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Local-equilibrium density operators built only from genuine conserved currents are free of pseudo-gauge and improvement ambiguities that affect spin-polarization estimates.

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2026-07-10 05:10 UTC pith:JE63XUFX

load-bearing objection Clean conceptual fix for residual pseudo-gauge/Zilch freedom in local-equilibrium density operators; solid formal work that closes a known technical gap without overclaiming.

arxiv 2607.08569 v1 pith:JE63XUFX submitted 2026-07-09 nucl-th hep-phhep-th

A symmetry-based resolution of pseudo-gauge ambiguities in local equilibrium

Carlos Hoyos This is my paper
classification nucl-th hep-phhep-th
keywords pseudo-gaugelocal thermal equilibriumspin polarizationspurious symmetriesenergy-momentum tensorspin hydrodynamicsimprovement termsheavy-ion collisions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

In relativistic heavy-ion collisions the matter produced is described by local thermal equilibrium, yet different ways of splitting total angular momentum into orbital and spin pieces (pseudo-gauges) change the density operator that defines that equilibrium and therefore change predicted spin polarizations. This paper shows that every such freedom, and every other improvement of a conserved current, can be rewritten as the addition of a chemical potential for a spurious symmetry whose total charge is identically zero. Once genuine currents are identified—quasi-primary operators in a conformal theory, or the symmetric energy-momentum tensor fixed by the broken-conformal Ward identity after relevant deformations—the local-equilibrium density operator is assembled solely from those currents. The resulting operator is invariant under the full set of pseudo-gauge and improvement transformations, including residual Zilch-type terms that earlier constructions left free. A sympathetic reader cares because the construction supplies a unique, physically grounded density operator for spin hydrodynamics and for the interpretation of polarization data.

Core claim

A local-equilibrium density operator constructed exclusively from genuine conserved currents (those associated with physical symmetries) is invariant under every transformation that adds identically conserved improvement terms to the energy-momentum tensor, spin current or any other conserved current; the residual ambiguities of earlier pseudo-gauge-invariant constructions are thereby eliminated.

What carries the argument

Spurious symmetries: identically conserved currents that are total divergences and therefore carry vanishing total charge. Their chemical potentials can be used to rewrite any improvement or pseudo-gauge transformation as a redefinition of the density operator that leaves the physical state unchanged once only genuine currents are retained.

Load-bearing premise

The rule that separates genuine from spurious currents in a conformal theory continues to work after the theory is deformed by mass terms or other relevant operators that break conformal invariance.

What would settle it

Construct an explicit free-field or interacting example in which two different choices of genuine energy-momentum tensor (both satisfying the broken-conformal Ward identity) produce unequal local-equilibrium density operators or unequal spin-polarization expectations; or show that a topological current wrongly classified as genuine alters a physical charge.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper reformulates pseudo-gauge and improvement ambiguities of the energy-momentum and spin tensors as the freedom to add identically conserved “spurious” currents (total divergences with vanishing total charge) to the local-equilibrium density operator. Genuine currents are identified as quasi-primary operators in a CFT, or as their continuations via the broken-conformal Ward identity T^µ_µ = Σ β_A O_A after relevant deformations. Once the density operator is written solely in terms of these genuine currents, the full class of improvement, pseudo-gauge and residual Zilch transformations become total derivatives (or total derivatives plus derivatives of the intensive variables) and leave the operator invariant. The construction recovers the Belinfante form and the earlier pseudo-gauge-invariant operator of Becattini & Hoyos while eliminating the residual Zilch freedom.

Significance. If the identification of genuine currents survives the relevant deformations of interest for QCD, the result supplies a symmetry-based, parameter-free definition of the local-equilibrium density operator that is free of the residual ambiguities that have affected spin-polarization estimates in heavy-ion collisions. The algebraic mapping of improvements onto total-derivative additions is clean and self-contained, and the free massive scalar and Dirac examples confirm that the same symmetric tensors continue to satisfy the Ward identity after mass deformation. The framework therefore offers a conceptually unified and practically usable resolution of a long-standing technical obstacle in spin hydrodynamics.

minor comments (3)
  1. The free-field examples in Sec. IV are helpful but brief; a short remark on how the same prescription applies to the interacting QCD energy-momentum tensor (or a pointer to the literature) would strengthen the claim that the construction covers the theories of phenomenological interest.
  2. Notation for the improvement potentials (Φ, M, Z) is introduced piecewise across Secs. II and V; a single consolidated definition early in Sec. II would improve readability.
  3. A few typographical inconsistencies appear (e.g., “naïve”, “VORTICITY” in the section heading, and occasional missing spaces around operators). These are purely cosmetic.

Circularity Check

1 steps flagged

No significant circularity: independent symmetry-based prescription recovers and extends prior result without reducing to it by construction.

specific steps
  1. self citation load bearing [Section V, paragraph discussing recovery of [15]; also Discussion]
    "At local equilibrium, the pseudo-gauge invariant operator constructed in [15] can be found in this approach following similar steps... This resolves the residual ambiguities present in the pseudo-gauge-invariant construction proposed in [15]."

    The paper cites its own prior construction [15] (Becattini & Hoyos) both to recover that operator as the Belinfante case and to claim resolution of its residual Zilch ambiguity. The citation is not load-bearing: the invariance proof itself uses only the independent genuine/spurious distinction and the total-derivative property of improvements; [15] is recovered rather than presupposed. Mild self-reference only.

full rationale

The paper's core derivation is self-contained. Genuine currents are identified independently via the quasi-primary conditions of a CFT (Eqs. 13) or their continuation through the broken-conformal Ward identity T^µ_µ = Σ β_A O_A (Eq. 15) after relevant deformations; free massive scalar and Dirac examples confirm the same symmetric tensors continue to satisfy the identity. Spurious currents are pure total divergences with vanishing charges (Eqs. 5–7). The local-equilibrium density operator is then written solely from the genuine currents (Eq. 10); any improvement/pseudo-gauge/Zilch transformation (Eqs. 12, 22, 23, 26, 27) adds only total-derivative terms (or total derivatives plus derivatives of the intensive variables) and therefore leaves the operator invariant by construction. The citation to the author's prior work [15] is used only to note that the earlier pseudo-gauge-invariant operator is recovered as the special case that employs the genuine (Belinfante) energy-momentum tensor, and that the residual Zilch freedom is now eliminated by the same total-derivative argument. That citation is not load-bearing for the invariance proof, which stands on the genuine/spurious distinction alone. No fitted parameters, self-definitional loops, uniqueness theorems imported from prior author work, or renaming of known empirical patterns appear. Score 1 reflects the single non-load-bearing self-citation; the central claim has independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

The central claim rests on standard QFT notions of conserved currents, improvements, and conformal quasi-primaries, plus the domain assumption that the same distinction survives conformal-breaking deformations. No free parameters or new dynamical entities are introduced.

axioms (4)
  • domain assumption Improvement terms of the form ∂_λ Φ^{λ,µν} (with Φ antisymmetric in the last two indices) leave the total charges unchanged and can be regarded as identically conserved spurious currents.
    Standard in relativistic field theory; invoked throughout Sections II–III.
  • domain assumption In a CFT, genuine conserved currents are quasi-primary operators while identically conserved currents are descendants.
    Standard CFT lore (Di Francesco et al.); used in Section IV to identify genuine currents.
  • domain assumption After relevant deformations that break conformal invariance, the genuine energy-momentum tensor continues to be the symmetric operator satisfying the Ward identity T^µ_µ = Σ β_A O_A.
    Standard for theories obtained by conformal deformation; applied to free massive scalar and Dirac fields in Section IV.
  • ad hoc to paper Local-equilibrium density operators should depend only on conserved charges of genuine symmetries.
    Philosophical/physical preference stated in Section III; not forced by the algebra alone.
invented entities (1)
  • spurious symmetries / spurious currents no independent evidence
    purpose: To rephrase improvement and pseudo-gauge transformations as the addition of identically conserved currents whose total charges vanish.
    Terminological and conceptual device; the underlying operators already exist in the literature as improvement terms. No new dynamical degrees of freedom are postulated.

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0 comments
read the original abstract

The total angular momentum current can be decomposed into orbital and spin contributions in different ways, known as pseudo-gauges. This freedom leads to ambiguities in the definition of local-equilibrium density operators, which in turn affect estimates of spin polarization in heavy-ion collisions. In this work, the pseudo-gauge ambiguity, together with other ambiguities associated with improvements of conserved currents, is reformulated in terms of spurious symmetries corresponding to conserved currents with vanishing total charge. A prescription for the unambiguous definition of a local-equilibrium density operator is introduced using the currents associated with genuine symmetries. The resulting density operator is invariant under transformations that add improvement terms to local currents, including the energy-momentum tensor.

discussion (0)

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Reference graph

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