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

TL;DR

This paper develops path integral and action formulations for relativistic quantum particles in noncommutative space-time, extending previous nonrelativistic approaches and connecting to noncommutative field theory propagators.

Contribution

It constructs $ heta$-modified actions for relativistic particles, including spinning particles, derived from noncommutative field theory propagators, and generalizes the Berezin-Marinov action.

Findings

Derived path integral representations for noncommutative Klein-Gordon and Dirac propagators.

Quantized the $ heta$-modified actions to recover noncommutative relativistic equations.

Extended Berezin-Marinov action to noncommutative spinor particles.

Abstract

It is known that actions of field theories on a noncommutative space-time can be written as some modified (we call them $\theta$-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 usual quantum mechanical features of the corresponding field theory. The $\theta$-modification for arbitrary finite-dimensional nonrelativistic system was proposed by Deriglazov (2003). In the present article, we discuss the problem of constructing $\theta$-modified actions for relativistic QM. We construct such actions for relativistic spinless and spinning particles. The key idea is to extract $\theta$-modified actions of the relativistic particles from path integral representations of the corresponding noncommtative field theory propagators. We consider 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 $\theta$-modified actions of the relativistic particles. To confirm the interpretation, we quantize canonically these actions. Thus, we obtain the Klein-Gordon and Dirac equations in the noncommutative field theories. The $\theta$-modified action of the relativistic spinning particle is just a generalization of the Berezin-Marinov pseudoclassical action for the noncommutative case.