Origin of the Covariant Wigner Operator as a Quantum Amplitude in QCD

TL;DR

This paper develops a Koopman-based phase space framework for the quark Wigner operator in QCD, interpreting it as a quantum amplitude that clarifies its nonclassical features and classical limit.

Contribution

It extends the Koopman-von Neumann approach to relativistic QCD, providing a unified phase space description of the Wigner operator as a spinor amplitude.

Findings

Wigner operator is isomorphic to a phase space spinor.

The framework clarifies the origin of negativity in the Wigner function.

Reproduces the classical limit of QCD within the Koopman approach.

Abstract

The Wigner function plays a central role in QCD as a phase space object encoding correlations among quarks, antiquarks, and gluons, yet its interpretation remains subtle due to its quasiprobabilistic nature and possible negativity. Recent work based on the Koopman-von Neumann-Sudarshan (KvNS) Hilbert space formulation of classical mechanics suggests the Wigner function arises as a quantum probability amplitude projected onto classical phase space, rather than a quasiprobability density (Bondar et al., 2013; McCaul et al., 2023). In the classical limit, this amplitude reduces to the classical Koopman wavefunction. In this work, we extend this perspective to relativistic QCD by constructing a Koopman description of the quark Wigner operator. We show that the Wigner operator is naturally isomorphic to a phase space spinor via an idempotent projection, providing a unified framework in which both classical and quantum dynamics are expressed. Within this formulation, the Wigner function retains its interpretation as an amplitude even in the relativistic regime. This viewpoint clarifies the origin of negativity and other nonclassical features, and provides a more transparent foundation for parton distribution functions in QCD. Remarkably, the relativistic Koopman framework reproduces the classical limit of QCD.