Hilbert space,the number of Higgs particles and the quantum two-slit experiment

Rigorous mathematical formulation of quantum mechanics requires the introduction of a Hilbert space. By contrast, the Cantorian E-infinity approach to quantum physics was developed largely without any direct reference to the afore mentioned mathematical spaces. In the present work we present a novel reinterpretation of basic ε^(∞) Cantorian spacetime relations in terms of the Hilbert space of quantum mechanics. In this way, we gain a better understanding of the physical and mathematical structure of quantum spacetime. In particular we show that the two-slit experiment required a definite topology which is consistent with a certain fuzzy Kähler manifold or more generally a Cantorian spacetime manifold. Finally by determining the Euler class of this manifold, we can estimate the most likely number of Higgs particles which may be discovered.     

Hilbert,Fock and Cantorian spaces in the quantum two-slit gedanken experiment

On the one hand, a rigorous mathematical formulation of quantum mechanics requires the introduction of a Hilbert space and as we move to the second quantization, a Fock space. On the other hand, the Cantorian E-infinity approach to quantum physics was developed largely without any direct reference to the afore mentioned mathematical spaces. In the present work we utilize some novel reinterpretations of basic E^(∞) Cantorian spacetime relations in terms of the Hilbert space of quantum mechanics. Proceeding in this way, we gain a better understanding of the physico-mathematical structure of quantum spacetime which is at the heart of the paradoxical and non-intuitive outcome of the famous quantum two-slit gedanken experiment.     

On the Sporadic 196884-Dimensional Group,Strings and e (∞) Spacetime

Some arguments are presented using the dimension of the Golay code and the center density of intersecting spheres in 24-dimensional lattices to show that the 196884 dimensions of the so-called monster group can be regarded as hierarchical and reduced by clustering to a quasi expectation value of four, in analogy to the E ^(∞) spacetime dimensional reduction. The analysis suggests that the conjectured DNA-like Cantorian spacetime may resemble a giant error correction code.     

From Arthur Cayley via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether and superstrings to Cantorian space–time

In this work we present a historical overview of mathematical discoveries which lead to fundamental developments in super string theory, super gravity and finally to E-infinity Cantorian space–time theory. Cantorian space–time is a hierarchical fractal-like semi manifold with formally infinity many dimensions but a finite expectation number for these dimensions. The idea of hierarchy and self-similarity in science was first entertain by Right in the 18th century, later on the idea was repeated by Swedenborg and Charlier. Interestingly, the work of Mohamed El Naschie and his two contra parts Ord and Nottale was done independently without any knowledge of the above starting from non- linear dynamics and fractals.     

Gravitational instanton in Hilbert space and the mass of high energy elementary particles

While the theory of relativity was formulated in real spacetime geometry, the exact formulation of quantum mechanics is in a mathematical construction called Hilbert space. For this reason transferring a solution of Einstein’s field equation to a quantum gravity Hilbert space is far of being a trivial problem.

On the other hand ^(∞) spacetime which is assumed to be real is applicable to both, relativity theory and quantum mechanics. Consequently, one may expect that a solution of Einstein’s equation could be interpreted more smoothly at the quantum resolution using the Cantorian ^(∞) theory.

In the present paper we will attempt to implement the above strategy to study the Eguchi–Hanson gravitational instanton solution and its interpretation by ‘t Hooft in the context of quantum gravity Hilbert space as an event and a possible solitonic “extended” particle. Subsequently we do not only reproduce the result of ‘t Hooft but also find the mass of a fundamental “exotic” symplictic-transfinite particle m1.8 MeV as well as the mass M_x and M (Planck) which are believed to determine the GUT and the total unification of all fundamental interactions respectively. This may be seen as a further confirmation to an argument which we put forward in various previous publications in favour of an alternative mass acquisition mechanism based on unification and duality considerations. Thus even in case that we never find the Higgs particle experimentally, the standard model would remain substantially intact as we can appeal to tunnelling and unification arguments to explain the mass. In fact a minority opinion at present is that finding the Higgs particle is not a final conclusive argument since one could ask further how the Higgs particle came to its mass which necessitates a second Higgs field. By contrast the present argument could be viewed as an ultimate theory based on the existence of a “super” force, beyond which nothing else exists.     

Jones Invariant,Cantorian Geometry and Quantum Spacetime

The relation between Jones knot polynominals and statistical mechanics is discussed in the light of Cantorian geometry. It is further shown that von Neumanns continuous geometry may be regarded as being a quantum spacetime akin to Cantorian space E ^(∞) and noncommutative geometry.     

On additive and multiplicative Hilbert cubes

Given subset E of natural numbers FS(E) is defined as the collection of all sums of elements of finite subsets of E and any translation of FS(E) is said to be Hilbert cube. We can define the multiplicative analog of Hilbert cube as well. E.G. Strauss proved that for every ε>0 there exists a sequence with density >1−ε which does not contain an infinite Hilbert cube. On the other hand, Nathanson showed that any set of density 1 contains an infinite Hilbert cube. In the present note we estimate the density of Hilbert cubes which can be found avoiding sufficiently sparse (in particular, zero density) sequences. As a consequence we derive a result in which we ensure a dense additive Hilbert cube which avoids a multiplicative one.     

A Note on the Subtle Mean in Physics

It is shown that the subtle mean, which is the third power of the Golden number, has some quite interesting properties. These properties connecting diverse fields such as knot theory, subfactors, noncummutative geometry, Cantorian spacetime and quasi crystals are discussed and illustrated. It is conjectured that the subtle mean is the mean dimension of actual spacetime at the resolution of quantum physics.     

Von Neumann Geometry and E(∞) Quantum Spacetime

In this paper, it is shown that von Neumann continuous geometry may be regarded as the first attempt towards formulating a general quantum spacetime geometry akin to that of Cantorian spacetime E^(∞) and noncommutative geometry.     

Characterizing Hilbert cube manifolds by their homological structure

A hierarchy of disjoint ?ech carriers properties is introduced; and each is shown to be characteristic of ANR's whose products with 2-cells are Hilbert cube manifolds.     

The Fine Structure Constant,Vacuum Polarization and the Geometry of e (∞) Space

This paper is a first attempt to derive the fine structure constant from the geometrical properties of the Cantorian manifold E ^(∞) which is assumed to model actual micro spacetime.     

Deriving the curvature of fractal-Cantorian spacetime from first principles

The paper gives various exact derivations of the curvature of spacetime manifold at different energy scales within the frame work of a fractal-Cantorian theory. It is argued that at a Hausdorff spacetime dimensionality equal 4 + ³ = 4.236067977 the unification fractal spacetime Cantorian manifold possesses a curvature equal to K = 26 + k = 26.18033989.     

The Golden Mean in Quantum Geometry,KnotTheory and Related Topics

The note summarizes some very recent and other not so well known results related tothe Golden Mean, φ, in theoretical physics. The subjects considered are knot theory,Noncommutative Geometry, four manifolds, Cantorian spacetime and quasi crystallography. Theinvolvement of the Golden Mean in these subjects is seen as an indication for an underlayingcoherent unity and unsuspected deep relationship which exists between various disciplines inphysics and which may appear when examined superficially to be unrelated although it is deeplyrelated.     

Noncommutative geometry,negative probabilities and Cantorian–Fractal spacetime

A straightforward explanation of the Young's two-slit experiment of a quantum particle is obtained within the framework of the Noncommutative Geometry associated with El Naschie's Cantorian–Fractal transfinite spacetime manifold.     

Cantorian Fractal Spacetime,Quantum-like Chaos and Scale Relativity in Atmospheric Flows

Cantorian fractal spacetime fluctuations characterize quantum-like chaos in atmospheric flows. The macroscale atmospheric flow structure behaves as a unified whole quantum system, where the superimposition of a continuum of eddies results in the observed global weather patterns with long-range spatiotemporal correlations, such as that of the widely investigated El Nino phenomenon. Large eddies are visualised as envelopes enclosing smaller eddies, thereby generating a hierarchy of eddy circulations, originating initially from a fixed primary small scale energising perturbation, e.g., the frictional upward momentum flux at the boundary layer of the Earths surface. In this paper, it is shown that the relative motion concepts of Einsteins Special and General Theories of Relativity are applicable to eddy circulations originating from a constant primary perturbation.     

P-Adic analysis and the transfinite E8 exceptional Lie symmetry group unification

In P-Adic analysis like in a fractal Cantorian space there is no absolute scale. P-Adic analysis with its prime numbers base is the mathematical quarks of the exceptional E8 and E-infinity. The P-Adic space permits the use of Weyl original spacetime gauge theory which is the rationale behind E-infinity.     

模糊紧空间在其一个自然的Hilbert方体紧化中的拓扑位置

以讨论模糊紧空间在其一个自然的Hilbert方体紧化中的拓扑位置为目的,利用Hilbert方体中伪边界的拓扑刻画,得出模糊紧空间是其Hilbert方体紧化的伪内部。     

Cremona transformations and the conundrum of dimensionality and signature of macro-spacetime

The issue of dimensionality and signature of the observed universe is analysed. Neither of the two properties follows from first principles of physics, save for a remarkably fruitful Cantorian fractal spacetime approach pursued by El Naschie, Nottale and Ord. In the present paper, the author's theory of pencil-generated spacetime(s) is invoked to provide a clue. This theory identifies spatial coordinates with pencils of lines and the time dimension with a specific pencil of conics. Already its primitive form, where all pencils lie in one and the same projective plane, implies an intricate connection between the observed multiplicity of spatial coordinates and the (very) existence of the arrow of time. A qualitatively new insight into the matter is acquired, if these pencils are not constrained to be coplanar and are identified with the pencils of fundamental elements of a Cremona transformation in a projective space. The correct dimensionality of space (3) and time (1) is found to be uniquely tied to the so-called quadro-cubic Cremona transformations – the simplest non-trivial, non-symmetrical Cremona transformations in a projective space of three dimensions. Moreover, these transformations also uniquely specify the type of a pencil of fundamental conics, i.e. the global structure of the time dimension. Some physical and psychological implications of these findings are mentioned, and a relationship with the Cantorian model is briefly discussed.