Path Integrals in Noncommutative Quantum Mechanics

TL;DR

This paper extends Feynman's path integral formalism to noncommutative quantum mechanics, providing a solvable framework for quadratic Lagrangians and illustrating it with specific quantum systems.

Contribution

It formulates a path integral approach for noncommutative quantum systems with quadratic Lagrangians, establishing relations with commutative cases and providing explicit solutions.

Findings

Derived general relations between commutative and noncommutative quadratic Lagrangians

Presented explicit noncommutative path integrals for quadratic systems

Applied the formalism to particle in a field and harmonic oscillator

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.