Eur. Phys. J. C (2009) 61: 111-139
https://doi.org/10.1140/epjc/s10052-009-0954-x

Regular Article - Theoretical Physics

Field on Poincaré Group and Quantum Description of Orientable Objects

¹ Instituto de Física, Universidade de São Paulo, Caixa Postal 66318-CEP, 05315-970, São Paulo, S.P., Brazil
² Moscow Institute of Radio Engineering, Electronics and Automation, Prospect Vernadskogo, 78, 117454, Moscow, Russia

^* e-mail: gitman@dfn.if.usp.br

Received: 24 December 2008
Revised: 21 January 2009
Published online: 24 March 2009

Abstract

We propose an approach to the quantum-mechanical description of relativistic orientable objects. It generalizes Wigner’s ideas concerning the treatment of nonrelativistic orientable objects (in particular, a nonrelativistic rotator) with the help of two reference frames (space-fixed and body-fixed). A technical realization of this generalization (for instance, in 3+1 dimensions) amounts to introducing wave functions that depend on elements of the Poincaré group G. A complete set of transformations that test the symmetries of an orientable object and of the embedding space belongs to the group Π=G×G. All such transformations can be studied by considering a generalized regular representation of G in the space of scalar functions on the group, f(x,z), that depend on the Minkowski space points x∈G/Spin(3,1) as well as on the orientation variables given by the elements z of a matrix Z∈Spin(3,1). In particular, the field f(x,z) is a generating function of the usual spin-tensor multi-component fields. In the theory under consideration, there are four different types of spinors, and an orientable object is characterized by ten quantum numbers. We study the corresponding relativistic wave equations and their symmetry properties.