Lagrangian Aspects of Quantum Dynamics on a Noncommutative Space

TL;DR

This paper explores the Lagrangian formulation of noncommutative quantum mechanics, establishing a connection between noncommutative and commutative systems, especially for quadratic Hamiltonians like harmonic oscillators.

Contribution

It derives a relationship between quadratic Lagrangians in noncommutative and commutative quantum systems, including a linear transformation for certain subclasses.

Findings

Quadratic Lagrangians in noncommutative systems can be related to commutative ones.

A subclass of Lagrangians allows a linear change of variables to connect regimes.

The approach facilitates evaluation of Feynman path integrals in noncommutative quantum mechanics.

Abstract

In order to evaluate the Feynman path integral in noncommutative quantum mechanics, we consider properties of a Lagrangian related to a quadratic Hamiltonian with noncommutative spatial coordinates. A quantum-mechanical system with noncommutative coordinates is equivalent to another one with commutative coordinates. We found connection between quadratic classical Lagrangians of these two systems. We also shown that there is a subclass of quadratic Lagrangians, which includes harmonic oscillator and particle in a constant field, whose connection between ordinary and noncommutative regimes can be expressed as a linear change of position in terms of a new position and velocity.