arXiv subjects
Publications and source records attributed to M. Lorente.
We explore the mathematical consequences of the assumption of a discrete space-time. The fundamental laws of physics have to be translated into the language of discrete mathematics. We find integral transformations that leave the lattice of any dimension invariant and apply these transformations to field equations.
We develop general expressions for the raising and lowering operators that belong to the orthogonal polynomials of hypergeometric type with discrete and continuous variable. We construct the creation and annihilation operators that correspond to the normalized polynomials and study their algebraic properties in the case of the Kravchuk/Hermite Meixner/Laguerre polynomials.
Following standard methods we explore the construction of the discrete Poincare group, the semidirect product of discrete translations and integral Lorentz transformations, using the Wigner-Mackey construction restricted to the momentum and position space on the lattice. The orbit condition, irreducibility and assimptotic limit are discussed.
Historically the starting point of wave mechanics is the Planck and Einstein-de Broglie relations for the energy and momentum of a particle, where the momentum is connected to the group velocity of the wave packet. We translate the arguments given by de Broglie to the case of a wave defined on the grid points of a space-time lattice and explore the physical consequences such as integral period, wave length, discrete energy, momentum and rest mass.
We continue the program, presented in previous Symposia, of discretizing physical models. In particular we calculate the integral Lorentz transformations with the help of discrete reflection groups, and use them for the covariance of Klein-Gordon and Dirac wave equation on the lattice. Finally we define the unitary representation of Poincar group on discrete momentum and configuration space, induced by integral representations of its closed subgroup.
We propose the assumption of quantum mechanics on a discrete space and time, which implies the modification of mathematical expressions for some postulates of quantum mechanics. In particular we have a Hilbert space where the vectors are complex functions of discrete variable. As a concrete example we develop a discrete analog of the one-dimensional quantum harmonic oscillator, using the dependence of the Wigner functions in terms of Kravchuk polynomials. In this model the position operator has a discrete spectrum given by one index of the Wigner functions, in the same way that the energy eigenvalues are given by the other matricial index. Similar picture can be made for other models where the differential equation and their solutions correspond to the continuous limit of some difference operator and orthogonal polynomial of discrete variable.
We review some modern theories about the structure of space and time, in particular those related to discrete space and time. Following an epistemological method we start from theories which discuss discrete space and time as a mathematical tool to solve physical models. Antother theories look for physical content of the discrete structure of space and time, based in relational theories of space and time which are derived from the relations of some fundamental entities. Finally we present some philosophical positions who try to find the ontological foundation of the relational theories os space and time.
We develope a difference calculus analogous to the differential geometry by translating the forms and exterior derivatives to similar expressions with difference operators, and apply the results to fields theory on the lattice [Ref. 1]. Our approach has the advantage with respect to other attempts [Ref. 2-6] that the Lorentz invariance is automatically preserved as it can be seen explicitely in the Maxwell, Klein-Gordon and Dirac equations on the lattice.
We present current theories about the structure of space and time, where the building blocks are some fundamental entities (yes-no experiment, quantum processes, spin net-work, preparticles) that do not presuppose the existence of space and time. The relations among these objects are the base for a pregeometry of discrete character, the continuous limit of which gives rise to the physical properties of the space and time.
Following recent assumptions to unify quantum mechanics and general relativity, the structure of spacetime is suppose to be a consequence of the relations among some fundamental objects, and its concept can be formulated without the reference to the intuition. As physical consequences the continuous laws should be translated in to difference equations and the lattice field theories should be interpreted as a realistic model.
We explore the connection between the symmetry transformations and conservation laws for the Klein-Gordon and Dirac fields on the lattice. The generators of the space time translations and Lorentz boost (defined on the lattice) are constants of motion and satisfy the standard commutation relations.
The main purpose of this work is to review the results obtained recently concerning the unitarization of highest weight representations for affine Kac-Moody algebras following the work of Jakobsen and Kac.
The study of unitarization of representations for non compact real forms of simple Lie Algebras has been achieved in the past decade by Jakobsen (JA81, JA83) and by Enright, Howe and Wallach (EH83) following different paths but arriving at the same final results. In order to discuss unitarity we need to introduce a scalar product. This is done in sections II and III introducing a sesquilinear form on the universal enveloping algebra (GL90). Such a sesquilinear form was introduced by Harish-Chandra (HC55), Gel'fand and Kirillov (GK69) and Shapovalov (SH72). The new developments given to Jakobsen method in GEL90 are contained in sections IV and V. In section VI we summarize the principal results due to Enright Howe and Wallach (EHW method). In section VII we give the possible places for unitarity including those for which the reduction level can't be higher than one and that were not considered in GEL90. We see also in this section and in a explicit way how the Jakobsen method and the EHW method give the same final results. This type of representations have found applications in physics for a long time (see GK75, 82; GL83, 89; LO86, 89 and references contained therein) In sections VIII and IX we consider the conformal and de Sitter algebras. In particular the construction of wave equations for conformal multispinors and the wave equations in de Sitter space are reviewed in sections X and XI. In the Appendix we give the representations of the classical algebras in the subspace Omega extending the results obtained in LG84 for the algebra A_l.
In a recent paper ([1],[2]) we have classified explicitely all the unitary highest weight representations of non compact real forms of semisimple Lie Algebras on Hermitian symmetric space. These results are necessary in order to construct all the unitary highest weight representations of affine Kac-Moody Algebras following some theorems proved by Jakobsen and Kac ([3],[4]).
Using Jakobsen theorems, unitarizability in Hermitian Symmetric Spaces is discussed. The set of all missing highest weights is explicitly calculated and the construction of their corresponding highest weights vectors is studied.
Using the method of finite differences a scheme is proposed to solve exactly the Klein-Gordon and Dirac free field equations, in a (1+1)-dimensional lattice. The hamiltonian of the Dirac field is translational invariant, hermitian, avoids fermion doubling, and, for the massless case, preserves global chiral symmetry. Coupling the fermion field to the electromagnetic vector potential we construct a gauge invariant vector current leading to the correct axial anomaly.