Eur. Phys. J. C (2008) 54: 325-332
https://doi.org/10.1140/epjc/s10052-007-0518-x

Regular Article - Theoretical Physics

Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin–Marinov action

¹ Instituto de Física, Universidade de São Paulo, 05315-970, São Paulo, SP, Brazil
² Physics Department, Tomsk State University, Tomsk, Russia

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

Received: 17 October 2007
Revised: 3 December 2007
Published online: 24 January 2008

Abstract

It is known that the actions of field theories on a noncommutative space-time can be written as some modified (we call them θ-modified) classical actions already on the commutative space-time (introducing a star product). Then the quantization of such modified actions reproduces both space-time noncommutativity and the usual quantum mechanical features of the corresponding field theory. In the present article, we discuss the problem of constructing θ-modified actions for relativistic QM. We construct such actions for relativistic spinless and spinning particles. The key idea is to extract θ-modified actions of the relativistic particles from path-integral representations of the corresponding noncommutative field theory propagators. We consider the Klein–Gordon and Dirac equations for the causal propagators in such theories. Then we construct for the propagators path-integral representations. Effective actions in such representations we treat as θ-modified actions of the relativistic particles. To confirm the interpretation, we canonically quantize these actions. Thus, we obtain the Klein–Gordon and Dirac equations in the noncommutative field theories. The θ-modified action of the relativistic spinning particle is just a generalization of the Berezin–Marinov pseudoclassical action for the noncommutative case.