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

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.