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

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.

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Cited by in corpus (17)

• A comparative review of four formulations of noncommutative quantum mechanics
• Dirac Equation in Noncommutative Space for Hydrogen Atom
• Position-dependent noncommutativity in quantum mechanics
• Noncommutativity due to spin
• Star products made (somewhat) easier
• Quantum mechanics with coordinate dependent noncommutativity
• Dynamics of a Dirac Fermion in the presence of spin noncommutativity
• Neutrino spin oscillations in gravitational fields in non-commutative Spaces
• Semiclassical and quantum description of motion on noncommutative plane
• Dynamical noncommutativity
• Self-quartic interaction for a scalar field in an extended DFR noncommutative spacetime
• Some aspects of quantum mechanics and field theory in a Lorentz invariant noncommutative space
• Quantum and pseudoclassical descriptions of nonrelativistic spinning particles in noncommutative space
• Quantum and pseudoclassical descriptions of nonrelativistic spinning particles in noncommutative space
• Gauge invariance and classical dynamics of noncommutative particle theory
• From Noncommutative Sphere to Nonrelativistic Spin
• The standard electroweak model in the noncommutative $DFR$ space-time