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Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin-Marinov action. (English) Zbl 1189.81127

Summary: It is known that the 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 the usual quantum mechanical features of the corresponding field theory. 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 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 \(\theta \)-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 \(\theta \)-modified action of the relativistic spinning particle is just a generalization of the Berezin-Marinov pseudoclassical action for the noncommutative case.

MSC:

81S40 Path integrals in quantum mechanics
81R60 Noncommutative geometry in quantum theory

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