A unifying perspective on the Moyal and Voros products and their physical meanings

TL;DR

This paper clarifies the relationship between Moyal and Voros products in non-commutative quantum mechanics, showing their mathematical equivalence but physical interpretability only in the Voros formulation.

Contribution

It demonstrates that Moyal and Voros formulations are mathematically equivalent but differ in physical interpretation, favoring the Voros formulation for consistent physical meaning.

Findings

Moyal and Voros are two representations of the same mathematical structure.

Only the Voros formulation admits a consistent physical interpretation.

Transition amplitudes differ between the two representations, highlighting interpretational differences.

Abstract

The Moyal and Voros formulations of non-commutative quantum field theory has been a point of controversy in the recent past. Here we address this issue in the context of non-commutative non-relativistic quantum mechanics. In particular we show that the two formulations simply correspond to two different representations associated with two different choices of basis on the quantum Hilbert space. From a mathematical perspective the two formulations are therefore completely equivalent, but we also argue that only the Voros formulaton admits a consistent physical interpretation. These considerations are elucidated by considering the free particle transition amplitude in the two representations.