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arXiv:hep-th/9410220 · Journal of Physics A Mathematical and General

Finite-difference equations in relativistic quantum mechanics

V Aldaya, J Guerrero

Published Feb 21, 1995• 1 citation
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Abstract

Relativistic quantum mechanics suffers from structural problems which are traced back. To the lack of a position operator x, satisfying (,p)=inl with the ordinary momentum operator p, in the basic symmetry group-the Poincare group. In this letter we provide a finite-dimensional extension of the Poincare group containing only one more generator pi (in l+l dimensions), satisfying the commutation relation (k, pi )=inl with the ordinary boost generator k. The unitary irreducible representations are calculated and the carrier space proves to be the set of Shapiro's wavefunctions. The generalized equations of motion constitute a simple example of exactly solvable finite-difference set of equations associated with infinite-order polarization equations.

DOI
10.1088/0305-4470/28/4/005

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Relativistic quantum mechanics suffers from structural problems which are traced back. To the lack of a position operator x, satisfying (,p)=inl with the ordinary momentum operator p, in the basic symmetry group-the Poincare group. In this letter we provide a finite-dimensional extension of the Poincare group containing only one more generator pi (in l+l dimensions), satisfying the commutation relation (k, pi )=inl with the ordinary boost generator k. The unitary irreducible representations are calculated and the carrier space proves to be the set of Shapiro's wavefunctions. The generalized equations of motion constitute a simple example of exactly solvable finite-difference set of equations associated with infinite-order polarization equations.
Relativistic Quantum Mechanics suffers from structural problems which are traced back to the lack of a position operator ˆx, satisfying [ˆx, ˆp] = i¯hˆ1 with the ordinary momentum operator ˆp, in the basic symmetry group – the Poincaré group. In this paper we provide a finite-dimensional extension of the Poincaré group containing only one more (in 1+1D) generator ˆπ, satisfying the commutation relation [ ˆ k, ˆπ] = i¯hˆ1 with the ordinary boost generator ˆ k. The unitary irreducible representations are calculated and the carrier space proves to be the set of Shapiro’s wave functions. The generalized equations of motion constitute a simple example of exactly solvable finite-difference set of equations associated with infinite-order polarization equations. Introduction: Higher-order Polarizations One essential difference between Geometric Quantization [1, 2, 3, 4] and a Group Approach to Quantization (GAQ) [5, 6] is the possibility of introducing in the latter higher-order polarizations made out of elements of the (left) enveloping algebra. These higher-order polarizations are especially suitable for those (anomalous) cases in which the
Time evolution of relativistic particle systems can be accomplished by means of a Schrödinger wave equation, provided the Hamiltonian of the N particle system satisfies some commutator equations involving the other generators of a suitable representation of the Poincaré group on the initial data space of the Schrödinger equation. This set of operator equations is solved in the N free particle case. Further, the structure of N interacting particle Hamiltonians is worked out to include second-order relativistic corrections, i.e., (1/c2)-terms. It is shown that Breit and Barker–O’Connell Hamiltonians (nonzero spin particles) and the Bažański Hamiltonian (zero-spin particles) are particular cases of this more general Hamiltonian.
"The problem of providing a physical interpretation of a relativistic quantum theory is discussed. It is shown that a single operator corresponding to a localized measurement on the states of the-theory is sufficient for a significant part (perhaps all) of this interpretation. The application of this method of interpretation to the relativistic quantum theory defined on the representation space of an irreducible representation of a higher symmetry group containing the Poincare group is discussed. It is concluded that the higher symmetry is, in fact, of little use for this interpretation. This .leads in a natural way to the problem of finding some of these ""quasilocal"" operators on the space of states of a given relativistic quantum theory. 1'/ This problem is specialized to the problem of constructing a model relativistic quantum theory of a certain type. The procedure used is to attempt to construct a representation of the generators of the Poincare group on the Fock space X of real scalarbosons in such a way that quasilocal observables are easily expressible in terms of the canonical creation-destruction operators. The structure of the model permits the construction of the generators to be approached by successively satisfying their structure relations in each of the spaces ~(n) specified by the eigenvalues of the free field particle number operator. These structure relations are satisfied in~(l) and~(2). The mathematical details of the problem increase rapidly with n, and no explicit solution is obtained for n > 2. Arguments are given which make it plausible that the structure relations in~(n), with n> 2, may be satisfied by a large class of generators. Several trial solutions in the space ~(3) are described, and the difficulties associated with them are discussed."
Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincar\'e quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group G5{\mathcal{G}}_{5}G5​ is suggested as the symmetry group of dynamics. G5{\mathcal{G}}_{5}G5​ contains both the nonrelativistic Galilei group and the Poincar\'e group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of G5{\mathcal{G}}_{5}G5​ by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ correspond to infinite towers of states with increasing spin.
The Hilbert space of the unitary irreducible representations of a Lie group that is a quantum dynamical group are identified with the quantum state space. Hermitian representation of the algebra are observables. The eigenvalue equations for the representation of the set of Casimir invariant operators define the field equations of the system. A general class of dynamical groups are semidirect products K *s N for which the representations are given by Mackey's theory. The homogeneous group K must be a subgroup of the automorphisms of the normal group N. The archetype dynamical group is the Poincare group. The field equations defined by the representations of the Casimir operators define the basic equations of physics; Klein-Gordon, Dirac, Maxwell and so forth. This paper explores a more general dynamical group candidate that is also a semi-direct product but where the 'translation' normal subgroup N is now the Heisenberg group. The relevant automorphisms of the Heisenberg group are the symplectic group. This together with the requirement for an orthogonal metric leads to the pseudo-unitary group for the homogeneous group K. The physical meaning and motivation of this group, called the quaplectic group, is presented and the Hermitian irreducible representations of the algebra are determined. As with the Poincare group, choice of the group defines the Hilbert space of representations that are identified with quantum particle states. The field equations that are the eigenvalue equations for the representation of the Casimir operators, are obtained and investigated. The theory embodies the Born reciprocity principle and a new relativity principle.
Representations of the proper Poincaré group on spaces of functions of the phase space variables q and p have been studied, by starting with the rotation subgroup SO(3), and carrying out an inducing procedure. A complete decomposition theory, for the physically interesting representations, have been developed. Explicit computations of the forms of the generators and certain systems of imprimitivity have been carried out. The role of the Newton–Wigner position and momentum operators, in the setting of phase space representations, has been discussed. The work is a continuation of an earlier paper on phase space representations, inspired ultimately by the possibility of developing a consistent relativistic quantum mechanics for single particles on phase space.