Path Integrals in Noncommutative Quantum Mechanics

Abstract

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.

9 pages,some misprints corrected

References in corpus (3)

• Feynman Path Integral on the Noncommutative Plane
• Non-Commutative GUTs, Standard Model and C,P,T
• Constant magnetic field and 2d non-commutative inverted oscillator

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• Time dependent transitions with time-space noncommutativity & its implications in Quantum Optics
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