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

• Published in: The European physical journal. C, Particles and fields Particles and fields, 2008-03, Vol.54 (2), p.325-332
• Main Authors: Gitman, D.M., Kupriyanov, V.G.
• Format: Article
• Language: English
• Subjects: Astronomy; Astrophysics and Cosmology; Elementary Particles; Field theory; Hadrons; Heavy Ions; Integrals; Mathematical analysis; Measurement Science and Instrumentation; Nuclear Energy; Nuclear Physics; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum mechanics; Regular Article - Theoretical Physics; Relativism; Relativistic effects; Relativistic particles; Relativity; Representations; Spacetime; String Theory
• ISSN: 1434-6044; 1434-6052
• DOI: 10.1140/epjc/s10052-007-0518-x

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.