Noncommutative geometry and spacetime gauge symmetries of string theory

We illustrate the various ways in which the algebraic framework of noncommutative geometry naturally captures the short-distance spacetime properties of string theory. We describe the noncommutative spacetime constructed from a vertex operator algebra and show that its algebraic properties bear a striking resemblence to some structures appearing in M Theory, such as the noncommutative torus. We classify the inner automorphisms of the space and show how they naturally imply the conventional duality symmetries of the quantum geometry of spacetime. We examine the problem of constructing a universal gauge group which overlies all of the dynamical symmetries of the string spacetime. We also describe some aspects of toroidal compactifications with a light-like coordinate and show how certain generalized Kac—Moody symmetries, such as the Monster sporadic group, arise as gauge symmetries of the resulting spacetime and of superstring theories.     

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.     

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.     

CR-structures and Lorentzian geometry

As a recent excellent example of mutual interplay between the Cauchy-Riemann structure and physical spacetime geometry, we present, in this paper, a few fresh ideas on this fruitful relationship with respect to the conformal geometry and the groups of motions of Lorentzian manifolds.     

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.     

Lorentzian geometry of globally framed manifolds

A new class of globally framed manifolds (carrying a Lorentz metric) is introduced to establish a relation between the spacetime geometry and framed structures. We show that strongly causal (in particular, globally hyperbolic) spacetimes can carry a regular framed structure. As examples, we present a class of spacetimes of general relativity, having an electromagnetic field, endowed with a framed structure and a causal spacetime with a nonregular contact structure. This paper opens a few new problems, of geometric/physical significance, for further study.     

Uniqueness of Kottler spacetime and the Besse conjecture

We establish a black hole uniqueness theorem for Schwarzschild–de Sitter spacetime, also called Kottler spacetime, which satisfies Einstein's field equations of general relativity with positive cosmological constant. Our result concerns the class of static vacuum spacetimes with compact spacelike slices and regular maximal level set of the lapse function. We provide a characterization of the interior domain of communication of the Kottler spacetime, which surrounds an inner horizon and is surrounded by a cosmological horizon. The proof combines arguments from the theory of partial differential equations and differential geometry, and is centered on a detailed study of a possibly singular foliation. We also apply our technique in the Riemannian setting, and establish the validity of the so-called Besse conjecture.     

Multipliers for the symmetry groups of p-adic spacetime

The consequences for particle classification of the Volovich hypothesis that spacetime geometry is non-archimedean at the Planck scale are explored. The multiplier groups and universal topological central extensions of the p-adic Poincaré and Galilean groups are determined. The text was submitted by the author in English.     

The energy—Momentum tensor as the source of the gravitational field and the effective Riemannian spacetime

It is shown in this paper that the assumption of the matter energy—momentum tensor is the source of the gravitational field leads naturally to an effective Riemannian geometry of spacetime.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 3, pp. 538–542, September, 1995.     

Nuclear Spacetime Theories,Superstrings, MonsterGroup and Applications

The present paper is basically a collection of notes and remarks on various nuclear-quantum spacetime theories and their possible application in physics. Several subjects are considered including quantum gravity, fractional quantum Hall effect, knot theory, superstrings, noncommutative geometry, partially ordered sets, Cantorian spaces, branched polymers and sporadic 196884 dimensional monster group.     

Recovering the Geometry of a Flat Spacetime from Background Radiation

We consider globally hyperbolic flat spacetimes in 2 + 1 and 3 + 1 dimensions, in which a uniform light signal is emitted on the r-level surface of the cosmological time for r → 0. We show that the frequency shift of this signal, as perceived by a fixed observer, is a well-defined, bounded function which is generally not continuous. This defines a model with anisotropic background radiation that contains information about initial singularity of the spacetime. In dimension 2 + 1, we show that this observed frequency shift function is stable under suitable perturbations of the spacetime, and that, under certain conditions, it contains sufficient information to recover its geometry and topology. We compute an approximation of this frequency shift function for a few simple examples.     

Fractal space,cosmic strings and spontaneous symmetry breaking

The present paper is conceived within the framework of El Naschie's fractal-Cantorian program and proposes to develop a model of the fractal properties of spacetime. We show that, starting from the most fundamental level of elementary particles and rising up to the largest scale structure of the Universe, the fractal signature of spacetime is imprinted onto matter and fields via the common concept for all scales emanating from the physical spacetime vacuum fluctuations. The fractal structure of matter, field and spacetime (i.e. the nature and the Universe) possesses a universal character and can encompass also the well-known geometric structures of spacetime as Riemannian curvature and torsion and includes also, deviations from Newtonian or Einsteinian gravity (e.g. the Rössler conjecture). The leitmotiv of the paper is generated by cosmic strings as a fractal evidence of cosmic structures which are directly related to physical properties of a vacuum state of matter (VSM). We present also some physical aspects of a spontaneous breaking of symmetry and the Higgs mechanism in their relation with cosmic string phenomenology. Superconducting cosmic strings and the presence of cosmic inhomogeneities can induce to cosmic Josephson junctions (weak links) along a cosmic string or in connection with a cosmic string (self) interactions and thus some intermittency routes to a cosmic chaos can be explored. The key aspect of fractals in physics and of fractal geometry is to understand why nature gives rise to fractal structures. Our present answer is: because a fractal structure is a manifestation of the universality of self-organisation processes, as a result of a sequence of spontaneous symmetry breaking (SSB). Our conclusion is that it is very difficult to prescribe a certain type of fractal within an empty spacetime. Possibly, a random fractal (like a Brownian motion) characterises the structure of free space. The presence of matter will decide the concrete form of fractalisation. But, what does it mean the presence of matter? Can there exist a spacetime without matter or matter without spacetime? Possibly not, but consider on the other hand a space far removed from usual matter, or a space containing isolated small particles in which a very low density matter can exist. Very low density matter might be influenced by a fractal structure of space, for example in the sense that it is subject also to fluctuations structured by random fractals. Diffraction and diffusion experiments in an empty space and very low density matter could provide evidence of a fractal structure of space. However, at very high (Planck) densities, and a spacetime in which fluctuations represent also the source of matter and fields (which is very resonable within the context of a quantum gravity), we can assert that Einstein's dream of geometrising physics and El Naschie's hope to prove the fractalisation (or Cantorisation) of spacetime are fully realised.     

Geometry of Normal Graphs in Euclidean Space and Applications to the Penrose Inequality in Minkowski

The Penrose inequality in Minkowski is a geometric inequality relating the total outer null expansion and the area of closed, connected and spacelike codimension-two surfaces \({{\bf \mathcal{S}}}\) in the Minkowski spacetime, subject to an additional convexity assumption. In a recent paper, Brendle and Wang A (Gibbons–Penrose inequality for surfaces in Schwarzschild Spacetime. arXiv:1303.1863, 2013) find a sufficient condition for the validity of this Penrose inequality in terms of the geometry of the orthogonal projection of \({{\bf \mathcal{S}}}\) onto a constant time hyperplane. In this work, we study the geometry of hypersurfaces in n-dimensional Euclidean space which are normal graphs over other surfaces and relate the intrinsic and extrinsic geometry of the graph with that of the base hypersurface. These results are used to rewrite Brendle and Wang’s condition explicitly in terms of the time height function of \({{\bf \mathcal{S}}}\) over a hyperplane and the geometry of the projection of \({{\bf \mathcal{S}}}\) along its past null cone onto this hyperplane. We also include, in Appendix, a self-contained summary of known and new results on the geometry of projections along the Killing direction of codimension two-spacelike surfaces in a strictly static spacetime.     

<Emphasis Type="Italic">p</Emphasis>-Adic mathematical physics: the first 30 years

p-Adic mathematical physics is a branch of modern mathematical physics based on the application of p-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of p-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.     

Structure,classification, and conformal symmetry of elementary particles over non-archimedean space-time

It is well known that at distances shorter than Planck length, no length measurements are possible. The Volovich hypothesis asserts that at sub-Planckian distances and times, spacetime itself has a non-Archimedean geometry. We discuss the structure of elementary particles, their classification, and their conformal symmetry under this hypothesis. Specifically, we investigate the projective representations of the p-adic Poincaré and Galilean groups, using a new variant of the Mackey machine for projective unitary representations of semidirect products of locally compact and second countable (lcsc) groups. We construct the conformal spacetime over p-adic fields and discuss the imbedding of the p-adic Poincaré group into the p-adic conformal group. Finally, we show that the massive and the so called eventually masssive particles of the Poincaré group do not have conformal symmetry. The whole picture bears a close resemblance to what happens over the field of real numbers, but with some significant variations.     

The standard model physical degrees of freedom interpretation of the electromagnetic fine structure coupling

The present short note gives for the first time a derivation of the inverse electromagnetic fine structure constant from the elementary particles content of the standard model plus graviton and the Higgs. It is the first derivation ever to interpret as the familiar N_f = (2)(48) = 96 Fermions and N_B = (2)(15) = 30 Bosons of the standard model plus the eleven dimensions D = 11 of super gravity spacetime . The exact theoretical value 137.082039325 and the accurate experimental results are also given clear mathematical derivation showing that all of the 137 and not only the 96 + 30 = 126 may be interpreted as physical particles so that in a sense elementary particles create and span spacetime.     

Stochastic geometry,random zero point field and quantum correction to the metric

The random zero point field induces the probabilistic aspect in the geometry of background spacetime. The corrections to the metric tensor for Riemannian or pseudo-Euclidean spaces are calculated by averaging over the ensemble of random A (x). This provides a cut-off procedure which yields a finite energy density for the vacuum state.