Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin–Marinov action |
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Authors: | DM Gitman VG Kupriyanov |
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Institution: | 1.Instituto de Física,Universidade de S?o Paulo,S?o Paulo,Brazil;2.Physics Department,Tomsk State University,Tomsk,Russia |
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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. |
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