On the notion of space in geometry

The development of the notion of space in geometry is traced from the early axiomatization in Euclid’s Elements over the discovery of non-Euclidean geometries to geometry of manifolds in relativity theory and in gauge and string theories in contemporary physics. The notion of space is considered in a historic-philosophical perspective including a short discussion of the contributions of artists to visualization of spatial objects.     

The Extended Relativity Theory in Clifford Phase Spaces and Modifications of Gravity at the Planck/Hubble Scales

We extend the construction of Born’s Reciprocal Phase Space Relativity to the case of Clifford Spaces which involve the use of polyvectors and a lower/upper length scale. The generalized polyvector-valued velocity and acceleration/force boosts in Clifford Phase Spaces is presented and we find an explicit Clifford algebraic realization of the velocity and acceleration/force boosts in ordinary phase space. Finally, we provide a Clifford Phase-Space Gravitational Theory based in gauging the generalization of the Quaplectic group and invoking Born’s reciprocity principle between coordinates and momenta (maximal speed of light velocity and maximal force). The generalized gravitational vacuum field equations are explicitly displayed. We conclude with a brief discussion on the role of higher-order Finsler geometry in the construction of extended relativity theories with an upper and lower bound to the higher order accelerations (associated with the higher order tangent and cotangent spaces).     

Extended Lorentz Transformations in Clifford Space Relativity Theory

Some novel physical consequences of the Extended Relativity Theory in C-spaces (Clifford spaces) were explored recently. In particular, generalized photon dispersion relations allowed for energy-dependent speeds of propagation while still retaining the Lorentz symmetry in ordinary spacetimes, but breaking the extended Lorentz symmetry in C-spaces. In this work we analyze in further detail the extended Lorentz transformations in Clifford Space and their physical implications. Based on the notion of “extended events” one finds a very different physical explanation of the phenomenon of “relativity of locality” than the one described by the Doubly Special Relativity (DSR) framework. A generalized Weyl-Heisenberg algebra, involving polyvector-valued coordinates and momenta operators, furnishes a realization of an extended Poincare algebra in C-spaces. In addition to the Planck constant ?, one finds that the commutator of the Clifford scalar components of the Weyl-Heisenberg algebra requires the introduction of a dimensionless parameter which is expressed in terms of the ratio of two length scales : the Planck and Hubble scales. We finalize by discussing the concept of “photons”, null intervals, effective temporal variables and the addition/subtraction laws of generalized velocities in C-space.     

Ground ring of <Emphasis Type="Italic">α</Emphasis>-generators and a sequence of Ramond-Neveu-Schwarz string theories

We construct a sequence of new nilpotent BRST charges in the Ramond-Neveu-Schwarz superstring theory based on the hierarchy of found local gauge symmetries. These gauge symmetries are in turn related to global α-symmetries in the space-time forming a noncommutative ring. The constructed BRST charges are presumably connected with the superstring dynamics in a curved space-time with an AdS-type geometry.     

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.     

Is quantum space-time infinite dimensional

The stringy uncertainty relations, and corrections thereof, were explicitly derived recently from the new relativity principle that treats all dimensions and signatures on the same footing and which is based on the postulate that the Planck scale is the minimal length in nature in the same vein that the speed of light was taken as the maximum velocity in Einstein's theory of Special Relativity. A simple numerical argument is presented which suggests that quantum space-time may very well be infinite dimensional. A discussion of the repercussions of this new paradigm in Physics is given. A truly remarkably simple and plausible solution of the cosmological constant problem results from the new relativity principle: The cosmological constant is not a constant, in the same vein that energy in Einstein's Special Relativity is observer dependent. Finally, following El Naschie, we argue why the observed D=4 world might just be an average dimension over the infinite possible values of the quantum space-time and why the compactification mechanisms from higher to four dimensions in string theory may not be actually the right way to look at the world at Planck scales.     

Spacetime and the Geometry behind it

This is the Leonardo da Vinci Lecture given in Milan in March 2006. It is a survey on the concept of space-time over the last 3000 years: it starts with Euclidean geometry, discusses the contributions of Gauss and Riemannian geometry, presents the dynamic concept of space-time in Einstein’s general relativity, describes the importance of symmetries, and ends with Calabi-Yau manifolds and their importance in today’s string theories in the attempt for a unified theory of physics. Leonardo da Vinci Lecture held on March 28, 2006     

The Mathematicians’ Happy Hunting Ground: Einstein’s General Theory of Relativity

There is hardly any doubt that for physics special relativity theory is of much greater consequence than the general theory. The reverse situation prevails with respect to mathematics: there special relativity theory had comparatively little, general relativity theory very considerable, influence, above all upon the development of a general scheme for differential geometry. —Hermann Weyl, “Relativity as a Stimulus to Mathematical Research,” pp. 536–537.     

Symmetry algebra of the AdS4×ℂℙ3 superstring

By direct calculation in the classical theory, we derive the central extension of the off-shell symmetry algebra for a string propagating in AdS ₄ ×?? ³ . It turns out to be the same as in the case of the AdS ₅ ×S ⁵ string. We consider the choice of the κ-symmetry gauge in detail and also explain how this gauge can be chosen without breaking the bosonic symmetries.     

The most intensive fluctuation in chaotic time series and relativity principle

Relativity principle in mechanics and principle of invariant speed of light lead to Einstein theory. The exponent of p-order momentum, derived from a piece of multi-scale chaotic time series, varies with the order p and cannot exceeds a maximum, so there exists the principle of scale relativity. Its special case is the same one as Lorenz transformation from Einstein theory.     

双标量-张量几何与标量-张量引力论变分原理

<正> §1.引言 自从1915年A.Einstein奠定了广义相对论的基础以来,曾出现过各种各样的引力理论;但似乎只有标量-张量引力理论可同广义相对论媲美.看来标量-张量理论同Einstein的广义相对论一样,是一种具有生命力的引力理论. 如所周知,Einstein的广义相对论实质上是引力现象的几何化理论,即是一种引力的度规张量理论.Einstein与Weyl的物理学之几何化思想对物理学的发展曾起过、并且将     

Geodesic equations for a charged particle in the unified theory of gravitational and electromagnetic interactions

A new model of gravitational and electromagnetic interactions is constructed as a version of the classical Kaluza-Klein theory based on a five-dimensional manifold as the physical space-time. The velocity space of moving particles in the model remains four-dimensional as in the standard relativity theory. The spaces of particle velocities constitute a four-dimensional distribution over a smooth five-dimensional manifold. This distribution depends only on the electromagnetic field and is independent of the metric tensor field. We prove that the equations for the geodesics whose velocity vectors always belong to this distribution are the same as the charged particle equations of motion in the general relativity theory. The gauge transformations are interpreted in geometric terms as a particular form of coordinate transformations on the five-dimensional manifold. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 517–528, June, 1999.     

Isovecteurs pour l'équation de Hamilton–Jacobi–Bellman,différentielles stochastiques formelles et intégrales premières en mécanique quantique euclidienneIsovectors for the Hamilton–Jacobi–Bellman equation,formal stochastic differentials and constants of motion in Euclidean Quantum Mechanics

We give a stochastic interpretation of the geometrical representation, from E. Cartan, of the heat equation, in terms of ideal exterior differential forms and isovectors generating the symmetries of this equation. The method can also be used to interpret as a stochastic deformation the contact geometry of first order ordinary differential equations and the search for infinitesimal symmetries of the associated Hamilton–Jacobi equation. We thus generalise, in an elegant and geometrical way, the results coming originally from long calculations of stochastic analysis. To cite this article: P. Lescot, J.-C. Zambrini, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 263–266.     

Integrability of Generalized (Matrix) Ernst Equations in String Theory

We elucidate the integrability structures of the matrix generalizations of the Ernst equation for Hermitian or complex symmetric (d×d)-matrix Ernst potentials. These equations arise in string theory as the equations of motion for the truncated bosonic parts of the low-energy effective action for the respective dilaton and d×d matrix of moduli fields or for a string gravity model with a scalar (dilaton) field, a U(1) gauge vector field, and an antisymmetric 3-form field, all depending on only two space-time coordinates. We construct the corresponding spectral problems based on the overdetermined 2d×2d linear systems with a spectral parameter and the universal (i.e., solution-independent) structures of the canonical Jordan forms of their matrix coefficients. The additionally imposed existence conditions for each of these systems of two matrix integrals with appropriate symmetries provide specific (coset) structures of the related matrix variables. We prove that these spectral problems are equivalent to the original field equations, and we envisage an approach for constructing multiparametric families of their solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 214–225, August, 2005.     

Twists of quantum groups and noncommutative field theory

We discuss the role of quantum universal enveloping algebras of symmetries in constructing the noncommutative geometry of space-time and the corresponding field theory. It is shown that in the framework of the twist theory of quantum groups, the noncommutative (super)space-time defined by coordinates with Heisenberg commutation relations is (super)Poincaré invariant, as well as the corresponding field theory. Noncommutative parameters of global transformations are introduced. Bibliography: 30 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 188–204.     

Eliminating gauge anomalies via a “point-less” fractal Yang–Mills theory

A Cantorian fractal extension of Yang–Mills theory can eliminate gauge anomalies caused by the frequent conflict between the classical and the internal symmetries.     

On Einstein’s super symmetric tensor and the number of elementary particles of the standard model

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.     

Ghost Symmetries and Multi-fold Darboux Transformations of Extended Toda Hierarchy

In this paper, the author constructs ghost symmetries of the extended Toda hierarchy with their spectral representations. After this, two kinds of Darboux transformations in different directions and their mixed Darboux transformations of this hierarchy are constructed. These symmetries and Darboux transformations might be useful in GromovWitten theory of CP¹.     

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.