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

​The Wigner function in QCD has traditionally been interpreted as a quasiprobability density, but recent work in Koopman–von Neumann (KvN) mechanics suggests it is more naturally understood as a probability amplitude. KvN is a classical statistical theory formulated within the Dirac–von Neumann axioms, complete with wavefunctions, a complex Hilbert space, the Born Rule, Hermitian operators, the eigenvalue problem, etc. It establishes continuity between classical and quantum Hilbert spaces, which allows the nonrelativistic Wigner function to be derived naturally as a wavefunction on classical phase space rather than a quasiprobability distribution, resolving its seemingly pathological features. Here we extend this reinterpretation to relativistic fermionic dynamics. Extending the KvN framework, the quark Wigner operator emerges naturally as a phase space spinor, i.e., spinorial amplitude rather than density. The meaning of a spinor in the classical limit is discussed. This formulation provides a transparent conceptual explanation for the negativity and other nonclassical features of the Wigner function: these arise naturally from the amplitude interpretation rather than appearing as artifacts of a quasiprobability. The framework reproduces classical QCD dynamics in the appropriate limit, establishing clean continuity between relativistic quantum and classical descriptions. Because parton distribution functions, generalized parton distributions, and transverse momentum distributions all derive from the Wigner operator, this reinterpretation may carry direct phenomenological consequences. In upcoming work, we will explore the experimental implications of treating the quark Wigner operator as an isomorphism to a spinorial amplitude.