On generalized Clifford groups II

The earlier study of the irreducible representations of the generalized Clifford groups G^m_n in the case where m is a prime number, is now extended to the case where m is any integer. The analysis of class structure and hence the construction of the irreducible representations of G^m_n for a non-prime integer m is found to be more complicated. This investigation also requires the properties of the generalized Clifford algebras C^m_n(I) which are studied in Section 2 of the paper. The case of infinite generalized Clifford group, i.e. G^∞_n involving the infinite- order root of unity as well as the physical relevance of the generalized Clifford groups are briefly dealt with.     

On generalized Yang-Mills theories and extensions of the standard model in Clifford (tensorial) spaces

We construct the Clifford-space tensorial-gauge fields generalizations of Yang-Mills theories and the Standard Model that allows to predict the existence of new particles (bosons, fermions) and tensor-gauge fields of higher-spins in the 10 Tev regime. We proceed with a detailed discussion of the unique D₄ − D₅ − E₆ − E₇ − E₈ model of Smith based on the underlying Clifford algebraic structures in D = 8, and which furnishes all the properties of the Standard Model and Gravity in four-dimensions, at low energies. A generalization and extension of Smith’s model to the full Clifford-space is presented when we write explicitly all the terms of the extended Clifford-space Lagrangian. We conclude by explaining the relevance of multiple-foldings of D = 8 dimensions related to the modulo 8 periodicity of the real Cliford algebras and display the interplay among Clifford, Division, Jordan, and Exceptional algebras, within the context of D = 26, 27, 28 dimensions, corresponding to bosonic string, M and F theory, respectively, advanced earlier by Smith. To finalize we describe explicitly how the E₈ × E₈ Yang-Mills theory can be obtained from a Gauge Theory based on the Clifford (16) group.     

Riemann surfaces,Clifford algebras and infinite dimensional groups

We consider functionals on one dimensional subshifts which have prescribed Randon-Nikodym derivative under transportation by conjugating homeomorphisms, and investigate their relation to Ruelle's transfer operator. In particular we show that two-sided functionals essentially are products of a functional which are supported on stable and unstable leaves. We also prove the meromorphicity of the Fourier transform of correlation functions for AxiomA follows in a more general setting.     

Riemann surfaces,Clifford algebras and infinite dimensional groups

We introduce a class of Riemann surfaces which possess a fixed point free involution and line bundles over these surfaces with which we can associate an infinite dimensional Clifford algebra. Acting by automorphisms of this algebra is a gauge group of meromorphic functions on the Riemann surface. There is a natural Fock representation of the Clifford algebra and an associated projective representation of this group of meromorphic functions in close analogy with the construction of the basic representation of Kac-Moody algebras via a Fock representation of the Fermion algebra. In the genus one case we find a form of vertex operator construction which allows us to prove a version of the Boson-Fermion correspondence. These results are motivated by the analysis of soliton solutions of the Landau-Lifshitz equation and are rather distinct from recent developments in quantum field theory on Riemann surfaces.     

Theory of the generalized collision integral. I

A diagram method previously suggested [1] for the construction of the integral collision in the single-particle kinetic equations is developed in this paper. The so-called diagram ladder summation, both with a variable external field and without it, is performed for systems with a strong short-range potential to any order in the density. It is shown that in a strong field the lowest order diagram in the density with preliminary ladder renormalization leads to a generalized Boltzmann collision integral.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 117–121, August, 1973.     

Second approximation of the generalized wkb method. I

Solutions are obtained for the one-dimensional Schrödinger equation corresponding to the second approximation of the generalized Wentzel-Kramers-Brillouin method (Petrashen-Miller-Good method). In the same approximation, the energy quantization conditions are found for bound particle states in a potential well.     

Anyons associated with the generalized braid groups

We discuss the inequivalent quantization of a physical system with a configuration space which is a certain orbit space of the Coxeter group. The framework for the generalization of the anyon is given. Also, we construct a gauge field whose holonomy gives rise to the statistical factor of the corresponding anyon.     

Some monodromy representations of generalized braid groups

A flat connection on the trivial bundle over the complement inC ⁿ of the complexification of the system of the reflecting hyperplanes of theB _n,D _n Coxeter groups is built from a simple Lie algebra and its representation. The corresponding monodromy representations of the generalized braid groupsXB _n,XD _n are computed in the simplest case.     

On the relationship between Twistors and Clifford algebras

Basis p-forms of a complexified Minkowski spacetime can be used to realize a Clifford algebra isomorphic to the Dirac algebra of matrices. Twistor space is then constructed as a spin space of this abstract algebra through a Witt decomposition of the Minkowski space. We derive explicit formulas relating the basis p-forms to index one twistors. Using an isomorphism between the Clifford algebra and a space of index two twistors, we expand a suitably defined antisymmetric index two twistor basis on p-forms of ranks zero, one, and four. Together with the inverse formulas they provide a complete passage between twistors and p-forms.     

On a class of generalized Poincaré groups: InhomogeneousSL(n,C)

A certain class of non semi-simple Lie groupsISL(n, C) based onSL(n, C) is investigated. Its Lie algebra and invariants are determined. The connection betweenISL(2,C) and the Poincaré group is discussed.On leave from Université de Marseille, Institut de Physique Théorique.     

Quantum groups and generalized statistics in integrable models

The paper deals with the integrable massive models of quantum field theory. It is shown that generalized statistics of physical particles is closely connected with the invariance under quantum groups. This invariance provides the possibility to construct quasi-local operators (parafermions) possessing generalized statistics which interpolates the physical particles. For the particular examples of SG, RSG models and scaling 3-state Potts model the parafermions are described completely (all their matrix elements in the space of states are presented).     

On Clifford representation of Hopf algebras and fierz identities

We present a short review of the action and coaction of Hopf algebras on Clifford algebras as an introduction to physically meaningful examples. Some q-deformed Clifford algebras are studied from this context and conclusions are derived.     

Quantum groups as generalized gauge symmetries in WZNW models. Part II. The quantized model

This is the second part of a paper dealing with the “internal” (gauge) symmetry of the Wess–Zumino–Novikov–Witten (WZNW) model on a compact Lie group G. It contains a systematic exposition, for G = SU(n), of the canonical quantization based on the study of the classical model (performed in the first part) following the quantum group symmetric approach first advocated by L.D. Faddeev and collaborators. The internal symmetry of the quantized model is carried by the chiral WZNW zero modes satisfying quadratic exchange relations and an n-linear determinant condition. For generic values of the deformation parameter the Fock representation of the zero modes’ algebra gives rise to a model space of U _q (sl(n)). The relevant root of unity case is studied in detail for n = 2 when a “restricted” (finite dimensional) quotient quantum group is shown to appear in a natural way. The module structure of the zero modes’ Fock space provides a specific duality with the solutions of the Knizhnik–Zamolodchikov equation for the four point functions of primary fields suggesting the existence of an extended state space of logarithmic CFT type. Combining left and right zero modes (i.e., returning to the 2D model), the rational CFT structure shows up in a setting reminiscent to covariant quantization of gauge theories in which the restricted quantum group plays the role of a generalized gauge symmetry.     

Representations of groups containing the Poincaré group. I

It is proved that in certain representations of the Poincaré group the mass operator must be identically zero.     

On generalized invariant transformations

The notion of generalized invariant transformations of variational problems in Lagrangian formulation is discussed in the framework of the variational theory in fibred manifolds. Necessary and sufficient conditions for generators of one-parameter groups of such transformations are derived, completing thus some previous results of A. Trautman on the theory of transformations.     

On generalized Lotka-Volterra lattices

Generalized matrix Lotka-Volterra lattice equations are obtained in a systematic way from a “master equation” possessing a bicomplex formulation. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

On a generalized metric

Various invariant theories concerning algebraic properties of four-dimensional space-times have been developed in which the theory of the Debever-Penrose directions and that of Petrov's or Synge's classification belong to the same category. These two algebraic problems have been dealt with in a generalized metric recently considered by the author in studying the wave solutions.