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arXiv:hep-th/0309204 · Theoretical and Mathematical Physics

Path Integrals in Noncommutative Quantum Mechanics

B. Dragovich, Z. Rakić

Published Sep 1, 2004• 13 citations
arXivHTMLPDF
DOI
10.1023/b:tamp.0000039834.84359.f8

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The gauge covariance of the wave function phase factor in noncommutative quantum mechanics (NCQM) is discussed. We show that the naive path integral formulation and an approach where one shifts the coordinates of NCQM in the presence of a background vector potential leads to the gauge non-covariance of the phase factor. Due to this fact, the Aharonov–Bohm phase in NCQM which is evaluated through the path-integral or by shifting the coordinates is neither gauge invariant nor gauge covariant. We show that the gauge covariant Aharonov–Bohm effect should be described by using the noncommutative Wilson lines, what is consistent with the noncommutative Schrödinger equation. This approach can ultimately be used for deriving an analogue of the Dirac quantization condition for the magnetic monopole.

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Extension of Feynman's path integral to quantum mechanics of noncommuting spatial coordinates is considered. The corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians) is formulated. Our approach is based on the fact that a quantum-mechanical system with a noncommutative configuration space may be regarded as another effective system with commuting spatial coordinates. Since path integral for quadratic Lagrangians is exactly solvable and a general formula for probability amplitude exists, we restricted our research to this class of Lagrangians. We found general relation between quadratic Lagrangians in their commutative and noncommutative regimes. The corresponding noncommutative path integral is presented. This method is illustrated by two quantum-mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.
The gauge covariance of the wave function phase factor in noncommutative quantum mechanics (NCQM) is discussed. We show that the naive path integral formulation and an approach where one shifts the coordinates of NCQM in the presence of a background vector potential leads to the gauge non-covariance of the phase factor. Due to this fact, the Aharonov–Bohm phase in NCQM which is evaluated through the path-integral or by shifting the coordinates is neither gauge invariant nor gauge covariant. We show that the gauge covariant Aharonov–Bohm effect should be described by using the noncommutative Wilson lines, what is consistent with the noncommutative Schrödinger equation. This approach can ultimately be used for deriving an analogue of the Dirac quantization condition for the magnetic monopole.