Phase Space Representation of the Density Operator: Bopp Pseudodifferential Calculus and Moyal Product

TL;DR

This paper explores the phase-space representation of quantum density operators using Bopp pseudodifferential calculus, highlighting its connection with the Moyal product and advancing deformation quantization techniques.

Contribution

It extends Bopp quantization to density operators, providing new insights into their role in phase-space quantum mechanics and deformation quantization.

Findings

Bopp calculus offers a new perspective on density operators.

Enhanced understanding of the Moyal product in quantum phase space.

Applications to mixed states in quantum mechanics.

Abstract

Bopp shifts, introduced in 1956, played a pivotal role in the statistical interpretation of quantum mechanics. As demonstrated in our previous work, Bopp's construction provides a phase-space perspective of quantum mechanics that is closely connected to the Moyal star product and its role in deformation quantization. In this paper, we both review and expand on our exploration of Bopp quantization, emphasizing its relationship with the Moyal product and its applications in elementary deformation quantization. Notably, we apply these constructions to the density operator, which represents mixed states in quantum mechanics, offering novel insights into its role within this framework.