A coherent-state-based path integral for quantum mechanics on the Moyal plane

TL;DR

This paper develops a path integral formulation for quantum mechanics on the Moyal plane using coherent states, revealing non-perturbative effects and spin-like phenomena induced by noncommutativity.

Contribution

It introduces a new coherent-state-based path integral approach for the Moyal plane that remains non-perturbative and connects noncommutativity with spin effects.

Findings

The kernel contains gaussian damping terms and is non-perturbative.

The free particle propagator oscillates with a period related to mass and .

Noncommutativity induces spin-like effects in quantum mechanics.

Abstract

Inspired by a recent work that proposes using coherent states to evaluate the Feynman kernel in noncommutative space, we provide an independent formulation of the path-integral approach for quantum mechanics on the Moyal plane, with the transition amplitude defined between two coherent states of mean position coordinates. In our approach, we invoke solely a representation of the of the noncommutative algebra in terms of commutative variables. The kernel expression for a general Hamiltonian was found to contain gaussian-like damping terms, and it is non-perturbative in the sense that it does not reduce to the commutative theory in the limit of vanishing $\theta$ - the noncommutative parameter. As an example, we studied the free particle's propagator which turned out to be oscillating with period being the product of its mass and $\theta$. Further, it satisfies the Pauli equation for a charged particle with its spin aligned to a constant, orthogonal $B$ field in the ordinary Landau problem, thus providing an interesting evidence of how noncommutativity can induce spin-like effects at the quantum mechanical level.