FINITE-DIFFERENCE EQUATIONS IN RELATIVISTIC QUANTUM MECHANICS 1

V. Aldaya, Julio Guerrero

Published Oct 28, 1994• 1 citation

Abstract

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

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