The Classical and Commutative Limits of noncommutative Quantum Mechanics: A Superstar \bigstar Wigner-Moyal Equation

TL;DR

This paper explores the relationship between quantum-classical and noncommutative-commutative limits in noncommutative quantum mechanics, deriving a new superstar Wigner-Moyal equation and analyzing the conditions under which classical mechanics emerges.

Contribution

The paper introduces the superstar Wigner-Moyal equation for NCQM and analyzes the conditions for classical limits, highlighting the role of the noncommutativity parameter and its relation to Planck's constant.

Findings

Classical limit exists only if the noncommutativity parameter approaches zero at least as fast as Planck's constant.

When noncommutativity is present, the classical limit does not reproduce Newtonian mechanics.

The classical and noncommutative-commutative limits exhibit qualitative differences, especially regarding the existence of classical mechanics.

Abstract

We are interested in the similarities and differences between the quantum-classical (Q-C) and the noncommutative-commutative (NC-Com) correspondences. As one useful platform to address this issue we derive the superstar Wigner-Moyal equation for noncommutative quantum mechanics (NCQM). A superstar $\bigstar$-product combines the usual phase space $\ast$ star and the noncommutative $\star$ star-product. Having dealt with subtleties of ordering present in this problem we show that the classical correspondent to the NC Hamiltonian has the same form as the original Hamiltonian, but with a non-commutativity parameter $\theta$-dependent, momentum-dependent shift in the coordinates. Using it to examine the classical and the commutative limits, we find that there exist qualitative differences between these two limits. Specifically, if $\theta \neq 0$ there is no classical limit. Classical limit exists only if $\theta \to 0$ at least as fast as $\hbar \to 0$, but this limit does not yield Newtonian mechanics, unless the limit of $\theta/\hbar$ vanishes as $\theta \to 0$. Another angle to address this issue is the existence of conserved currents and the Noether's theorem in the continuity equation, and the Ehrenfest theorem in the NCQM context.