by the work of Richard Feynman and its development by the Canadian Physicist G. Ord and French Astrophysicist L. Nottale [7] [12] [13]. ...
... The curves in 3 1 E have relationships with relativity in physics. Timelike, spacelike and null curves correspond to the path of a particular moving at less than the speed of light, faster than the speed of light and at a speed of light, respectively [13]. ...
In this study, a new type of timelike general helix associated to a main non-lightlike curve is introduced in Minkowski 3-space. This new helix is called an associated timelike helix. Some special types of associated timelike helices are introduced and also by considering the conditions that main curve is a non-lightlike helix or a spacelike slant helix, the position vectors of these new timelike helices are determinate. MSC: 53A04, 53C40.
... and (9) In other words in physical terms there is no difference between union and intersection in space and obtaining a which way information based on probability theory is fundamentally not possible and such a space is said to fundamentally and irreducibly nonlocal as discussed in numerous previous publications in the past twenty years or so [29][30][31][32][33][34][35][36][37][38][39]. The novel point however is the similarity to the platonic arguments applied many centuries ago using pure reason. ...
The four basic building blocks of the cosmos hypothesized by the Athenian philosopher Plato and his corresponding theory are replaced by the backbone golden mean based scaling pertinent to each of the four kinds of blocks. In the course of doing this we stretch and generalize Plato’s philosophical ideas to ultimately find out that it is essentially the deep philosophical origin of the golden mean number system of E-infinity Cantorian spacetime theory of high energy physics and quantum cosmology. Subsequently we use this new platonic form of E-infinity quantum set theory to uncover a remarkably rich Kaluza-Klein fractal version of Gross et al’s ingenious Heterotic superstring theory.
... Particularly intriguing is the mathematical possibility of a negative metric. Now it is extremely interesting that there is a geometry in which two separated points may still have a zero distance analogous to the corresponding to a null geodesic [7]. §2. ...
A Smarandache multi-space is a union of spaces ⋃ i=1 n S i for an integer n≥1. In this paper, we deduce the spacetime geodesics of a Schwarzschild space, i.e., a Smarandache Schwarzschild space with n=1, by using Lagrangian equations. A deformation retract of this space is presented. The relation between folding and deformation retraction of this space is elucidated.
... Minkowski space is originally from the relativity in physics. In fact, a time like curve corresponds to the path of an observer moving at less than the speed of light, a light like curves correspond to moving at the speed of light and a space like curves moving faster than light [17]. ...
Our aim in the present article is to introduce and study new types of retractions of hyperbola in Minkowski 3-space. Types of the deformation retracts of hyperbola in Minkowski 3-space are presented. The relations between the retraction and the deformation retract of hyperbola are deduced types of minimal retractions of hyperbola in Minkowski 3-space are also presented. Also, the isometric and topological folding in each case and the relation between the deformation retracts after and before folding have been obtained. New types of homotopy maps are deduced.
... to look something like: aa(1), ab (2), ac (3), t(4), ba (5), bb (6), bc (7), ca (8), cb (9), cc (10), ad (11), ae (12), bd (13), be (14), cd (15), ce (16), dd (17), de (18), da (19) Table 15). ...
In 2007, A. Garrett Lisi published a paper on "An Exceptionally Simple Theory of Everything". In response, this paper will explore the application of even-ranked Exceptional Lie algebras to the search for a Theory of Everything (TOE). Particular attention is given to the classic E8, and quasi-Exceptional, semi-simple E10, E12, and E8 × H4 algebras. The complete skeletal framework of a TOE including fermion, boson and scalar boson content is presented for these algebras.
... Minkowski space is originally from the relativity in physics. In fact, a timelike curve corresponds to the path of an observer moving at less than the speed of light, a null curve correspond to the moving at a speed of light and a spacelike curves to moving faster than light [26]. The importance of the study of null curve and its presence in the physical theories is clear from the fact that: the classical relativistic string is a surface or world-sheet in Minkowski which satisfies the Lorentzian analogue of minimal surface equation [27]. ...
In this paper, we introduce the notion of the principal (binormal)-directional curve and the principal (binormal)-donor curve of the Frenet curve in the Minkowski space E13. Furthermore, we study some associated curves of a given Frenet curve to classify the general helices and the slant helices. Also, we introduce the natural representation of general helices and slant helices from the natural representation of plane curves and general helices, respectively. As an example, we give the natural representation of the closed slant helices constructed by circles.
... We note that massless particles, such as the photon, move on null geodesies. This can be interpreted as saying that in a 4-dimensional space, the photon does not move and that for a photon, time does not pass, intriguing is the mathematical possibility of a negative metric [19,20]. Now, it is extremely interesting that there is a geometry in which two separated points may still have a zero aistar.ee ...
Our aim is to introduce and study a new type of metric, namely chaotic Ricci metric. Types of the chaotic retractions of the chaotic Ricci space are presented. The geodesics of chaotic Ricci space will be deduced. The connection between the deformation retract and the folding of chaotic Ricci space is presented. Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.
... The notion of transfinite discreteness is homomorphic to fuzzy topology, foliation, and fractal geometry (El Naschie 2005; Nottale 2007). There is thus an interesting convergence between the rediscovery of the role of time in science – as factor of irreversibility – and this manner for man to imagine that human individual destiny, that of the human nervous system, of the encounters that have brought humanity about must subsist with humanity itself, anticipating a resurrection that would mean infinitely more than some angelization. ...
... However, for such a dream to come true, new physical and mathematical concepts must be used such as, cooperation of local, non–local and long–range correlations in complex systems. A new complementarity is developed fundamental local physical interactions and global ordering physical process including fractal space–time, or fuzzy space and non–communicative geometry or wild topologies [15], [36], [38], [46], [50], [84], [87]. In this direction and in order to remark the new state we can summarize by the following phrase of Castro: It is reasonable to suggest that there must be a deeper organizing principle, from small to large scales, operating in Nature which might be based in the theories of complexity, non linear dynamics and information theory where dimensions, energy and information are intricately connected [22]. ...
During the last two decades our scientific group has developed new nonlinear
methods of analysis applied to various physical systems. In this review study
we present our scientific contribution to nonlinear science, including also some novel
concepts as for the constructive role of complexity in modern physical theory. The
experimental verification of chaos existence in physical systems remains one of the
most signi�cant problems of non linear science and complexity. The extended chaotic
algorithm presented in the following as well as the results concerning its application at
di�erent experimental time series reveal the universal character of the complexity theory
for the far from equilibrium dynamics of spatially extended physical systems. The
developed methodology that was used compromises different types of computational
tools as well as theoretical concepts for the physical interpretation of the experimental
information. As we present here the strong dispute and criticism of chaos hypothesis
in physical systems during the last two decades was fruitful and challenged us to
develop a novel composition of experimental and theoretical knowledge of universal
character for the far from equilibrium dynamics. The solar and magnetospheric dynamics
included in space plasma processes, the environmental and seismic dynamics,
the human brain or the on{chip workload are distinct systems which were studied
by our group revealing common chaotic characteristics and chaotic phase transition
processes. Moreover, the intellectual struggle for the comprehension of the theoretical
presuppositions of the experimentally observed universal chaotic character of spatially
distributed systems lead us to the fundamentals of complexity theory as manifested
at the macroscopic and microscopic level of physical reality. From this point of view,
some common characteristics of macroscopic and microscopic complexity included in
the scientific knowledge of the recent two or three decades can be used as a road
for the physical theory unification. That is complexity, scaling, chaos, quanticity and
fractality could be supported as different manifestations of a unified physical law from
the microscopic to the macroscopic and cosmological level. As we can argue, determinism
and probabilism can also be unified through chaoticity. Moreover, the rising
of new physical knowledge reveals that under the macroscopic or the microscopic
physical phenomena there exist a fundamental and multilevel acting unit physical
process that produces physical reality rather than a fundamental essence or simple substance from which cosmos can be build.
... We should mention here that El Naschie have discussed in E-infinity theory random Cantor sets with the golden mean dimension which is very important in quantum physics [9][10][11][12]. ...
The problem of “rate of change” for fractal functions is a very important one in the study of local fields. In 1992, Su Weiyi has given a definition of derivative by virtue of pseudo-differential operators [Su W. Pseudo-differential operators and derivatives on locally compact Vilenkin groups. Sci China [series A] 1992;35(7A):826–36. Su W. Gibbs–Butzer derivatives and the applications. Numer Funct Anal Optimiz 1995;16(5&6):805–24.
[2] and [3]]. In Qiu Hua and Su Weiyi [Weierstrass-like functions on local fields and their p-adic derivatives. Chaos, Solitons & Fractals 2006;28(4):958–65. [8]], we have introduced a kind of Weierstrass-like functions in p-series local fields and discussed their p-adic derivatives. In this paper, the 3-adic Cantor function on 3-series field is constructed, and its 3-adic derivative is evaluated, it has at most order. Moreover, we introduce the definition of the Hausdorff dimension [Falconer KJ. Fractal geometry: mathematical foundations and applications. New York: Wiley; 1990. [1]] of the image of a complex function defined on local fields. Then we conclude that the Hausdorff dimensions of the 3-adic Cantor function and its derivatives and integrals on 3-series field are all equal to 1.There are various applications of Cantor sets in mechanics and physics. For instance, E-infinity theory [El Naschie MS. A guide to the mathematics of E-infinity Cantorian spacetime theory. Chaos, Solitons & Fractals 2005;25(5):955–64. El Naschie MS. Dimensions and Cantor spectra. Chaos, Solitons & Fractals 1994;4(11):2121–32. El Naschie MS. Einstein’s dream and fractal geometry. Chaos, Solitons & Fractals 2005;24(1):1–5. El Naschie MS. The concepts of E infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22(2):495–511.
[9],
[10],
[11] and [12]] is based on random Cantor set which takes the golden mean dimension as shown by El Naschie.
In this study, we determine subplanes of Hall projective plane coordinatized
by elements of a left nearfield of order 9 by using an algorithm (implemented in C#).
In the present short letter we aim at deriving the cosmic ordinary effective quantum gravity energy density as well as that of dark energy from the SO (10) Lie symmetry group of grand unification. Remarkably the derivation makes no use of quantum mechanics and remains largely classical except for nonclassical geometry and topology. Finally our main conclusions and results are reinforced using a nonlocal classical elastic field theory.
A straightforward simple proof is given that dark energy is the natural conse-quence of a quantum disentanglement physical process. Thus while the ordinary energy density of the cosmos is equal to half that of Hardy’s quantum probability of Entanglement i.e. where , the density of cosmic dark energy is consequently one minus divided by two i.e. . This result is in full agreement with all the numerous previous theoretical predictions as well as being in remarkable agreement with the overwhelming majority of cosmic accurate measurements and observations.
The topological speed of light which may be used to compute the density of ordinary energy and dark energy of the cosmos is replaced by dimensionless quantity taken from Special Relativity. The said quantity may be interpreted as akin to time dilation ergo a notion topologically equivalent to the speed of the passing of time or the difference of elapsed time between two events in Einstein’s Relativity Theory. This results via Newton’s kinetic energy into the well-known observationally confirmed and accurately measured 4.5 and 95.5 percent of ordinary and dark Cosmic Energy density respectively.
The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacetime be four-dimensional on all scales. It is true that on our classical scale, the 4D decouples into 3D plus one time dimension and that on very large scale only the curvature of spacetime becomes noticeable. However the critical problem is that such spacetime must remain 4D no matter how small the scale we are probing is. This is something of crucial importance for quantum physics. The present work addresses this basic, natural and logical requirement and shows how many contradictory results and shortcomings of relativity and quantum gravity could be eliminated when we “complete” Einstein’s spacetime in such a geometrical gauge invariant way. Concurrently the work serves also as a review of the vast Literature on E-Infinity theory used here.
In a sense Feynman's path integral and the transfinite sums over infinite dimensions used in the E-Infinity Cantorian spacetime theory are two complimentary ways of describing the outcome of the two slit experiment. In the present work an idealized two-slit experiment is envisaged in which a hypothetical experimental set-up is constructed in such a way as to resemble a toy model giving information about the structure of quantum space-time itself. Thus starting from a very simple equation which may be interpreted as a physical realization of Gödel's undecidability theorem, we proceed to show that space-time is very likely to be akin to a fuzzy Kähler-like manifold on the quantum level. This remarkable manifold transforms gradually into a classical space-time as we decrease the resolution in a way reversibly analogous to the processes of recovering classical space-time from the Riemannian space of general relativity. The main message of the present paper is to emphasize that the two-slit experiment, as explained via Feynman's path integral, could be given a different interpretation by altering our classical concept of space-time geometry and topology. This would be in line with the development in theoretical physics since special and general relativity. In the final analysis it would seem that we have two different yet, from a positivistic philosophy viewpoint, completely equivalent alternatives to view quantum physics. Either we insist on what we see in our daily experiences, namely, a smooth four-dimensional space-time, and then accept, whether we like it or not, things such as probability waves and complex probabilities. Alternatively, we could see behind the charade of classical space-time the real far more elaborate and highly complex fuzzy space-time with infinite hierarchical dimensions as is the case with the so-called Fuzzy K3 or E-Infinity space-time. The reward for this imaginative picture is that we can return to real probabilities without a phase and an almost classical picture with the concept of a particle's path restored. We say almost classical because non-linear dynamics and deterministic chaos have long shown the central role of randomness in classical mechanics and that there can be no return to absolute determinism. This fact is reinforced once more in our model which is directly related not to Newtonian motion, but rather to a diffusion-like random walk similar to that used with great skill by Einstein and later on by Nagasawa and particularly the English-Canadian physicist Garnet Ord.
This short letter, in fact, this short telegram is mainly intended to point out a recent and quite unexpected realization that E-Infinity space time (E-infinity) theory (M. S. El Naschie,Chaos, Soliton & Fractals, 29 pp. 209-236, 2004) could be of a considerable help in deciphering one of the greatest secrets and impenetrable questions of our own existence, namely what is consciousness and how does it relate to the brain(G. M. Edelman. Consciousness. Penguin Books, London,2000).
Abstract
Eringen nonlocal elasticity is extended to an elastic Cosserat-like model of spacetime and
a nano/micro scale charged jet in Bubbfil spinning process, respectively, to explain El
Naschie’s dark energy equation E = mc2
(21/22) and the hierarchy motion in the
electrospinning process.
Keywords: Nonlocal elasticity, discrete element method, effective quantum gravity,
modified relativity, dark energy, semi-classical field theory, instability, electrospinning,
size-effect (nano-effect)
What is the reason for Einstein's failure to unify gravity with electromagnetism? Now El-Naschie gives a positive answer. Space and time are discontinuous, 4 dimensional spacetime is not enough for Einstein's unified field theory. El-Naschie's E-infinity theory is mathematically perfect and beautiful as well as geometrically simple, it is an art, poetry, and philosophy as well.
The objective of living is the enjoyment of beauty and promoting life in all its wonderful manifestations. This short note takes the reader on a very short but intensive trip from King Ludwig's magical Room of Mirrors via Cocteau's tunnelling to the magnificent topological models of Thurston, with the help of which we are trying to understand our universe.. In this connection we briefly discuss the recent important work of Luminet and lovane which were triggered by the new WMAP cosmological data.
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JQIS> Vol.5 No.1, March 2015
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A Fractal Rindler-Regge Triangulation in the Hyperbolic Plane and Cosmic de Sitter Accelerated Expansion
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DOI: 10.4236/jqis.2015.51004 1 Downloads 3 Views
Author(s) Leave a comment Mohamed S. El Naschie*
Affiliation(s)
Department of Physics, University of Alexandria, Alexandria, Egypt.
ABSTRACT
The well known finite elements Regge calculus is transformed to a triangulation in the hyperbolic plane using fractal Rindler wedges as tiling elements. The final result is an expanding de Sitter hyperbolic, i.e. Gauss-Bolyai-Lobachevsky universe with dark energy and ordinary energy densities in full agreement with cosmic observations and measurements. In the course of obtaining this vital result, the work addresses fundamental points connected to a host of subjects, namely Hardy’s quantum entanglement, an extension of Turing’s machine to a transfinite version, the phenomenon of measure concentration in the context of Banach-like spaces with high dimensionality as well as the pioneering work on the relation between quantum entanglement and computational efficiency.
KEYWORDS
Component, Hyperbolic Regge Calculus, Finite Elements in Cosmology, de Sitter Universe, E-Infinity Theory, Transfinite Turing Golden Mean Computer, Rindler Triangulation, Endophysics, Anti-Bethes Poof, Topological Quantum Entanglement, Gauss-Bolyai-Lobachevsky Geometry
Cite this paper
Naschie, M. (2015) A Fractal Rindler-Regge Triangulation in the Hyperbolic Plane and Cosmic de Sitter Accelerated Expansion. Journal of Quantum Information Science, 5, 24-31. doi: 10.4236/jqis.2015.51004.
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[22] El Naschie, M.S. (2011) Quantum Entanglement as a Consequence of a Cantorian Micro Spacetime Geometry. Journal of Quantum Information Science, 1, 50-53.
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[23] Amendola, L. and Tsujikawa, S. (2010) Dark Energy: Theory and Observations. Cambridge University Press, Cambridge.
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[27] El Naschie, M.S. (2008) Transfinite Harmonization by Taking the Dissonance out of the Quantum Field Symphony. Chaos, Solitons & Fractals, 36, 781-786.
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[28] El Naschie, M.S. (2004) A Review of E-Infinity and the Mass Spectrum of High Energy Particle Physics. Chaos, Solitons & Fractals, 19, 209-236.
http://dx.doi.org/10.1016/S0960-0779(03)00278-9
[29] El Naschie, M.S. (2006) Elementary Number Theory in Superstring Loop Quantum Mechanics, Twistors and E-Infinity High Energy Physics. Chaos, Solitons & Fractals, 27, 297-330.
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[30] El Naschie, M.S. (2006) Intermediate Prerequisites for E-Infinity Theory. Chaos, Solitons & Fractals, 30, 622-628.
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[31] El Naschie, M.S. (2006) Advanced Prerequisites for E-Infinity Theory. Chaos, Solitons & Fractals, 30, 636-641.
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[32] El Naschie, M.S. (2006) Topics in the Mathematical Physics of E-Infinity Theory. Chaos, Solitons & Fractals, 30, 656-663.
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[33] El Naschie, M.S. (2009) The Theory of Cantorian Spacetime and High Energy Particle Physics (an Informal Review). Chaos, Solitons & Fractals, 41, 2635-2646.
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[34] El Naschie, M.S., Olsen, S, He, J-H., Marek-Crnjac, L., Nada, S.I. and Helal, M.A. (2012) On the Need for Fractal Logic in High Energy Quantum Physics. Fractal Spacetime & Noncommutative Geometry in Quantum & High Energy Physics, 2, 80-92.
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http://dx.doi.org/10.1016/S0960-0779(02)00587-8
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We find all space-like loxodromes on rotational surfaces which have space-like meridians or time-like meridians, respectively by using a relevant Lorentzian angle in Minkowski 3-space. To understand loxodromes better, we draw some pictures of them via Mathematica computer program. (c) 2013 Elsevier Inc. All rights reserved.
Based on the connection between Tsallis nonextensive statistics and
fractional dimensional space, in this work we have introduced, with the aid of
Verlinde's formalism, the Newton constant in a fractal space as a function of
the nonextensive constant. With this result we have constructed a curve that
shows the direct relation between Tsallis nonextensive parameter and the
dimension of this fractal space. We have demonstrated precisely that there are
ambiguities between the results due to Verlinde's approach and the ones due to
fractional calculus formalism. We have shown precisely that these ambiguities
appear only for spaces with dimensions different from three. Hence, we believe
that this is a result in favor of our three dimensional world.
An orbital picture depicts the path of an object under semi-group of transformations. The concept initially given by Barnsley [3] has utmost importance in image compression, biological modeling and other areas of fractal geometry. In this paper, we introduce superior iterations to study the role of linear and nonlinear transformations on the orbit of an object. Various characteristics of the computed figures have been discussed to indicate the usefulness of study in mathematical analysis. Modified algorithms are given to compute the orbital picture and V-variable orbital picture. An algorithm to calculate the distance between images makes the study motivating. A brief discussion about the proof of the Cauchy sequence of images is also given.
V-variable fractals and superfractals have recently been introduced by Barnsley [Barnsley Michael, Hutchinson John, Stenflo Örjan. A fractal valued random iteration algorithm and fractal hierarchy. Fractals 2005;13(2):111–46 [MR2151094 (2006b:28014)]] to the world of mathematics and computer graphics. In this paper, we introduce superior iterates to study the role of contractive and non-contractive operators in relation to superfractals. A modified algorithm along with details of computer implementation is also provided to compute V-variable fractals. A brief discussion about the various aspects of the computed figures indicates usefulness of the study.
The paper is written partially very informal with one formal section about hyper Kähler manifolds of the E-infinity fuzzy type. We discuss spinors and the particle content of minimally extended super-symmetric standard model.
The present work constitutes a guided tour through the mathematics needed for a proper understanding of E-infinity theory as applied to high energy physics and quantum gravity.
The present work is in response to a recent short note by Prof. W. Martienssen regarding the wave-particle duality. It is also dedicated to his pending 80th birthday for which not only the author, but a whole generation of physicists who have the privilege to call themselves his students, would like to congratulate him with the hope that his wave function may go on propagating for a very long time.
An analysis of the manner of establishing a relativistic micro-universe with respect to the motion of soliton-like defects in elastic media is performed. It is demonstrated that the change of variables in the elastic-dynamic equations holding the motion of a screw dislocation must be complemented by the contraction law for the displacement vector and that a theory based on Lorentz's transformations is not the only possible framework for representing the motion of soliton-like defects. (c) 2005 Elsevier Ltd. All rights reserved.
R/S method is widely used to estimate long-range dependence of a time series, but few papers do research on how to use the R/S method to determine the periodicity. In this paper, the log(h)-log((R/S)h) figures and the log(h)-Vh figures are further studied by lots of numeral simulations, which shows that the two figures of a periodic time series both have obviously similar structures. Based on these structures, a new method, similar figure method (SFM), is established to estimate whether a time series has periodicity and determine the length of the periodicity. SFM is tested with a disturbed nonlinear time series, an actual monthly runoff series and a random series. The results show that SFM is effective. This method is an extension to the R/S analysis.
We present an invariant characterization of a curved super symmetric spacetime in terms of scalers constructed from the Riemannian and metric tensors of general relativity. The corresponding number of independent components found that way leads to results consistent with the number of massless states of the Heterotic string theory, namely 8064 as well as the phenomenology at the energy scale of the standard model, i.e. 60 experimentally verified particles plus two conjectured particles.
In this article we introduce types of chaotic spheres in chaotic space-like Minkowski space time Mn+1. The variations of the density functions under the folding of these chaotic spheres are defined. The foldings restriction imposed on the density function are also discussed. The relations between the folding of geometry and pure chaotic manifolds are deduced. Some theorems concerning these relations are presented.
At Kharrouba (near Tunis City in northern Tunisia), on the southern margin of the Tethyan realm, the Paleocene-Eocene (P-E) transition interval deposition is continuous and complete. Based on high-resolution analysis and quantitative data of planktonic and benthic foraminifera at the Kharrouba section, this transition interval records expanded deposition of the relevant standard planktonic foraminiferal biozones with indicative index species i.e.: Morozovella velascoensis for the latest Paleocene P5 zone, and Acarinina sibaiyaensis for the earliest Eocene E1 zone, Pseudohastigerina wilcoxensis for the E2 zone, Morozovella marginodentata for the E3 zone and Morozovella formosa for the E4 zone. This complete section contains benthic foraminiferal assemblages which include calcareous and agglutinated cosmopolitan deep-water species (DWBF). Among the calcareous deep benthic foraminifera Aragonia velascoensis Anomalinoides rubiginosus, Oridorsalis umbonatus, Nuttallides truempyi, Pullenia coryelli and Tappanina selmensis, are relatively abundant. These species are the main representatives of the Velasco fauna indicative of a bathyal-abyssal environment. Moreover, within this section, the agglutinated species e.g., Glomospira charoides, Karrerulina horrida, Rzehakina epigona, Ammodiscus spp. and Gaudryina pyramidata, assumed to be restricted to deep-sea palaeoenvironments, constitute an important proportion of the benthic foraminiferal assemblages. Therefore, during the Paleocene-Eocene transitional period, the Kharrouba area hosted many cosmopolitan deep-sea benthic foraminiferal species as was the case at Zumaya and several DSDP sites. The depth range tolerances of these deep-marine taxa, both with calcareous and agglutinated test, indicate that close to the P/E boundary, the Kharrouba area was located in the lower bathyal environment in the southern Tethys margin.
During the last two decades, low dimensional chaotic or selforganized criticality (SOC) processes have been observed by our group in many different physical systems such as space plasmas, the solar or the magnetospheric dynamics, the atmosphere, earthquakes, the brain activity as well as in informational
systems. All these systems are complex systems living far from equilibrium with strong self-organization and phase transition character. The theoretical interpretation of these natural phenomena needs a deeper insight into the fundamentals of complexity theory. In this study, we try to give a synoptic description of complexity
theory both at the microscopic and at the macroscopic level of the physical reality. Also, we propose that the self-organization observed macroscopically is a phenomenon that reveals the strong unifying character of the complex dynamics which includes thermodynamical and dynamical characteristics in all levels of the physical reality. From this point of view, macroscopical deterministic and stochastic processes are closely related to the microscopical chaos and self-organization. In this study the scientific work of scientists such as Wilson, Nicolis, Prigogine, Hooft, Nottale, El Naschie, Castro, Tsallis, Chang and others is used for the development of a unified physical comprehension of complex dynamics from the microscopic to the
macroscopic level.
The theory of space-charge-limited currents (SCLC) is briefly reviewed and the spectroscopic character of the method is discussed. Shapes of current–voltage characteristics of thin films strongly depend on temperature, presence of charge carrier traps and their distribution in energy, spatial inhomogeneity of the sample and electrode configuration. The presence of traps influences generation-recombination noise which can be used as additional information for the determination of density-of-states function. The use of the technique is illustrated with experimental results obtained on thin films of phthalocyanines.
In this paper, we will show the consequences of the link between ε(∞) and H(∞). Starting from El Naschie’s ε(∞) nature shows itself as an arena where the laws of physics appear at each scale in a self–similar way, linked to the resolution of the observations; while Hilbert’s space H(∞) is the mathematical support to describe the interaction between the observer and dynamical systems.The present formulation of space–time, based on the non-classical, Cantorian geometry and topology of the space–time, automatically solves the paradoxical outcome of the two-slit experiment and duality. The experimental fact that a wave-particle duality exists is an indirect confirmation of the existence of ε(∞). Another direct consequence of the fact that real space–time is the infinite dimensional hierarchical ε(∞) is the existence of the scaling law R(N). The present author proposed it as a generalization of the Compton wavelength. This rule gives an answer to segregation of matter at different scales; it shows the role of fundamental constants like the speed of light and Plank’s constant h in the fundamental lengths scale without invoking the methodology of quantum mechanics.In addition, we consider the genesis of E-Infinity. A Cantorian potential theory can be formulated to take into account the geometry and topology of ε(∞) in the context of gravitational theories. Consequently, we arrive at the result of the existence of gravitational channels.
It was shown recently for a chaotic and a nonchaotic slow-moving dynamical system that the Newtonian and special relativistic trajectories eventually disagree completely. Here, it is argued that once the Newtonian trajectory is completely different from the relativistic trajectory, Newtonian mechanics can no longer be used to study the trajectory of the slow-moving system. Special relativistic mechanics must be used instead. For a nonchaotic system, Newtonian mechanics can still be used for a very long time. However, for a strongly chaotic system, Newtonian mechanics cannot be used essentially from the outset.
In this paper, we study the position vectors of a timelike and a null helix (or a W-curve), i.e. curve with constant curvatures in the Minkowski 3-space . We give some characterizations for timelike and null helix whose image lies on the Lorentzian sphere or pseudo-hyperbolical space by using the positions vectors of the curve.
In this paper, starting from El Naschie E-infinity Cantorian space–time, we consider the link between the Brownian motion presented by the first author in some earlier papers in the context of cosmology and the multiresolution analysis based on the wavelet transform. We will consider the hypothesis that the present hierarchical segregated Universe is the effect of an expansion, which follows a scaling law. We will give a mathematical method to support El Naschie’s picture of the resolution dependence of the observations. In other words, this confirms the vision in which everything we see or measure is resolution dependent.
Chaos, Solitons & Fractals
Volume 19, Issue 1, January 2004, Pages 209–236
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A review of E infinity theory and the mass spectrum of high energy particle physics
M.S. El Naschie
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doi:10.1016/S0960-0779(03)00278-9
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Abstract
The essay outlines the basic conceptual framework of a new space–time theory with application to high energy particle physics. Both achievements and limitations are discussed with direct reference to the mass spectrum problem.
1. The main purpose of the present work
In what follows we would like to give a short account of the so-called E infinity (ε(∞)) theory, the main application of which has been so far in determining coupling constants and the mass spectrum of the standard model of elementary particles. I am afraid I will have to make a long story (which took many years of work) quite short with all of what this entails in reading it. The results of E infinity are at present contained in dozens of published papers too numerous to refer to them all, but for the purpose of filling the gaps in the present summary, half a dozen papers which are mentioned at the end may offer good help in overcoming the inevitable shortcomings in a condensed presentation (see Ref. [1], [2], [3], [4], [5], [6] and [7]).
2. An outline of the conceptual framework of the theory
2.1. General remarks
The main conceptual idea of my work (which is encoded in Fig. 1 and Fig. 5) is in fact a sweeping generalisation of what Einstein did in his general theory of relativity, namely introducing a new geometry for space–time which differs considerably from the space–time of our sensual experience. This space–time is taken for granted to be Euclidean. By contrast, general relativity persuaded us that the Euclidean 3 + 1 dimensional space–time is only an approximation and that the true geometry of the universe in the large, is in reality a four dimensional curved manifold. In E infinity we take a similar step and allege that space–time at quantum scales is far from being the smooth, flat and passive space which we use in classical physics [1], [2] and [3]. On extremely small scales, at very high observational resolution equivalent to a very high energy, space–time resembles a stormy ocean [1]. The picture of a stormy ocean is very suggestive and may come truly close to what we think the high energy regime of the quantum world probably looks like (see Fig. 1, Fig. 2, Fig. 3 and Fig. 4). However such a picture is not accessible to mathematical formulation, let alone an exacting solution. The crucial step in E infinity formulation was to identify the stormy ocean with vacuum fluctuation and in turn to model this fluctuation using the mathematical tools of non-linear dynamics, complexity theory and chaos [1], [8] and [9]. In particular the geometry of chaotic dynamics, namely fractal geometry is reduced to its quintessence, i.e. Cantor sets (see Fig. 5) and employed directly in the geometrical description of the fluctuation of the vacuum. How this is done and how to proceed from there to calculating for instance the mass spectrum of high energy elementary particles is what I will try to explain and summarise in the following essay.
Full-size image (112 K)
Fig. 1.
Tiling the plane using Klein’s modular curve in the Beltrami–Poincare representation. E infinity theory alleges that the quantum gravity of space–time is a hyperbolic fractal on a Klein modular group akin to what is shown in the figure. The relevance to high energy physics is more direct than one may suspect. For instance, the Weinberg mixing parameter sin2θw at the electroweak scale is given by the cosine of the 3π/7 angle of Klein’s modular curve as cos(3π/7)≅0.2225 in excellent agreement with the experimental measurement. Similarly the number of triangles gauged in our mass norm gives the approximate value for the constituent mass of the fundamental light quarks. Since we have 168 automorphisms, then the number is (2)(168) and the masses are given by mu*=md*=336 MeV for the up and down quarks. In addition the 42 point unique orbit of the original Klein curve corresponds to the inverse of the quantum gravity couplings constant in the non-supersymmetric case.
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Fig. 2.
A depiction of T. Right’s cosmos as a form of sphere packing on all scales. This is very similar to the author’s S(∞). This space is related, but by no means identical to the Cantorian E infinity space. One notes the similarity with Fig. 4. However Fig. 4 was published for the first time in Chaos, Solitons & Fractals 11 (2000) 453–464. By contrast, T. Right’s cosmos was conceived by him in the middle of the 18th century.
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Fig. 3.
A fractal-like universe, with clusters of clusters ad infinitum as envisaged by the Swedish astronomer C. Charlier who lived between 1862 and 1934. This work was clearly influenced by the work of the Swedish astrophysicist A. Swedenborg (1688–1772).
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Fig. 4.
An artistic impression of E infinity space–time published by the author almost two and a half centuries after the work of Swedenborg and without any conscious knowledge of this or similar work of the same period. The figure represents a form of space made up of turbulent disorderly packed 3D spheres. E infinity space is similar only it has infinitely more dimensions.
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Fig. 5.
An alternative two dimensional construction of a topological equivalent to the Cantor set using pairs of circles. The E infinity limit set is very similar but has infinitely many dimensions and not only two like here.
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As is well known, special relativity fused time and space together, then came general relativity and introduced a curvature to space–time. Subsequently Kaluza and later on Klein added one more dimension to the classical four in order to unify general relativity and electromagnetism. From this time on, the dimensionality of space–time played a paramount role in the theoretical physics of unification leading to the introduction of the 26 dimensions of string theory, the 10 dimensions of super string theory and finally the Heterotic string theory [10] dimensional hierarchy 4, 6, 10, 16 and 26. This is all apart from the so-called abstract or internal dimensions of various symmetry groups used, for instance, the 8 dimensions of the SU(3) of the strong interaction.
By contrast, in E infinity theory we admit formally infinite dimensional “real” space–time [1] and [2]. However this infinity is hierarchical in a strict mathematical way and we were able to show that although E infinity has formally infinitely many dimensions, seen from a distance, i.e. at low resolution or equivalently at low energy, it mimics the appearance of a four dimensional space–time manifold which has only four dimensions. Thus the four dimensionality is a probabilistic statement, a so-called expectation value. It is remarkable that the Hausdorff dimension of this topologically four dimensional-like “pre” manifold is also a finite value equal to 4+φ3, where with the remarkable self-similar continued fracture representation (which is in a sense self-similar as Fig. 1, Fig. 2 and Fig. 5):
There are various ways for deriving this result which was given in detail in numerous previous publications. However maybe the simplest and most direct way is to proceed from the mathematical definition of E infinity.
2.2. Definition of the E infinity space
Definition
E infinity refers to the limit set of a pre-geometry model of the transfinite extension of a projective Borel hierarchy [11].
From the definition of the above and in particular the definition of Borel sets and projective hierarchy [11], it follows that if the sets involved in the Borel set are taken themselves to be transfinite Cantor sets (see Fig. 5), then the Hausdorff dimension of E infinity could be written as [1], [2], [8] and [9]
where d(0)c is the Hausdorff dimension of the involved transfinite sets where the superscript refers to the Menger–Urysohn dimension of the one dimensional Cantor set, namely 1−1=0. Now there is a well known theorem due to Mauldin and Williams which states that with a probability equal to one, a one dimensional randomly constructed Cantor set will have the Hausdorff dimension [1] and [2] equal to , i.e. the golden mean φ. Setting dc(0)=φ one finds
as anticipated. It is now instructive to contemplate the following. The intersection rule of sets shows that [1] and [2] we can lift dc(0) to any dimension n as follows
Thus taking d(0)c=φ and n=4, one finds
In other words, we have
which shows that the expectation value of the Hausdorff dimension of E infinity is 4+φ3 but its intrinsic embedding “expectation” dimension is exactly 4 and that although the formal dimension is infinity. In fact the expression ∑∞0n(dc(0))n may be regarded as the sum of the weighed n=1, n=2, n=3, … dimensions where the weights are the golden mean and its power. That is why E infinity is hierarchical. Note that intrinsic embedding is just another name for the Menger–Urysohn dimension and that our intuitive embedding dimension for dc(0) is not zero, but one. Similarly for dc(4) it is 5 and not 4.
2.3. The limit set, Kleinian groups and Penrose tiling
It then turns out that the limit set of any Kleinian-like group is a set which is best described in terms of chaotic Cantor sets (see Fig. 1 and Fig. 5) and E infinity [9]. This fact is clear from the work of Mumford et al. [12]. Another surprise was the realisation that
is just twice the isomorphic length of the so-called Penrose-hyperbolic fractal tiling [1] and [2]
where ϱ is the radius of the circular region considered.
In other words if one projects the space–time of vacuum fluctuation on a Poincare circle we will see a hyperbolic tesselation of this circle with predominantly Klein curve-like geometry [13] which ramifies at the circular boundary exactly as in many of the famous pictures of the Dutch artist M. Echer. It is an important part of our thesis that actual quantum space–time strongly resembles the hyperbolic geometry of the ramified χ(3π/7) Klein curve (see Fig. 1).
We started with the picture of a turbulent sea which we took to model vacuum fluctuation then moved to model the space–time of the vacuum using infinite numbers of unions and intersections of an elementary random Cantor set only to find at the end that this is the limit set of the well known Möbius–Klein transformation of space [9], [12] and [13] which may be represented using the Beltreram–Poincare methods of hyperbolic geometry [1]. In other words, quantum space–time is conceived here as a hyperbolic fractal in which the low resolution major part is the original Klein curve while the high resolution part at the circular boundary may be considered a transfinite correction which may be superimposed to it following certain rules, just like in a perturbation analysis of a weakly non-linear problem. For instance, and as will be shown later on, the dimension SU(5)=24 will be replaced by 24−φ9=24−0.013155617=23.98684438 and similarly Dim E8⊗E8=496 will become 496−k2=45.96747752 where k=φ3(1−φ3) and φ9 are examples of such transfinite “corrections”.
We will see later on that the Heterotic string hierarchy is also imbedded in E infinity and that the theory is clearly related to A. Conne’s non-commutative geometry [14] as well as the four dimensional fusion algebra and M. Friedman’s theory of four dimensional topological manifolds [14]. It is worthwhile to note that in all the preceding theories, our Hausdorff dimension plays a pivotal role [14].
2.4. The string connection and KAM theorem
Having mentioned string theory, we should mention that in string theory particles are perceived as highly localised vibration of Planck length strings, so that strictly speaking, within string theory there is no essential difference between a resonance particle and say a meson or an electron. The nice thing about the geometrical–topological picture which E infinity theory offers is that the string picture may be retained yet in another form, namely as a sizzling Cantor set [2], simulating string vibration and that such strings are embedded in E infinity as will be shown. Thus the two theories remain reasonably compatible. However we are running ahead of the logical sequence and we should return to the Cantorian hyperbolic geometry of quantum space–time. It is well known that hyperbolic geometry is highly distortive. Taking the original Klein curve as an example [13], all triangles are essentially the same, yet they are distorted and the further away from the center of the Beltrami–Poincare projection we are, the larger the distortion is. It is here that our basic conjecture regarding the mass spectrum of high energy particle physics come to the fore. We will show that all particles are just different scaling of all other particles as long as we disregard all other aspects and concentrate on energy. E infinity has a set of scaling exponents which distorts the “figure” of any particle so that it all depends on the region which the particle inhabits, or said in another way, it all depends on the way we probe space–time (see Fig. 1).
Exactly as in Einstein’s general relativity and even far more so, space–time topology decides on the mass spectrum which we observe. Seen that way one should really expect infinite numbers of elementary particles but this is clearly not the case. It is one of the main pillars of the E infinity theory to hold that the whole issue is that of stability. Only stable particles could be observed. Again it is one of the most important results of E infinity theory to reason that the question of stability of elementary particles is closely related to KAM theory, Arnold diffusion and the vague attractor of Kolomogorove [2] and that the most irrational number that exists, namely is the secret of the stability of certain elementary particles [9] and [13]. Vibration simulating particles which do not have a sufficiently irrational winding number dissipate as fast as they are produced.
3. Dimensions and coupling constants
3.1. The fine structure constant and the special orthogonal group SO(n)
Now we need to show some quantitative results. We start by deriving an important dimensionless quantity namely the low energy fine structure constant α0. Within E infinity theory this α0 is no different from the expectation value of the Hausdorff dimension of E infinity, namely
and its inverse may therefore be regarded as a dimension while α0 itself may be thought of as a probability in the fashion of the interactive theory dealing with massinger particles rather than in the theory of electromagnetic field. In fact in E infinity theory all coupling constants α0, αs and so on will be regarded as probabilities to absorb or emit the corresponding massinger particle. Let us give first a formal derivation of α0. We know that the special orthogonal group SO(32) is similar to the E8⊗E8 group of strings which is the only theory which has a graviton arising naturally from its basic formulation. Now the dimension of SO(32) is equal to that of E8⊗E8 and is given simply by [10] and [13]
Next we take n to be the non-integer
Consequently one finds the expectation value
Taking 20 copies of that one finds
where k0=φ5(1−φ5)=0.082039325⋯
This is the value of the global which may be written more conveniently as
where
Clearly we need not justify setting 20⊗SO(4+φ3) equal to as long as we can show that it gives the right physics and is consistent. Nevertheless, we will return later on to give a more intuitive justification beyond the fair numerical agreement with our “physical” reality. This is done by linking to the so-called expectation π meson [2] and [13].
3.2. Embedding of strings dimensions in E infinity
Next let us show how the dimensional hierarchy of the Heterotic string is imbedded in our E infinity and scale using the golden mean. That way we find [14]
Setting k=φ3(1−φ3)=0.18033989 equals zero one finds
Note that can be shown to be the non-super symmetric coupling constant of the unification of all fundamental forces. In the super symmetric case we have .
3.3. Isospin and symmetry groups
Since the introduction of isospin theory by W. Heisenberg, groups theoretical considerations play a prominent role in high energy particle physics. However, strictly speaking we just became used to group theoretical arguments although they remain as abstract as ever, for instance to say we have 12 massless gauge bosons because [2] and [14] the total dimensions of all the symmetry groups involved in the standard model (SM)
is not justified by any intuitive physics, only pure mathematics and experimental verification. The situation in string theory is even more abstract. In string theory we accept that we have 496 massless gauge bosons without experimental evidence because
In E infinity theory we compliment groups by sets in a manner of speaking. Seen at a low resolution, our set is a Kleinian modular group but it is the set character which is important in E infinity and as we will see, it leads to exactly verifiable results regarding the mass spectrum [2] and [14]. Note that the ratio between in number of mass less gauge bosons in the standard model (SM) and quantum gravity defines the non-super symmetric coupling constant .
Thus in E infinity theory the dimension corresponding to 496 is in fact 496−k2 where k=φ3(1−φ3)=0.18033989 which we can neglect for all practical purposes, but not in principle. In fact, ignoring small numbers can lead as we learnt from chaotic non-linear dynamics to disastrous inaccuracies in certain cases. Nevertheless, 496−k2≃496 is not only a dimension but is much more than that as will be shown shortly. It is related to the expectation of the mass of the K meson.
4. The expectation mesons and the mass of the nucleons
4.1. The π mesons
The dimension 496−k2 may be obtained from the dimension 137+k0 by scaling as follows
The deep meaning of the above is the following. It was Sidharth [32] who showed using classical mechanics and non-classical Cantorian space–time that the mass of a π meson is given by [2] and [14]
Sidharth’s calculation is only approximately true because we do have two π mesons
and
To obtain the accurate result we postulate the existence of an expectation π meson given by as follows
Subsequently we can show that mπ± and mπ0 are just different scaling of another fundamental non-composite particle, say the electron. However before showing that we show that the neutron mass may be obtained as a scaled i.e. in fractal hyperbolic space–time of E infinitely distorted π meson [2]
That means the scaling exponent in this particular case is
and
which is almost exactly equal to the experimental value. As we define an expectation meson
which came very close to the expected value found from taking the average mass of mπ± and mπ0
one could define an expectation K meson by scaling 〈mπ〉 using λ=3+φ
4.2. The mass of the neutron and the mass of the proton
Now by scaling 〈mK〉 we can find again mN as follows [2]
That means scaling this time is
where 10=D(10) is the dimension of the core of super string embedded in E infinity and 19−φ6=b−2 which is a Betti number. Consequently one finds
To obtain the mass of the proton all what we need is to change k2=(0.18033989)2 to 4k and find
which is almost exactly the best know experimental result. Now we calculate the mass of the electron again as a scaling, this time of the proton by writing
where and and the scaling is .
Now we can determine the mπ± and mπ0 as scaling of the electron
and
in excellent agreement with the experimental evidence.
5. Linking some scaling exponents to string theory and non-commutative geometry
5.1. Hyper tensor
It may be of deep theoretical interest into the structure of the super string theory [10] to note that
plays a profound role in a necessary condition for anomaly cancellation, namely that the number of hyper tensor and vector multiples satisfies the following condition [10]
This anomaly cancellation is what made Schwarz and Green develop super string by adding super symmetry and gravity to the bosonic string. Seen that way, E infinity theory can give string theory practical predictivity power with considerable accuracy at least as far as the vital mass spectrum is concerned.
Now one could, with considerable justification, ask why did we take these particular scaling exponents and whether we could take any arbitrary factors as a scaling. The answer to this and similar questions, is the following.
5.2. Non-commutative geometry and the golden mean
We could take any exponent and there are very many of them, but there are restrictions. We have a large set of admissible factors but they must be looked at carefully to be taken from the topology of E infinity space–time. First the golden mean and all its powers multiplications and additions may be taken as valid scaling as long as they come out from Connes dimensional function and its extension to higher dimensions. The two dimensional function for instance is given in non-commutative geometry by [2], [8], [9] and [13]
D=a+b(1/φ),a,b∈Z
Second, all the transfinite version of the Heterotic super string dimensions and their combinations are valid scaling provided the corresponding vibration can be shown to be stable. Thus we may use 10, 6+k, 6, 16, 16+k, 26+k, 26 and so on. In addition the following dimensions are extremely important and are drawn upon continuously in E infinity theory:
The last expression gives the dimension of the SU(5) symmetry group needed for GUT i.e. grand unification of all non-gravitational forces which is due to Glashow and Georgi [10].
6. Stability considerations, scaling and the quantum
6.1. The Planck length and the Bohr radius
Besides the preceding conditions the stability condition must be established [2]. Clearly if we know a particle with a certain mass which we have just calculated using a certain scaling, then this particle really exists, otherwise we will not be sure. The problem is that KAM stability and Arnold diffusion in higher dimensions [2] (more than 4) are almost impossible to solve at present. Thus the more direct and obvious the scaling is, the more confidence we will have that such particles will be sufficiently stable to be observed. An example of a direct scaling is for instance the isomorphic length. As we mentioned this length is directly proportional to 4+φ3
where ϱ is a radius which can be any number. Consequently 4+φ3 and 4 are obvious fundamental scaling exponents in E infinity. To show that this is true we give a simple but striking example of how 4+φ3 and 4 connects the Plank length (which is related to the quantum h by h=(lp)2 cm2 in natural units) with the semi-classical scale par excellence, namely the Bohr radius [2] and [10].
While for the related stony length one finds
Thus 4 and 4+φ3 are the scaling of the classical (h=0) to the quantum (h≠0) and visa versa. Other scaling transformation have a direct and obviously appealing physical interpretation and inspire a direct confidence even without experimental verification. An example of this kind is the following coupling equation
Thus
where is the super symmetric coupling constant of quantum gravity, k=φ3(1−φ3), k0=φ5(1−φ5) and . Clearly is the coupling between the graviton field represented by the string group, namely the Lie expectational group E8⊗E8 and the electromagnetic field as represented by the quasi-dimension . The factor 2 is analogus to of the φ scaling of to give the Heterotic string dimensional hierarchy discussed earlier and may be interpreted indirectly as a kind of Bose condensation of a Cooper particle at the extremely high energy of some (10)19 GeV [10].
6.2. Geometry and topology of the vacuum and quantum gravity
Maybe we still need to explain the deeper origin of the preceding relation. At least historically the relation goes back to the sigma model. In this model and as is explained for instance by ‘t Hooft, it is the squares of the masses which must be compared [15], a situation which is similar to the Regge trajectories. For the π meson and the K meson, the correct comparison has to use the expectation π and K meson which are defined in E infinity as theoretical intermediate and probably totally unstable particles which need not really exist and the ratio comes indeed near to 14. In fact it is exactly 13.09016995. This value happens to be exactly half of the value of the theoretical super symmetry quantum gravity coupling constant. Thus
Now at the beginning of any new theory, the most difficult things is the new concept. Once this is established then mathematics takes over. Let us clear the concept a little more because it is not immediately obvious how we move from a Hausdorff dimension to mass. The chain is not long. We know that entropy is a measure for complexity. Likewise the Hausdorff dimension is a measure for complexity. This is how the work of Schlögel and Beck should be understood because the Hausdorff dimension is related to thermodynamics. Consequently the Hausdorff dimension is related to energy via thermodynamics and since energy is related through special relativity to mass, the connection of Hausdorff dimension to mass becomes clearer. Now the Hausdorff dimension is predominantly a geometrical–topological devise and the afore mentioned connections mean that geometry is indirectly connect to temperature and mass. In essence there is nothing new in all of that, it is general relativity seen from another maybe deeper view point. On a deep level geometry is paramount and decides on everything including energy and thus mass. The preceding simple thoughts were the basis for a relatively recent work in which the author derived the temperature for a drawing by Pablo Picasso [16], [17] and [36].
6.3. Complexity theory
Having said all that, one should not confuse disorder with complexity (see Ref. [37]). In fact hyper-disorder may be regarded as a form or ergodicity and ergodicity is a completely uniform disorder which has a complexity zero, exactly as complete disorder. Innovation in nature takes place somewhere between the two extremes and I was not astonished to find out that the VAK state has a maximum complexity. Consequently the vacuum has a maximum complexity which is the reason why it is so rich giving rise to quantum physics. I was later informed that maximum complexity is connexted approximately to the number 0.273. Prof. Alan MacKay, a leading British crystalographer was particularly intrigued by this because this number appears continuously in the E infinity theory, (i.e. ).
So far we have discussed the inter-scaling relationship between π mesons, K mesons, electrons and nucleon, but what about the quarks model for hadrons. Could this model by of any use in E infinity theory? The answer is yes it is and treating the same particles using a combination of the quarks model and scaling gives a deeper understanding of the theory which we do next. In fact some scientists regard the electron as a kind of quark and that was used in our earlier analysis. [33] and [39].
7. Constructing the neutron and the proton from quarks
7.1. The mass of the quarks
First we give here without derivation, the current and constituent masses of the light and heavy quarks which are consistent with E infinity theory. Needless to say, that these masses are in excellent agreement with the majority of the scarce and difficult to obtain data about the mass of the quarks. It takes only one look at these values for anyone to realise that they form a harmonic musical ladder. In fact, particle physics seen through the eyes of E infinity must resemble a cosmic symphony, even for the most hard nosed so-called realist. Here are the values [18]:
(a) Current mass [14] and [18]
(b) Constituent mass [14] and [18]
7.2. Constructing the nucleon from quarks
The point is that we know from the classical quark theory that a nucleon is supposed to be made up from three confined light quarks. For the neutron these are two down quarks and one up and for the proton, two up quarks and one down. That way one finds that [16] and [18]
where λN is a scaling given by
Thus
where n1≃496 is the expectation value for the number of massless gauge bosons in the quantum gravity field while is the expectation value of the number of massless bosons in the standard model.
That means
which is exactly the value we obtained previously and which agrees completely with the experimental results. Now we look at the proton which is electrically charged and must therefore be made up of two up quarks and one down quark [16] and [17]
where λp is the scaling
Thus we have
in full agreement with the experimental evidence namely mp=938.279 MeV. However we gain from the previous equation a great deal of insight into the relation between mN and mp within E infinity. We notice in the last equation that mp is a projection of mN=939.57427 MeV. The projection is due to a rotation of an angle equal to π divided by (D(10))(D(6)−k)=60 where D(6)=6+k and D(10)=10 as we have known from the φ scaling of corresponding to the dimension of our transfinite version of the Heterotic string theory. One could ask why this rotation? Formally one could answer it is exactly equivalent to the internal rotation of the isospin theory of Heisenberg only more tangible and it gives the right result. However using our hyperbolic distortion picture (see Fig. 1) of the Cantorian E infinity space, we can give the deeper answer that this is the angle at which we look at a neutron and conceive it as a proton as far as the mass is concerned. It is the geometry and topology of space–time all over again. In string theory we know that the mass equations of the “particles” lives in the 6 dimensional part of the 10 dimensional space of the string core embedded in E infinity. This is one of the important results of the theory of super strings [10].
8. Deriving the mass of the meson from the “vibration” of the light quarks
8.1. The expectation π meson
Now we would like to derive the expectation π meson mass (which was never observed experimentally until this point of time, but may be found in the future) using the quarks model. We know that the meson consists of two quarks. For that purpose we take one up and one down quark and find
The reassuring thing here is that we find the scaling to be exactly λπ=10. Thus the 〈mπ〉 is ten copies, (to use the terminology of the 10 dimensional super string) of the sum of the two light quarks
In such cases it is instructive to see the calculation as going forward and backward from higher to lower dimensionality and visa versa. That means the masses of the quarks which we perceive are the ones measured here in our 3 + 1 dimensional projection of E infinity. However the combination we talk about takes place in this case in the 10 dimensional super string core of E infinity so that the value we measure in our 3 + 1 projection is the 10-fold of the simple sum of the mass of mu and md.
8.2. Nested and fractal vibration
So far we have made no direct quantitative reference to the vibrational interpretation of E infinity and that is what we will touch upon now.
Consider a simple two degrees of freedom linear vibration consisting of two masses connected by linear elastic springs and hanging on the ceiling. Setting the masses and the spring constant equal to unity, we obtain a quadratic secular equation with two frequencies as the solution, namely [14] and [18]
These are indeed the golden mean again. If we now imagine an infinite collection of such two degrees of freedom unit cells connected sequentially and in parallel at random, then we need only to introduce a so-called wired hierarchy in the architecture of our neural network like structure and we would have some reasonable mechanical realisation of E infinity space. In fact, such an infinite collection of possibly nested oscillators has already been considered by L. Crnjac [5] and also by S. Wolfram in his recent book “A new kind of science”. I have used in this context the well known Eigen value theorems of Southwell and Dunkerly to show that the expected hierarchy of frequencies of vibration are simple or complex function of the golden mean and may add that many of the results obtained within the theory of N. Wiener and its modern recasting in the theory of spontaneous self-organisation (see Ref. [37]) are of great relevance to E infinity and reproduce partly some of our arguments. This could however take us too far from our present limited objective of an introduction to E infinity and will not be discussed in detail. The important point which we gain from the preceding “N. Wiener” picture is that when we add say two Hausdorff dimensions, for instance
we can regard this also as adding two frequencies to find a joint frequency. Similarly we have in the sequential net the second variant of adding two frequencies and that would be
Thus
This is obviously trivial but things can get quite sophisticated and our approach requires knowledge of advanced modern geometry of the Kähler manifold [10] in particular the so-called K3. To explain this point let us take a concrete example. Very frequently when writing a scaling exponent using the main dimensions of the Heterotic string we would write something like −26+10=−16 and we justify this by saying that the 10 dimensions of D(10)=10 are moving to the right while the 26 dimensions of D(26)=26 are moving to the left. This situation is not as mad as it initially sounds. The point is somewhat similar to what we encounter frequently in the general theory of diffusion where we have a process defined at least mathematically via two diffusion equations, one running forward and the second backward in time. This is a special case of what I have introduced as the complex conjugate time of the quantum world [19]
Something similar is used in the theory of Heterotic super strings where we introduce a so-called Minkowski analytical continuation and end with a holomorphic field and anti-holomorphic field. We use then the synonyms for left moving for holomorphic and right moving for anti-holomorphic [10].
9. Quantization and transfinite discretization
9.1. The work of G. Ord
This brings us now to what we should have explained at the beginning but deliberately postponed until this stage. The theory of E infinity would have remained without a strong theoretical foundation if it had not been for the work of the English–Canadian physicist, Garnet Ord [3]. Ord set out to take the mystery from analytical continuation. We should recall that analytical continuation is what converts an ordinary diffusion equation into a Schrödinger equation and a telegraph equation into a Dirac equation. Analytical continuation is thus the short cut quantization. However what really happened is totally inexplicable. It was Ord who showed, using his own (invented) quantum calculus, that analytical continuation is not needed if we work in a fractal-like setting, a fractal space–time if you want. In fact it was Ord who introduced the expression fractal space–time in a formal paper in the eighties. Only recently Ord’s work has gained acceptance in Physic Review Letters and so one is hopeful that his message will be widely understood; it is the transfinite geometry and not quantization which produces the equations of quantum mechanics. Quantization is just a very convenient way to reach the same result fast, but understanding suffers in the process of analytical continuation [3].
9.2. Complex time and transfiniteness
Ord has accepted the limited validity of 0±it as dual equations and that quantum mechanics [19] for instance is governed not only by one Schrödinger equation but by a second conjugate complex Schrödinger equation as pointed out by the author [35]. However he has written that this is not going as far as one should in demystifying analytical continuation and replacing it with a deep geometrical understanding. The further development of E infinity take Ord’s point completely which he acknowledged in several of his recent papers. So, our slogan for E infinity could be
‘Do not quantize and do not merely discretize. You should discretize transfinitely’. This is the right way from M. Kac to P. Dirac.
Once this is done, we are in the middle of hyperbolic Cantor sets and E infinity. Now we come naturally to a totally justified question, namely what happens to h. Ord never needed to look into this question thoroughly because he regards his equation as being totally dimensionless and setting C=h=1 are his natural units system. Later on once he arrived at his Schrödinger and Dirac equations, he restores the situation and h appears again. In my E infinity, I do not need to dwell on h directly, but it is built in there for sure. This is because the dimension D(26)=26+k≃26 is at the same time the value at which all differences between all fundamental forces completely disappear and we have then one force, the super force so to speak. This situation takes place at an energy of around 1019 GeV. This energy is in turn related to the Planck length and to the length at which complete unification takes place. The Planck length and the total unification length are connected via this coupling constant, namely
On the other hand, h is nothing but the square of the Planck length when measured in centimeters
That is where h is hiding in E infinity. In other words, once we have found h experimentally and once we have accepted it as fundamental and final, we should have at once given up the smooth Euclidean space in favour of something more in harmony with
such as E infinity space time. Now we may return again back to our main concern, the mass spectrum. We have so far converted some particles into others by means of scaling as far as mass is concerned, but we never really explained where mass came from in the first place. In the standard model for instance which E infinity accepts as a valid approximation, mass is explained using the Higgs mechanism. However no one has ever seen a Higgs experimentally and could not be sure that this Higgs really exists. None the less, this is not an argument against the Higgs because no one has ever seen a quark either, I mean a single quark moving freely in space and none the less, we accept the existence of quarks. If the Higgs particle exists, then one could ask again where did the Higgs particle get its mass from? In addition how could the Higgs field hide away its gravitational attraction which should in principle be detectable even with today’s technology as emphasized continuously by Veltman.
10. Unification and the mass of the electron
10.1. The unification π meson
Now all these questions are answered within E infinity theory in a fundamentally different way. In E infinity the particles acquire their mass at the unification of all fundamental forces. To explain this point we would like to calculate here the mass of the electron from the condition of unification.
All fundamental forces are unified when all the three fundamental coupling constants intersect with that of gravity [20] at one point in the space where αi. stands for the coupling constants of the weak force, the strong force and the electromagnetic force as well as the dimensionless coupling of gravity while E stands for the corresponding energy. Steven Weinberg gave a highly simplified and lucid account of this subject in the Millennium Edition of Scientific American [21] and one could see from his clearly presented coloured figures that assuming super symmetry the unification coupling constant lies indeed near which is very close to our theoretical result . Now we remember that we calculated a theoretical intermediate particle which we called an expectation π meson and found it equal to MeV. Remember we also obtained D(26)=26.18033 as a scaling of () namely .
10.2. The unification electron and the experimental mass of the electron
Thus in analogy with that we would like to introduce formally a hypothetical particle with a mass equal to 26.18 MeV which we will call the unification π meson
However we should keep in mind that this point is a point at which there is no difference what so ever between gravity and consequently mass and energy and electromagnetic charge nor nuclear forces. Now we know that the dimensionless electric charge is given by . Consequently we may deduce analogically a unification charge equal to . However this has to be lifted to 10 dimensions as we have seen before so that the correct expression would seem to be
The above relation as it stands is unfortunately not right. It would have been right if it would not have been for a remarkable duality known in string theory as the Olive–Montonen duality, where we have to take the reciprocal value when moving from large scale to the ultra small scale and the correct expression is the reciprocal value
This value measured as 〈mπ〉 in MeV and is what we call the unification electron mass.
To obtain our 3 + 1 electron mass we have to project onto 3 + 1 dimensions and find using D(10) and D(6):
The experimentally found value for me is, as is well known, me=0.511 MeV.
11. The experimental fine structure ‘constant’ and the electroweak ‘particles’
11.1. The Sommerfeld α0
Now some may feel uneasy about the introduction of the string dualities [10] as well as the projection but both manoeuvres are routinely used in string theory and we are basing our self on it. One may find more than adequate and detailed coverage in the concerned monumental literature on the subject of strings, which we basically, globally accept as excellent approximation of what is the case in high energy physics. With E infinity theory however, we need not think of projection as more than special scaling to account for the distortion caused by infinite dimensional hyperbolic and fractal topology of quantum space–time. There is also a vital meaning for the procedure of projection connected to the low energy inverse fine structure constant . We have found to be
but the very accurately measured is really
so what is the meaning of this slight but important difference. The explanation is as follows. The is a global and is a true constant. By contrast, is a projection in 3 + 1 and may therefore vary slightly with space and time. To obtain the experimental we project it using the “quantized” projection angles (see Fig. 1) in this case θ=π/α0 and one finds [2] and [6]
in complete agreement with the experimental value.
11.2. The electroweak theory
In fact the cosine of, “quantized” angles plays a very important role in E infinity and may be thought of as a diffraction-like effect such as that found in crystallography. For instance, the Weinberg mixing parameter
is identified in E infinity theory with the cosine of the angles of the triangles which make up the original Klein curve χ(3π/7) which forms the major part of E infinity as seen in the hyperbolic Poincare–Beltrami disc (see Fig. 1)
With this value at our disposal, we can determine the masses of the W± and Z0 of the electroweak. For this purpose we look at the mW± as a scaled m*t, that is to say the constituent mass of the top quarks where
and
That way one finds
The best experimental value is mW=80.4 GeV. To obtain the Mz we use the same formalism of the Glashow–Salam–Weinberg theory but use cos(3π/7) instead of sin2θw and one finds
The best experimental value is 91.18 GeV. Incidentally the coupling constant of the electroweak is also easily found from
which agrees with the P-adic expansion of , namely
Thus 128 may be interpreted as being at the electroweak scale
while 8 is the inverse of the strong coupling . The one left may be related to quantum gravity in the P-adic theory.
The relation between P-adic numbers, fractal and E infinity was discussed by many authors. We should also note that 137 is the 33 prime number while 127 is the 31 prime number. We may also note that since the mass of our theoretical π meson is MeV we could interpret 128, 8 and 1 as masses measured in MeV.
12. Continued fraction and stability
There is an important point which we did not discuss so far and which is important for our vital quarks model interpretation of E infinity. We have reason to think that in our E infinity theory we must have
This is indeed the case as can be verified from
This relation is extremely important because all permanent matter is made of m+p and m0N i.e. protons and neutrons. Therefore in any realistic model protons must be the most stable particle. In string terms as well as in E infinity terms, this must be the most stable “string” vibration. Since according to KAM this will entail the most irrational frequency ratio possible, the ratio of mu and md must be the most irrational number possible which is the golden mean, as is well known from number theory [12].
Now we should contemplate the following. The proton is the most stable composite particles we know of and this particle is made of two mu and one md, so we have
By contrast, for the unstable neutron we have
The question we hope someone can answer precisely one day is the following. Is it possible simply from looking at the continued expansion of the ratio of two elementary or sub-elementary particles or the logarithmic scaling of the ratio to judge the relative stability of the concerned particle from a criterion connected to the continued fractional expansion of these quantities.
We have a strong feeling that such criteria may be possible and this would simplify KAM theory and Arnold diffusion calculation beyond our present hopes, something which is, as far as we are aware, completely lacking at present.
13. Present mathematical limitations imposed on a general theory
Maybe it is now the place where we should discuss the limitations of our present theory. The customary thing to do in classical physics is either to establish the differential equation using Newtonian physics or what is completely equivalent to find a variational principle for which a Lagrangian is needed. This standard procedure is kept, as far as possible also in the standard model. However, as we can see, inconsistencies force us to give up smooth space–time and it was the French astrophysicist, Nottale [7] who investigated the consequence of giving up differentiability and came to his by now reasonably well known theory of scale relativity and conclusions similar to ours. Nottale of course did not give up continuity but only differentiability and this was difficult enough [7]. However in our case we have to find a way to integrate infinitely many times something which is classically non-integrable in order to find the stationary points corresponding to our Eigen values, i.e. masses and coupling constants, then we have to find the first variation (i.e. we differentiate) of the non-existing Lagrangian and set it equal to zero
δ(L(VAK))=0
However if this could be done, we still do not know anything about the stability of this solution unless we require that we take the second variation (if it exists) and insist that
δ2(L(VAK))>0
for stability and only then can we find the stable particles which could be observed.
It is clear that we would need for this program a mathematics which does not exist yet and the only hope for an exact solution would be a super, super computer, i.e. a quantum computer.
E infinity theory may be regarded thus as an attempt to go around all these difficulties as far as possible and extract as much exact information as we can using a flexible strategy of applying almost everything we have in mathematics simultaneously. In particular we do not use only group theoretical consideration but also set theory and number theory as well physical considerations such as the nested vibration model [5], which was probably inspired by the pictures of self-similar universe due to Swedenborg, Charlier and Right (see Fig. 2, Fig. 3, Fig. 4 and Fig. 5).
14. The meuon and the mass spectrum
Now we may turn our attention once again to the calculation of the mass spectrum and it is time to consider the meuon. Being an electron in every respect except for being 206 times more heavy, the meuon should be regarded within the classical quarks theory as non-composite. Such notions, I mean “non-composite”, are only relative within E infinity theory and depend on the resolution which is used but for all practical purposes, we may regard the meuon as non-composite. The best is then to regard it as scaling (distortion) of a quark and we take it to be the up quark. That way we may write
where
Consequently
On the other hand the meuon is clearly a scaled electron
where
Consequently
Setting for mμ the value found earlier namely mμ=105.6656315 MeV, the electron mass is readily found to be
exactly as expected. Now we may need to discuss the scalings λμ. Most of the time these scaling involve the ratio of the largest symmetry group we have, namely that which contains super strings and consequently gravitation, . The second value 24−φ9≃24 could be interpreted in different ways. First it is the dimension of SU(5) of the GUT unification.
The φ9 is the transfinite so-called “correction” which reminds us that we are dealing with Cantor limit sets. On the other hand, if we take the 496 to be the number of massless gauge bosons of string theory, then the 24 should be taken to be the number of instantions which is equal to the second Chern class for K3⊗T2 as well as E8⊗E8. That means [10]
Multiplication with 10 is taking it to the 10 dimensions of the super string core. I do of course appreciate that the preceding explanation itself needs explanation but this would take us deep into the topology of super strings and string field theory which is definitely not the purpose of the present introduction.
15. Possible experiments
A question of great interest for any true physicist is obviously the following: could we have direct experimental verification for E infinity theory? The answer is probably yes but probably also not so direct. In my opinion, if it can be done at all, calculating the Hausdorff dimension of a quantum path may be our best bet [33]. Such an experiment should at the end say that the Hausdorff dimension of a quantum path is larger than the classical topological dimensional one. To find that the average Hausdorff dimension is exactly two would be a definite confirmation for all the postulates of E infinity theory. Although such sophisticated experiments are completely outside my range of expertise, I have given this question some though and think it will involve reconstruction of quasi-phase space using Ruell Takens method as well as deep laser cooling but this is still too vague to write about it here [33]. It seems that H. Kröger [34] in Canada attempted to find the Hausdorff dimension of a quantum path experimentally but so far, no real experiments were ever made.
Another possibility is to find a deviation in Newton’s gravity law which could not be explained except with the existence of five and more dimensions for space–time. Such a possibility is being pursued by a team in CERN [31]. Two more predictions of E−∞ could be tested experimentally. First the existence of the expectation π meson 〈mπ〉=137.08 MeV as well as the expectation K meson 〈mK〉=495.9674 MeV.
16. Additional points of interest––the mass of the neutrino
We hope the preceding discussion helps to clarify the basic idea behind our approach although we have ignored some important aspects related to sphere packing in higher dimensionality, Leech lattices, quantum calculus and scale relativity as well as loop quantum mechanics and knot theory. Some of these subjects were discussed by the writer and other authors in many previous publications [22], [23], [24], [25], [26], [27], [28] and [29], for instance Saniga [26], consider the relation between E infinity and projective geometry whereas Agop et al. [29] considers super conductivity and E infinity. Discussing all these aspects would take a considerable space and we reserve them for coming occasions but one more remark regarding the neutrino may be essential. Any new theory for particle physics is tacitly expected to say something about the mass of the neutrino. E infinity can do that and predict the mass of the neutrino on the basis of the energy of the microwave background radiation energy [30] to be of the order of 10−4 eV which agrees well with the scarce experimental evidence [30]. We also should draw attention to a recent interesting paper by Koschmieder [22]. A work which is similar in it’s philosophy is that of Sternglass [39].
17. Intermediate summary of the results
Let us summarise the most important formulas found so far for the different masses to reassure ourselves of their simplicity and elegance which excludes any possibility of interpreting these as any thing but true.
18. Symmetry breaking of E8⊗E8, the fundamental mass norm and
We could arrive at via unification argument. Such an argument relies heavily upon quantum field theory and strings formulation and readers not familiar with both subjects may just disregard the reasoning of this section and note only the final conclusion.
One of the accepted scenarios for moving from E8⊗E8 with its 496 massless bosons to the standard model SU(3) SU(2) U(l) with its initially 12 massless gauge boson is to assume that E8⊗E8 brakes into the smaller exceptional Lee group E6⊗E6
where .
Let us recall first the approximate integer value of the fine structure constant, namely . Thus we may write that
where 19 may be interpreted as the Bitti number of K3, namely . Our symmetry breaking may thus be written symbolically as
Now recall that the mass of the two intermediate “theoretical” particles, namely the expectation meson 〈mπ〉 and the expectation Kaon were given by
and
so that the following theoretical “decay” is suggested by the preceding symmetry breaking
We see that the electromagnetic fine structure constant for a Cooper pair arises naturally from the preceding symmetry breaking and in addition we have
Consequently this may be interpreted as
but we also know that
Consequently the dimension-like value corresponding to 1 MeV. This may be a cumbersome way to state a trivial but deeply surprising and immensely useful fact. In E infinity space every dimension corresponds to 1 MeV in the mass space.
19. The Higgs and E infinity
We have already mentioned that our approach to the mass problem is quite different from that of the standard model and the Higgs mechanism. However the Higgs picture could be in general interpreted in a way useful for E infinity. The mere fact that if we do not involve self-interaction of the Higgs field in order to give the Higgs particle mass, then we must assume that there is a second Higgs field which gives the particles of the first their mass and so on indefinitely is a statement about fractalness. In this picture and as mentioned by Veltmann, the Higgs would be just a new level of finer description of particle physics. (see Fig. 1 and Fig. 5)
19.1. The fine structure constant revisited
General remarks and alternative rationalisation
The reader may have long noticed the central role played by the fine structure constant α0 in ε(∞). One could say that the value range second in the line of importance just after the Hausdorff dimension
A well meaning critic which I take very seriously for more than many very good reasons besides being one of the best theoretical physicists of the past century, remarked that he expected α0 to come at the end of a general theory as a final conclusion and not at the beginning. This remark hits the nail on the head. Indeed, this is the point. In order to be able to achieve what we set out to do, I had to turn the classical way of attacking the problem on its head. The rationale behind this reversed strategy is found in the topological–geometrical conception of E infinity theory. Once we take the topologicalization program seriously, then α0 follows from its interpretation as a probability. In our case as a geometrical–topological probability. It is this deceptively simple move which made everything fall into place and laid bare the deep harmony underlying the golden mean mass spectrum of high energy particle physics. To explain this let us start ab initio.
We have already established that E infinity is a kind of probability space. However E infinity is strictly speaking a “prespace” and therefore we should be very careful in using words like space and probability. All the same, we need to define what we mean with probability. In our particular case we have a formaly infinite dimensional Cantor set with unaccountably infinitely many Cantor points in a “prespace” without a metric because the Lebesgue measure of E infinity is zero. As a consequence of this situation, combinatorial probability can be ruled out because the probability in all events will be
In such a case one would usually attempt to define probability geometrically but also in this case we find
The only way left is to attempt to define a topological probability using the Hausdorff dimension and the embedding dimension
This means
In other words, we have,
For a Mauldin–William random Cantor set one finds
That means that the multiplication and addition theorems may be applied to PTop=dc(0)=φ. For instance
P=φ3=φ⊗φ⊗φ
is the probability that event with a probability φ takes place three times simultaneously. On the other hand the probability that only one event of three events takes place is given by
P=3φ=φ⊗φ⊗φ
Applying these elementary ideas to reality we Consider once more the fine structure constant . We interpret as in atomic physics in a quite elementary fashion as a cross-section for the interaction of two electrons. A cross-section is a nuclear engineering conception but is actually a marvellous geometrical concept ideally suited to the entire philosophy and structural concepts of E infinity. Thus α0 would be thought of primarily and in contrast to the classical definition of α0 as a probability. It is the probability for two electrons to interact for instance. It is also a probability for an electron to emit or absorb a photon. Thus it is a measure of the strength of the electromagnetic field interactivity. Now in the strain of positivistic philosophy we could define at will to be
We disregard for the moment the slight difference from the experimental value
The only thing we need to show is that defining in this way leads to rational and particularly economic way of describing physical phenomena without contradicting well established other theories nor of course contradicting well established experimental facts. However at least as far as the present author is concerned, this positivistic operational philosophy is not entirely satisfactory and we would like to give a deeper explanation as to why we fine tune to be . Now (1/φ)4 could be interpreted as
(1/φ)4=(1/φ)(1/φ)(1/φ)(1/φ)
Since φ is a probability of finding a Cantorion point in a one dimensional Cantor set, then φ4 is also a probability. It is the probability of finding a Cantor point, a so-called Cantorion in all four topological dimensions simultaneously. That means or a part of is totally an inextricably connected to four dimensional space. However E infinity goes further than that. We have the 10 dimensional core of the super string space D(10)=10 which we have shown to be part of E infinity and embedded in it. Now the probability (1/φ)4 penitrates into the “string space” through the non-massive section, namely the (26+k)−(6+k)=20 dimensions and that on the basis of the addition theorem, so that the total fine structure constant becomes
There are numerous ways to convince oneself with the inevitability of setting . However they all have a feel of ad hocness to them. For instance one could see as the intersection of 3+φ3 with 4+φ3=3+φ3+1 living in the union of the D(10)=10. That means
19.2. The E infinity interpretation of the 26 dimensions of super strings
One must have noticed by now that the philosophy of D. Finkelstein and his school regarding that a process is more fundamental than space–time and that a particle creates its own space time has at least some indirect application in E infinity theory. In a sense that is the reason why particle masses and dimensions are so much interrelated within this theory. It is here where E infinity theory can give an intuitive rationalisation for the need for some 26 dimensions for space–time. To explain this I may use a lucid and clear presentation of the number of the free parameters. Such a presentation was given to the author on request by A. Goldfain and is a follows:
First we have three coupling constant of SU(3), SU(2), U(1). Second, we have the two parameters of the Higgs mass and its vacuum expectation value. Then we have the mixing angle of the instanton contributions. That brings us to six parameters. Next we have (Nq2+1) quark parameters made up of 2Nq quark masses for Nq generations and (Nq−1)2 mixing angles (Cabbibo) and phases. For Nq=3 we have the 10 parameters in addition to the previous 6 making the 16 free parameters. Finally we add another (Nl2+1) lepton parameters giving for generation number Nl=3 another 10 parameters and consequently we end with 26 free parameters. This may be found directly from the formula 2(Nf2+1)+6 when setting Nf=3.
The situation is just like in elementary linear algebra where 26 equations are needed to find 26 unknowns. Consequently we need 26 degrees of freedom in our Lagrangian and these 26 degrees of freedom are our quasi-dimension.
The attentive reader may have noticed that we made use of b2−=19−φ6≃19 in our mass formula.
Now this is a geometrical quantities related to the Betti number of the Kahler manifold K3. However it could also be interpreted as the number of quasi-dimensions when we set in our standard model massless neutrinos and dispose of the leptonic mixing angle. This is again a valid approximation depending on the resolution in keeping with the basic philosophy and concepts of E infinity. Finally in the so-called minimal model we could reach the minimum number of 18 free parameters which is what is commonly quoted in text books. By contrast in string theory one normally reads the sentence that the standard model possesses about 20 free parameters.
Thus from our E infinity view point we think that we should think of the 26 dimensions of string theory as being the expectation value of the number of needed free parameters for a consistant theory. This alone should give us yet another strong argument to believe that the neutrinos must have a non-vanishing mass. Finally we should link the hierarchical structure of the mass spectrum with the number of dimensions and the fact that the volume of n dimensional sphere vanished as n goes to infinity.
Now, in E infinity theory the exact value is an expectation value namely 26.18033989 rather than just 26. In addition the number of fundamental forces is not just 5 for electric force, magnetic force, weak force, strong force and gravity, but an expectation value 5+φ3=5.236067977. Consequently the total number of free parameters is
n=(5+φ3)(26.18033989)=137.082039325
In other words we have gained yet another derivation and interpretation of as dimension and number of free parameter at a higher resolution namely
20. Symmetry breaking and the Higgs field
There seems to be some misunderstanding about the role of symmetry breaking in connection with the Higgs field which we would like to explain briefly.
If we restrict ourselves to the very elementary case of a skeleronomic and holonomic conservative system described by a potential energy then there are only three types of symmetry breaking bifurcation points. The stable symmetric, the unstable symmetric and the asymmetric or Poincare exchange of stability. Even here we do not have the case of indifferent equilibrium which must therefore be classified as unstable. Thus the massless particles are indifferent to any “potential” and thus unstable. Once the particle “absorbs” a Higgs from the surrounding Higgs field, then it puts on a weight, i.e. it acquires a mass. This mass will in the ball analogy lead to either a stable or an unstable position depending on the shape of the potential. Thus we are not really dealing with a true symmetry breaking bifurcation, neither in the sense of Poincare nor Hopf nor in fact that of Pexito structural stability. We may note on passing that the field associated with E infinity is a fractal field. Thus it is not a classical field like the scalar field of the Higgs nor the Vector field of Electromagnetism. It is also not a tensor field like in general relativity. It is far more complex.
21. Cantor space and Newton’s non-dimensional gravity constant
For a projective Borel hierarchy, one comes to the notion of a Cantor space as follows:
Definition
Let the space AN be viewed as the product of infinitely many copies of A with discrete topology, be completely metrizable and countable. In the case of A=2={0,1} and A=N, we call the space
ε=2N
a Cantor space. The amazing and amusing fact which follows from the above is the following. Taking heuristically N to be
which is the 31 prime number, then one finds
ε=(2)127≅(1.7)(10)38
The dimensionless Newton gravity constant is given by
The agreement between ε and αG is remarkable and although suggesting and indicative of the deep relation between physics and E infinity theory, we do not want to over estimate its importance nor are we at present in a position to give a rational consistent physical explanation for it. We have of course some intuition for the problem derived for instance from comparing the square of the Planck mass and the proton mass which leads to the same pure number when squared
While mpl/mp≃(1.3)(10)19 when measured in GeV gives us the unification scale of quantum gravity. Note also that [(αG)(GeV)2]−1 gives us the right gravitational constant order of magnitude, namely
The experimental value is GN=(6.70784)(10)−39 GeV−2. Similar conclusions were reach in a couragious work by the prominent Stanford experimental physicist Noyes [38].
22. The mass spectrum revisited
Having established the mass norm relating quasi-dimensions to MeV units in ε(∞) it is an amazingly simple task to estimate the masses of the some 200 or so known elementary particles. Here we will restrict our attention to only some of the more well known particles and resonances.
The following particles are simply multiples of MeV and the results are in excellent agreement with experiments
The experimental values are 548.8, 957.5, 9460.3 MeV. Particularly interesting is the mass for the expectation Σ particle. This is given by
The experimental value is 11932.8 MeV. The mass of the well known J/ψ can also be found easily as
The experimental value is 3096.9 MeV. For we also have a simple formula
The experimental value is 1115.63 MeV. Similarly for Δ (1232), ms (770) and mω (783) one finds
and
The experimental values are 1230, 770 and 782 MeV respectively. A particularly neat expression is found for the tau particle by scaling the proton using
Proceeding that way one finds
The experimental value is 1777 MeV according to D. Perkins. We could come to a similarly accurate estimation by scaling the expectation π meson, K meson or the constituent mass of the t quark m*t:
In conclusion we may give the mass of the Exi minus and Exi plus particles
and
The experimental values are 1321.32 and 1314.9 MeV.
23. A brief history of ideas leading to the E infinity concept
If we focus our attention on hierarchy and self-similarity (see Fig. 2, Fig. 3, Fig. 4 and Fig. 5) rather than on mathematical transfiniteness, then one may be surprised to see an unsuspected long history of ideas which bear a striking resemblance to the geometrical concept of E infinity.
The idea of hierarchy and self-similarity in science first started in cosmology before moving to the realm of quantum and particle physics. It is quite possible that a clergyman, T. Right was the first to entertain such ideas (see Fig. 2). Later on the idea reappeared in the work of the Swedish scientist Emanular Swedenborg and then much later and in a more mathematical fashion, in the work of another Swedish astrophysicist, Carl Charlier (see Fig. 3). This Swedish school may have inspired the work of the eminent British scientist Lord Kelvin on the space–time foam and then in turn together with the work of the Swedish school may have reached Zyldovich in the former Soviet Union.
My own work was done independently and until very recently without any knowledge of the above starting from non-linear dynamics as applied to turbulence (see Fig. 4). Subsequently I became acquainted with Wheeler space–time foam as well as the work of G. Ord and then L. Nottale, K. Svozil, B. Sidharth and finaly the Swedish School.
24. Conclusions
Seen through the eyes of transfinite sets and the golden mean renormalization groups the mass spectrum of high energy particles resembles a non-linear dynamical symphony where everything fits with everything else. We could start virtually any where and drive everything form everything else. Once we manage to familiarise ourselves with the mass norm, everything falls into place. In fact it takes very little effort to be able to memorise the masses of the most important particles and derive the corresponding formulas with remarkable ease. If we take the words of E. Mach seriously, that understanding may be equated with “denk” economy, then one could say that E infinity theory furnishes us with such economy of thoughts and thus understanding of the mass spectrum.
Acknowledgements
The author is deeply indebted to Prof. Dr. W. Martienssen as well as to Prof. E. Fredkin for stimulating discussions. I am also very grateful for a pleasant visit to the Royal Institut of Technology in Stockholm, Sweden during which the present work was written. Finally I am thankfull to Prof. D. Mumford for permission to use some material from his publication cited in the references.
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Address: P.O. Box 272, Cobham, Surrey, UK
Copyright © 2003 Elsevier Ltd. All rights reserve
Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies
The paper is an elementary introduction to the concepts of E Infinity of quantum physics. In the first two paragraphs the main concepts of E Infinity theory and some of the principle results have been explained. Subsequently we address the problem of the mass spectrum of the standard model from various viewpoints, in particular, from that of gravitational instanton.Particular attention is paid throughout the paper to giving an intuitive grasp for an extended theory of vacuum fluctuation based on Dirac's hole theory and E Infinity theory. It is further shown how the golden mean and logarithmic scalings can be used to understand quantum gravity and how the new transfinite Dirac's theory can explain certain anomalous positron production which was observed by several experimental groups worldwide.
Die Elementarteilchen können mit den regulären Körpern in Platos “Timaios” verglichen werden. Sie sind die Urbilder, die Ideen der Materie. (Werner Heisenberg)A careful counting routine of all experimentally confirmed elementary particles plus the theoretically conjectured ones needed for a sound formulation of a mathematically consistent field theory is undertaken within a minimal N=1 super symmetric extension of the standard model of high energy physics. The number arrived at is subsequently linked to certain massless on shell representations connected to the quantized gravity interaction. Finally with the help of number theoretical arguments arising from a rigorous application of the formalism of transfinite Heterotic super string and E-infinity theory, we show that the proposed scheme would lack mathematical consistency and elegant simplicity unless we retain a postulated triplet which is logically identified as the H+, H− and H0 Higgs particles. Connections to the 11 dimensional M theory and Harari's extended “sub-quarks” theory is also discussed.
Unification of the fundamental forces via supersymmetry is shown to yield valuable information about the number of particles and that of the Higgs particles which we could still discover experimentally within a reasonable distance from the electroweak energy scale.
- Ms The El Naschie
- Higgs
El Naschie MS. The Higgs. Chaos, Solitons & Fractals 2004;22:1199-209.