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Open AccessArticle

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

by Chueng-Ryong Ji and Daniel W. Piasecki

Symmetry 2026, 18(6), 1018; https://doi.org/10.3390/sym18061018 - 12 Jun 2026

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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. 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, 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. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry in Quantum Chromodynamics (QCD))

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20 pages, 388 KB

Open AccessArticle

Koopman–von Neumann and Weyl–Wigner Phase-Space Formulation of Inviscid Euler Flows

by Sandor M. Molnar and Joseph R. Godfrey

Entropy 2026, 28(4), 416; https://doi.org/10.3390/e28040416 - 7 Apr 2026

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Abstract

We develop a unified Koopman–von Neumann (KvN) operator and Weyl–Wigner phase-space framework for inviscid ideal (barotropic) Euler flows. Our approach reformulates the nonlinear fluid dynamics as a linear KvN evolution on an enlarged field phase space, thereby enabling us to apply tools developed for quantum mechanics (Weyl quantization, Moyal ⋆-products, and Wigner functionals) to a classical fluid. We construct the appropriate KvN generator (including the required Jacobian term for unitarity) and derive the evolution equation for the corresponding Wigner functional. This framework clarifies when the classical Liouville (Vlasov) description is exact—namely, in quadratic or linear regimes where the Moyal bracket reduces to the Poisson bracket—and when higher-order quantum-like corrections become significant in fully nonlinear regimes. As an analytic example, we obtain a closed-form Wigner solution for a one-dimensional Burgers flow (pressureless Euler) and verify, term by term, that it reproduces the expected Liouville transport (with distributional contributions at the shock). We also compare the phase-space approach with a kinetic (Vlasov–monokinetic) formulation and outline the extension of the framework to three-dimensional flows using a Clebsch variable representation. Full article

(This article belongs to the Section Multidisciplinary Applications)

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13 pages, 333 KB

Open AccessArticle

Reframing Classical Mechanics: An AKSZ Sigma Model Perspective

by Thomas Basile, Nicolas Boulanger and Arghya Chattopadhyay

Universe 2025, 11(6), 196; https://doi.org/10.3390/universe11060196 - 19 Jun 2025

Viewed by 988

Abstract

The path-integral re-formulation due to E. Gozzi, M. Regini, M. Reuter, and W. D. Thacker of Koopman and von Neumann’s original operator formulation of a classical Hamiltonian system on a symplectic manifold M is identified as a gauge slice of a one-dimensional Alexandrov–Kontsevich–Schwarz–Zaboronsky [...] Read more.

T^{*} (T [1] M \times R [1])

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(This article belongs to the Section Field Theory)

32 pages, 448 KB

Open AccessEditor’s ChoiceArticle

Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density

by David Chester, Xerxes D. Arsiwalla, Louis H. Kauffman, Michel Planat and Klee Irwin

Symmetry 2024, 16(3), 316; https://doi.org/10.3390/sym16030316 - 6 Mar 2024

Cited by 8 | Viewed by 2961

Abstract

We generalize Koopman–von Neumann classical mechanics to poly symplectic fields and recover De Donder–Weyl’s theory. Compared with Dirac’s Hamiltonian density, it inspires a new Hamiltonian formulation with a canonical momentum field that is Lorentz-covariant with symplectic geometry. We provide commutation relations for the classical and quantum fields that generalize the Koopman–von Neumann and Heisenberg algebras. The classical algebra requires four fields that generalize spacetime, energy–momentum, frequency–wavenumber, and the Fourier conjugate of energy–momentum. We clarify how first and second quantization can be found by simply mapping between operators in classical and quantum commutator algebras. Full article

(This article belongs to the Section Physics)

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13 pages, 282 KB

Open AccessArticle

Contextuality in Classical Physics and Its Impact on the Foundations of Quantum Mechanics

by Fritiof Wallentin

Entropy 2021, 23(8), 968; https://doi.org/10.3390/e23080968 - 27 Jul 2021

Cited by 3 | Viewed by 2861

Abstract

It is shown that the hallmark quantum phenomenon of contextuality is present in classical statistical mechanics (CSM). It is first shown that the occurrence of contextuality is equivalent to there being observables that can differentiate between pure and mixed states. CSM is formulated in the formalism of quantum mechanics (FQM), a formulation commonly known as the Koopman–von Neumann formulation (KvN). In KvN, one can then show that such a differentiation between mixed and pure states is possible. As contextuality is a probabilistic phenomenon and as it is exhibited in both classical physics and ordinary quantum mechanics (OQM), it is concluded that the foundational issues regarding quantum mechanics are really issues regarding the foundations of probability. Full article

(This article belongs to the Special Issue Quantum Probability and Randomness III)

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