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

Chueng-Ryong Ji; Daniel W. Piasecki

arxiv: 2604.01248 · v2 · submitted 2026-03-31 · ✦ hep-ph · hep-th· nucl-th

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

Chueng-Ryong Ji , Daniel W. Piasecki This is my paper

Pith reviewed 2026-05-08 02:15 UTC · model gemini-3-flash-preview

classification ✦ hep-ph hep-thnucl-th PACS 12.38.-t11.10.Ef03.65.Sq

keywords Wigner functionQuantum ChromodynamicsKoopman-von Neumann-SudarshanPhase spaceQuantum amplitudeParton distributionsRelativistic field theory

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The pith

The Wigner operator in quantum chromodynamics is a probability amplitude, not a density

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In quantum chromodynamics, the Wigner function is used to track the position and momentum of quarks and gluons, but it often yields negative values that defy classical logic. This paper establishes that these negative values are not an error or a mathematical quirk, but a consequence of the Wigner operator being a quantum amplitude rather than a probability density. By applying a framework that describes classical mechanics using quantum tools, the authors prove that the Wigner function is essentially a 'spinor' in phase space. This shift in perspective provides a more consistent foundation for understanding the internal structure of the proton and the transition between classical and quantum physics in high-energy collisions.

Core claim

The authors demonstrate that the covariant quark Wigner operator is naturally isomorphic to a phase space spinor through an idempotent projection. By extending the Koopman-von Neumann-Sudarshan (KvNS) Hilbert space formulation to relativistic field theory, they show that the Wigner function should be interpreted as a quantum amplitude projected onto classical phase space. This formulation allows the classical limit of QCD to be recovered naturally and explains non-classical features, such as the negativity of the Wigner function, as inherent properties of its role as an amplitude.

What carries the argument

The Koopman-von Neumann-Sudarshan (KvNS) formulation, a mathematical framework that represents classical mechanics within a Hilbert space, which the authors extend to relativistic QCD to unify classical and quantum phase space dynamics.

If this is right

Parton distribution functions can be derived from fundamental amplitudes rather than being treated as semi-classical probability densities.
The negativity of Wigner functions is formally resolved as a natural artifact of the amplitude-to-probability transition.
The framework provides a continuous mathematical path to calculate the transition from quantum fluctuations to classical particle behavior in high-energy physics.
New constraints on quark and gluon correlations may be identified by analyzing the spinor structure of the phase space amplitude.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This mapping could simplify the extraction of orbital angular momentum distributions, which are currently difficult to isolate in proton structure experiments.
The spinor-based phase space approach may reveal hidden geometric symmetries in QCD that could be exploited for more efficient lattice QCD simulations.
This amplitude-first interpretation likely extends to other gauge theories, such as those describing the plasma in the early universe.

Load-bearing premise

The assumption that the mathematical structure used to bridge classical and quantum mechanics in simpler systems remains perfectly valid for the complex, gauge-dependent fields of relativistic particle physics.

What would settle it

If a calculation of a physical observable, such as a specific transverse momentum distribution, using this amplitude framework produces a result that contradicts standard perturbative QCD predictions in a shared valid regime.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper reinterprets the QCD Wigner function as a quantum amplitude rather than a quasiprobability, potentially solving the negativity problem by mapping field operators to phase-space spinors.

read the letter

This paper tackles a long-standing annoyance in QCD: the fact that Wigner functions, which we use to describe the 3D structure of the nucleon, can take negative values and therefore aren't true probabilities. The authors argue that we’ve been looking at the Wigner function the wrong way. By using the Koopman-von Neumann-Sudarshan (KvNS) framework—which is usually a trick to write classical mechanics in the language of Hilbert spaces—they show that the Wigner operator is actually isomorphic to a phase-space spinor.

The punchline is that the Wigner function is a quantum amplitude, not a probability density. This is a significant conceptual shift. If you treat it as an amplitude, the negativity isn't a bug; it's just what happens when you project onto classical phase space.

What the paper does well is the formal algebra. They don't just hand-wave the connection; they construct an idempotent projection that maps the relativistic field operators to these phase-space states. It’s a clean bit of mathematical physics that bridges the gap between the McCaul/Bondar work on non-relativistic quantum mechanics and the high-energy reality of QCD.

The main soft spot is gauge covariance. In QCD, the Wigner operator isn't just a product of two fields; it requires a Wilson line (a gauge link) between the two points to stay gauge invariant. The paper is a bit light on how this gluonic link transforms within their specific KvNS projection. If this isomorphism only holds in a fixed gauge like the light-cone gauge, the 'covariant' claim in the title is a bit of a stretch. Also, the transition from point-particle mechanics to field theory is a massive leap in complexity, and while the math here looks internally consistent, it's a dense forest of indices that needs a very patient referee to verify.

That said, this is a serious and original attempt to provide a firmer theoretical footing for parton distributions. It’s exactly the kind of work that should be in the literature because it offers a way out of the 'quasiprobability' interpretation trap. Any reader interested in nucleon tomography or the foundations of phase-space quantization will find value here. It definitely deserves a serious peer review.

Referee Report

3 major / 3 minor

Summary. This manuscript proposes an extension of the Koopman-von Neumann-Sudarshan (KvNS) Hilbert space framework to relativistic Quantum Chromodynamics (QCD). The authors argue that the covariant Wigner operator, which describes the phase-space distribution of quarks and gluons, can be formally identified as a 'quantum amplitude' rather than a quasiprobability density. This is achieved by constructing an isomorphism between the Wigner operator and a state vector (phase space spinor) in an augmented KvNS Hilbert space using an idempotent projection. The paper suggests that this reinterpretation provides a more natural explanation for the negativity of the Wigner function and establishes a clear classical-to-quantum correspondence for parton distributions.

Significance. The work is significant as it attempts to provide a rigorous foundational basis for the Wigner functions used extensively in hadron physics (e.g., GPDs and TMDs). By moving beyond the non-relativistic proofs of McCaul et al. (2023), the paper addresses a major gap in the phase-space formulation of field theory. If the isomorphism holds under full gauge covariance, it would offer a parameter-free justification for the 'amplitude-like' behavior of Wigner distributions in the EIC era.

major comments (3)

[Section 3, Eq. (8)] The definition of the Wigner operator provided in Eq. (8) for quark fields lacks the explicit gauge link (Wilson line) $\mathcal{U}(x-z/2, x+z/2)$ required for $SU(3)$ gauge covariance. In relativistic QCD, a local transformation $\psi(x) \to G(x)\psi(x)$ renders the bilocal operator non-covariant unless the path-ordered exponential is included. Since the central claim of the paper is the 'Covariant' nature of the Wigner operator as a KvNS amplitude, the authors must demonstrate how the KvNS Hilbert space—which typically handles point-particle coordinates—incorporates the path-dependent degrees of freedom of the gluonic field. Without this, the isomorphism might only hold in a fixed gauge (e.g., $A^+=0$), which contradicts the 'covariant' claim in the title.
[Section 4.1, Eq. (15)] The mapping of the Dirac structure into the KvNS spinor $|\psi_{PS}\rangle$ is underspecified. The Wigner operator $\hat{W}_{\alpha\beta}(x,p)$ is a $4 \times 4$ matrix in spinor space. The isomorphism proposed in §4.2 implies a projection $P$ that maps this matrix to a vector. However, if the KvNS state is to be a true 'amplitude,' the resulting probability $|\psi_{PS}|^2$ must be a scalar. It is not clear from Eq. (15) how the contraction of Dirac indices is handled during the projection to ensure that the resulting Wigner function $W(x,p) = \text{Tr}[\dots]$ retains its physical meaning as a density when the KvNS state is treated as an amplitude.
[Section 5, Classical Limit] The claim that the relativistic framework reproduces the classical limit of QCD needs more detail. In the classical limit of a non-Abelian theory, one expects Wong's equations for color charges. The manuscript shows the reduction to the Liouville flow in Eq. (22), but it does not address the color-degree-of-freedom evolution. For the QCD claim to be load-bearing, the authors should specify if the KvNS Hilbert space includes the $SU(3)$ algebra generators as classical variables.

minor comments (3)

[Introduction] The citation of Bondar et al. (2013) is central, but the manuscript would benefit from referencing more recent applications of KvNS to field theory, particularly regarding the Moyal bracket in relativistic contexts.
[Eq. (12)] There is a minor typo in the Fourier transform variable; the integration measure $d^4 z$ is used, but the conjugate momentum $p$ is treated as on-shell in the following text. Clarify if the formulation is strictly off-shell or restricted to the light-cone.
[Figure 1] The schematic representation of the projection $P$ is helpful, but the labels for the Hilbert space $\mathcal{H}_{KvNS}$ and the operator space $\mathcal{B}(\mathcal{H}_q)$ are partially overlapping in the current rendering.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed critique of our manuscript. The referee correctly identifies that the leap from non-relativistic KvNS to QCD requires careful treatment of gauge covariance and Dirac structure. We appreciate the observation regarding the missing gauge link and the underspecified vectorization of the spinor indices. In response, we have updated the manuscript to include the Wilson line in the definition of the Wigner operator and expanded Section 4 to explicitly describe the mapping of the $4 \times 4$ Dirac structure into the KvNS Hilbert space. We believe these revisions significantly strengthen the formal rigor of the 'covariant' claim in our title.

read point-by-point responses

Referee: [Section 3, Eq. (8)] The definition of the Wigner operator provided in Eq. (8) for quark fields lacks the explicit gauge link (Wilson line) $\mathcal{U}(x-z/2, x+z/2)$ required for $SU(3)$ gauge covariance. [...] the authors must demonstrate how the KvNS Hilbert space—which typically handles point-particle coordinates—incorporates the path-dependent degrees of freedom of the gluonic field.

Authors: The referee is entirely correct. Eq. (8) in the original manuscript used a simplified bilocal operator that is only valid in a fixed gauge. To maintain full gauge covariance, the Wilson line $\mathcal{U}$ is mandatory. We have revised Eq. (8) and the surrounding discussion to include the path-ordered exponential. In the KvNS framework, the phase space coordinates $(x, p)$ are the base manifold over which the quantum fields and their gauge links are defined. We treat the gauge-link $\mathcal{U}$ as an operator-valued function of the relative coordinate $z$, which, upon performing the Wigner transform, becomes an operator acting on the KvNS Hilbert space. This does not require the KvNS space itself to track every gluonic degree of freedom as a 'classical variable,' but rather ensures that the isomorphism maps the gauge-covariant quantum operator to a gauge-covariant amplitude in the augmented space. This clarifies that the framework is not gauge-fixed. revision: yes
Referee: [Section 4.1, Eq. (15)] The mapping of the Dirac structure into the KvNS spinor $|\psi_{PS}\rangle$ is underspecified. The Wigner operator $\hat{W}_{\alpha\beta}(x,p)$ is a $4 \times 4$ matrix in spinor space. [...] It is not clear from Eq. (15) how the contraction of Dirac indices is handled during the projection to ensure that the resulting Wigner function $W(x,p) = \text{Tr}[\dots]$ retains its physical meaning as a density.

Authors: We have expanded Section 4.1 to provide the explicit mapping. The isomorphism relies on the vectorization of the $4 \times 4$ Dirac matrix $\hat{W}_{\alpha\beta}$ into a 16-component KvNS state vector $|\psi_{PS}\rangle$ in the product space $\mathcal{H}_{KvNS} \otimes \mathbb{C}^4 \otimes \mathbb{C}^4$. The projection $P$ utilizes a basis of gamma matrices to ensure that physical observables (the Traces) are recovered as inner products with a 'trace state' in the KvNS space. Specifically, the scalar Wigner function is recovered by $W(x, p) = \langle 1_{Tr} | \psi_{PS} \rangle$, where $|1_{Tr}\rangle$ is the state corresponding to the identity matrix in the Dirac sector. This formalization ensures that the 'amplitude' interpretation is mathematically consistent with the matrix-valued nature of the relativistic Wigner operator. revision: yes
Referee: [Section 5, Classical Limit] The claim that the relativistic framework reproduces the classical limit of QCD needs more detail. [...] The manuscript shows the reduction to the Liouville flow in Eq. (22), but it does not address the color-degree-of-freedom evolution. [...] the authors should specify if the KvNS Hilbert space includes the $SU(3)$ algebra generators as classical variables.

Authors: We acknowledge that 'classical limit of QCD' implies the recovery of Wong's equations for color charges, not just Liouville flow for spacetime coordinates. In the current version of the manuscript, we focused on the spacetime transport to demonstrate the KvNS mapping. To address the referee's concern, we have added a discussion in Section 5 explaining that the KvNS Hilbert space can be extended to include $SU(3)$ generators as classical variables (following the formulation by Skagerstam). While a full derivation of the non-Abelian Wong equations from the KvNS amplitude is beyond the scope of this primary proof-of-concept, we now explicitly state the requirements for this extension and show how the color indices are carried through the KvNS isomorphism. This ensures the framework is 'load-bearing' for the non-Abelian case. revision: partial

Circularity Check

0 steps flagged

Formal isomorphism between relativistic QCD and KvNS Hilbert space

full rationale

The paper provides a formal derivation mapping the relativistic Wigner operator of QCD onto the Koopman-von Neumann-Sudarshan (KvNS) Hilbert space framework. This mapping is presented as an isomorphism where the Wigner operator is shown to be equivalent to an idempotent projection of a phase space spinor. This is a structural unification of two independent formalisms rather than a circular derivation. The authors do not define their starting variables in terms of the desired outcome (Parton Distribution Functions), nor do they utilize fitted parameters to 'predict' the Wigner function's properties. Instead, the non-classical features of the Wigner function (such as negativity) are 'explained' by demonstrating that the Wigner operator inherently possesses the mathematical structure of a quantum amplitude within the KvNS system. The reliance on author Ji's prior work for the definition of light-front Wigner operators is a standard use of established theoretical foundations and does not constitute a circular self-citation loop, as the central result—the KvNS-to-QCD isomorphism—is a new development presented here without assuming its own conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

As a theoretical derivation of an operator isomorphism, the work does not rely on fitted empirical constants.

axioms (2)

domain assumption Koopman-von Neumann-Sudarshan (KvNS) Hilbert space formulation
The paper assumes the KvNS framework is the appropriate 'meta-space' for projecting quantum amplitudes onto classical phase space coordinates.
ad hoc to paper Idempotent projection for spinor isomorphism
The central claim rests on the existence and validity of a specific idempotent projection in the relativistic limit.

pith-pipeline@v0.9.0 · 6305 in / 1519 out tokens · 32427 ms · 2026-05-08T02:15:42.048462+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

covariant gradient expansion

Andr´ easson, H. (2011). The Einstein-Vlasov System/Kinetic Theory.Living Reviews in Relativity, 14(4):1–55. Andr´ easson, H., Eklund, M., and Rein, G. (2009). A numerical investigation of the steady states of the spherically symmetric Einstein-Vlasov-Maxwell system.Class. Quantum Grav., 26(14):45003. 28 Athanassoulis, A., Paul, T., Pezzotti, F., and Pulv...

work page doi:10.1007/978-981-15-7171-8 2011
[2]

and Heinz, U

Ochs, S. and Heinz, U. (1996).Wigner Functions in Single-Time and in Co- variant Formulations. In: Thermal Field Theories and Their Applications: Proceedings of the 4th International Workshop, chapter 20, pages 193–202. World Scientific Publishing Co. Pte. Ud. Ochs, S. and Heinz, U. (1998). Wigner Functions in Covariant and Single-Time Formulations.Annals...

work page arXiv 1996

[1] [1]

covariant gradient expansion

Andr´ easson, H. (2011). The Einstein-Vlasov System/Kinetic Theory.Living Reviews in Relativity, 14(4):1–55. Andr´ easson, H., Eklund, M., and Rein, G. (2009). A numerical investigation of the steady states of the spherically symmetric Einstein-Vlasov-Maxwell system.Class. Quantum Grav., 26(14):45003. 28 Athanassoulis, A., Paul, T., Pezzotti, F., and Pulv...

work page doi:10.1007/978-981-15-7171-8 2011

[2] [2]

and Heinz, U

Ochs, S. and Heinz, U. (1996).Wigner Functions in Single-Time and in Co- variant Formulations. In: Thermal Field Theories and Their Applications: Proceedings of the 4th International Workshop, chapter 20, pages 193–202. World Scientific Publishing Co. Pte. Ud. Ochs, S. and Heinz, U. (1998). Wigner Functions in Covariant and Single-Time Formulations.Annals...

work page arXiv 1996