Remarks on the Formulation of Quantum Mechanics on Noncommutative Phase Spaces

TL;DR

This paper develops a formalism for noncommutative quantum mechanics in phase space, linking classical and quantum descriptions, and demonstrates that the distribution function exhibits twisted Galilean symmetry.

Contribution

It introduces a novel approach to formulating noncommutative quantum mechanics in phase space, establishing connections with Hilbert space formalism and classical mechanics.

Findings

Reproduction of quantum conditions via noncommutative distribution functions

Derivation of noncommutative quantum mechanics formalism in phase and Hilbert spaces

Identification of twisted Galilean symmetry in the distribution function

Abstract

We consider the probabilistic description of nonrelativistic, spinless one-particle classical mechanics, and immerse the particle in a deformed noncommutative phase space in which position coordinates do not commute among themselves and also with canonically conjugate momenta. With a postulated normalized distribution function in the quantum domain, the square of the Dirac delta density distribution in the classical case is properly realised in noncommutative phase space and it serves as the quantum condition. With only these inputs, we pull out the entire formalisms of noncommutative quantum mechanics in phase space and in Hilbert space, and elegantly establish the link between classical and quantum formalisms and between Hilbert space and phase space formalisms of noncommutative quantum mechanics. Also, we show that the distribution function in this case possesses 'twisted' Galilean symmetry.