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.     

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.     

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.     

A Cantorian potential theory for describing dynamical systems on El Naschie’s space–time

In this paper we analyze classical systems, in which motion is not on a classical continuous path, but rather on a Cantorian one. Starting from El Naschie’s space–time we introduce a mathematical approach based on a potential to describe the interaction system-support. We study some relevant force fields on Cantorian space and analyze the differences with respect to the analogous case on a continuum in the context of Lagrangian formulation. Here we confirm the idea proposed by the first author in dynamical systems on El Naschie’s ϵ^(∞) Cantorian space–time that a Cantorian space could explain some relevant stochastic and quantum processes, if the space acts as an harmonic oscillating support, such as that found in Nature. This means that a quantum process could sometimes be explained as a classical one, but on a nondifferential and discontinuous support. We consider the validity of this point of view, that in principle could be more realistic, because it describes the real nature of matter and space. These do not exist in Euclidean space or curved Riemanian space–time, but in a Cantorian one. The consequence of this point of view could be extended in many fields such as biomathematics, structural engineering, physics, astronomy, biology and so on.     

Hilbert cube model for fractal spacetime

A three-dimensional Hilbert cube has exactly three dimensions. It can mimic our spatial world on an ordinary observation scale. A four-dimensional Hilbert cube is equivalent to Elnaschie Cantorian spacetime. A very small distance in a very high observable resolution is equivalent to a very high energy spacetime which is inherently Cantorian, non-differentiable and discontinuous. This article concludes that spacetime is a fractal and hierarchical in nature. The spacetime could be modeled by a four-dimensional Hilbert cube. Gravity and electromagnetism are at different levels of the hierarchy. Starting from a simple picture of a four-dimensional cube, a series of higher dimensional polytops can be constructed in a self-similar manner. The resulting structure will resemble a Cantorian spacetime of which the expectation of the Hausdorff dimension equals to 4.23606799 provided that the number of hierarchical iterations is taken to infinity. In this connection, we note that Heisenberg Uncertainty Principle comes into play when we take measurement at different levels of the hierarchy.     

Spectral reciprocity and matrix representations of unbounded operators

We study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. For an infinite discrete set X, we consider operators acting on Hilbert spaces of functions on X, and their representations as infinite matrices; the focus is on ?²(X), and the energy space HE. In particular, we prove that these operators are always essentially self-adjoint on ?²(X), but may fail to be essentially self-adjoint on HE. In the general case, we examine the von Neumann deficiency indices of these operators and explore their relevance in mathematical physics. Finally we study the spectra of the HE operators with the use of a new approximation scheme.     

Loop quantum mechanics and the fractal structure of quantum spacetime

We discuss the relation between string quantization based on the Schild path integral and the Nambu-Goto path integral. The equivalence between the two approaches at the classical level is extended to the quantum level by a saddle-point evaluation of the corresponding path integrals. A possible relationship between M-Theory and the quantum mechanics of string loops is pointed out. Then, within the framework of “loop quantum mechanics”, we confront the difficult question as to what exactly gives rise to the structure of spacetime. We argue that the large scale properties of the string condensate are responsible for the effective Riemannian geometry of classical spacetime. On the other hand, near the Planck scale the condensate “evaporates”, and what is left behind is a “vacuum” characterized by an effective fractal geometry.     

The Fibonacci code behind super strings and P-Branes. An answer to M. Kaku’s fundamental question

In “Beyond Einstein” the leading string theoretician and notable science writer Michio Kaku referred to what he labelled the ‘strange’ link between the E₈ exceptional Lie group and the various dimensionalities of strings and super string theories and commented on that by saying “If we could understand why the numbers 8, 10 and 26 continually crop up in super string theory, perhaps we could understand why the universe is four dimensional”.In the present work we demonstrate the existence of a Fibonacci code-like connection between the various coupling constants, charges and dimensionalities of super strings and P-Brane theories. This code is based on the Fibonacci numbers and the golden mean and in the final analysis, may be attributed to the deterministically chaotic nature of the hyperbolic Cantorian sets fixing the geometry and topology of quantum spacetime.     

Mathematical problems of stochastic quantum mechanics: Harmonic analysis on phase space and quantum geometry

In this paper we review the mathematical methods and problems that are specific to the programme of stochastic quantum mechanics and quantum spacetime. The physical origin of these problems is explained, and then the mathematical models are developed. Three notions emerge as central to the programme: positive operator-valued (POV) measures on a Hilbert space, reproducing kernel Hilbert spaces, and fibre bundle formulations of quantum geometries. A close connection between the first two notions is shown to exist, which provides a natural setting for introducing a fibration on the associated overcomplete family of vectors. The introduction of group covariance leads to an extended version of harmonic analysis on phase space. It also yields a theory of induced group representations, which extends the results of Mackey on imprimitivity systems for locally compact groups to the more general case of systems of covariance. Quantum geometries emerge as fibre bundles whose base spaces are manifolds of mean stochastic locations for quantum test particles (i.e., spacetime excitons) that display a phase space structure, and whose fibres and structure groups contain, respectively, the aforementioned overcomplete families of vectors and unitary group representations of phase space systems of covariance.Work supported in part by the Natural Science and Engineering Research Council of Canada (NSERC) grants.     

Self-similar and oscillating solutions of Einstein's equation and other relevant consequences of a stochastic self-similar and fractal Universe via El Naschie's ε(∞) Cantorian space–time

In this paper we make some suggestion regarding the unification of the fundamental forces and the age of the Universe in the context of the a stochastic self-similar and fractal Universe using El Naschie's ε^(∞) Cantorian space–time. We also show how Einstein's equation can admit for the scale factor a(t) a self-similar solution in agreement with our stochastic self-similar, fractal Universe and El Naschie's ε^(∞) Cantorian space–time. In addition, this solution is found to be oscillating one. Thanks to the first quantization it is possible to recast the equations in a Schrödinger-like form. Consequently, the presently observed large scale structure reflects the phenomenology of the Early Universe or of the microscopic world. Again it appears clear that the Universe and the structures inside must have a memory of its quantum origin as conjectured sometime ago.     

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.     

Cantorian space–time,Fantappie’s final group,accelerated universe and other consequences

In this paper we introduce Mohamed El Naschie’s ϵ^(∞) Cantorian space–time in connection with stochastic self-similar processes to give a possible explanation of the segregation of the Universe at fixed scale; then by considering the Fanntappie’s transformation group we show how the universe could appear accelerated on Cantorian space–time.     

On the similarity problem for locally compact quantum groups

A well-known theorem of Day and Dixmier states that any uniformly bounded representation of an amenable locally compact group G on a Hilbert space is similar to a unitary representation. Within the category of locally compact quantum groups, the conjectural analogue of the Day–Dixmier theorem is that every completely bounded Hilbert space representation of the convolution algebra of an amenable locally compact quantum group should be similar to a ?-representation. We prove that this conjecture is false for a large class of non-Kac type compact quantum groups, including all q-deformations of compact simply connected semisimple Lie groups. On the other hand, within the Kac framework, we prove that the Day–Dixmier theorem does indeed hold for several new classes of examples, including amenable discrete quantum groups of Kac-type.     

The mean uncertainty and distance dimension of multidimensional cantorian spaces

Several ways of describing the internal structure of infinite point sets and determining a corresponding average dimension are outlined. Possible relevance to discretisation of quantum space-time are discussed. Finally, a method for determining the average internal distance between different Cantor points is proposed.The main conclusion is that a micro-scale Cantorian space-time may be based on an average Cantorian distance d_l⁽⁰⁾ ⋍ 0.629, while the smooth space-time is retrieved when d_l⁽⁰⁾ → 0.5, which means d_l⁽⁰⁾ tends towards the average distance of a continuous smooth line at a vanishing resolution. This correspond to the case when space is viewed from very far, at the macro-scale of classical physics.     

Improved gradient method for monotone and lipschitz continuous mappings in banach spaces

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {A_n}_(n∈N) be a family of monotone and Lipschitz continuos mappings of C into E~*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [10] for solving the variational inequality problem for{A_n} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.     

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.     

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.     

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.     

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.