Deformation Quantization: Quantum Mechanics Lives and Works in Phase-Space

TL;DR

This paper introduces deformation quantization, a phase-space formulation of quantum mechanics using Wigner functions, highlighting its independence from traditional Hilbert space and its applications across various physics and signal processing fields.

Contribution

It provides an overview of deformation quantization as a complete, alternative formulation of quantum mechanics in phase-space, emphasizing its conceptual and practical advantages.

Findings

Furnishes a third, independent formulation of quantum mechanics

Demonstrates applications in quantum optics, nuclear physics, and signal processing

Highlights the logical completeness and self-standing nature of the phase-space approach

Abstract

Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It is also of importance in signal processing. Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations. In this logically complete and self-standing formulation, one need not choose sides--coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle. This is an introductory overview of the formulation with simple illustrations.