Noncommutative quantum mechanics: uniqueness of the functional description

TL;DR

This paper investigates the phase space path integral formulation of noncommutative quantum mechanics using the generalized Weyl transform, demonstrating the uniqueness of the Feynman kernel in the zero time-slice limit despite initial non-uniqueness.

Contribution

It proves that the $ extalpha$-dependent contributions in the noncommutative quantum mechanics path integral vanish as the time slice approaches zero, ensuring a consistent kernel.

Findings

The $ extalpha$-dependent terms disappear in the zero time-slice limit.

The antisymmetry of the noncommutativity matrix is crucial for cancelation.

The representation of the Feynman kernel is unique in the continuum limit.

Abstract

The generalized Weyl transform of index $\alpha$ is used to implement the time-slice definition of the phase space path integral yielding the Feynman kernel in the case of noncommutative quantum mechanics. As expected, this representation for the Feynman kernel is not unique but labeled by the real parameter $\alpha$. We succeed in proving that the $\alpha$-dependent contributions disappear at the limit where the time slice goes to zero. This proof of consistency turns out to be intricate because the Hamiltonian involves products of noncommuting operators originating from the non-commutativity. The antisymmetry of the matrix parameterizing the non-commutativity plays a key role in the cancelation mechanism of the $\alpha$-dependent terms.