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arxiv: 2604.15926 · v1 · submitted 2026-04-17 · ✦ hep-th · nucl-th

Recognition: unknown

Exact expectation values in a boost-invariant fluid of Dirac fermions with finite spin density

Andrea Palermo , Daniele Roselli
Authors on Pith no claims yet

Pith reviewed 2026-05-10 07:53 UTC · model grok-4.3

classification ✦ hep-th nucl-th
keywords boost-invariant expansionDirac fermionsspin polarizationspin hydrodynamicsMilne coordinatespartition functionnon-equilibrium density operatorquark-gluon plasma
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The pith

In a boost-invariant fluid of non-interacting Dirac fermions, spin polarization arises only from a finite spin potential, while shear-induced polarization and the spin Hall effect are absent.

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

The authors examine an expanding system of free Dirac fermions kept exactly boost-invariant while carrying a nonzero canonical spin potential. They solve the Dirac equation in Milne coordinates, diagonalize the full density operator, and extract the partition function along with expectation values for energy density, pressures, spin density, spin torque, and polarization. The calculation yields an analytic partition function at finite spin potential and confirms that thermodynamic relations between it and the thermodynamic functions remain valid. The central result is that polarization requires the spin potential as its source; neither the velocity gradients from shear nor spin Hall mechanisms generate it in this free theory.

Core claim

We obtain an analytic expression for the partition function at finite spin potential and show that thermodynamic relations hold. In a boost-invariant system both shear-induced polarization and the spin Hall effect are absent, so a non-vanishing polarization can arise only from a finite spin potential in a free theory. We derive analytic expressions for the spin polarization in special cases and compute its exact numerical value otherwise, together with the energy density, longitudinal and transverse pressures, spin density, and spin torque.

What carries the argument

Exact diagonalization of the non-equilibrium density operator in Milne coordinates after solving the Dirac equation for boost-invariant fermions with finite spin potential.

If this is right

  • The partition function is analytic in the spin potential and satisfies the usual thermodynamic relations even out of equilibrium.
  • Spin polarization is strictly a function of the spin potential with no shear or Hall contribution.
  • Spin torque appears explicitly as the source term for spin non-conservation.
  • The results supply an exact benchmark for free-theory limits of relativistic spin hydrodynamics.

Where Pith is reading between the lines

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

  • Adding interactions would likely allow shear-induced polarization to appear, providing a testable distinction from the free case.
  • Observed polarization in expanding systems such as the quark-gluon plasma would then require additional mechanisms beyond boost-invariant free streaming.
  • The same exact-diagonalization approach could be applied to other coordinate systems or initial conditions to map where the absence of shear effects breaks down.

Load-bearing premise

The fermions are non-interacting and the expansion remains exactly boost-invariant at all times.

What would settle it

A nonzero value for the spin polarization expectation value when the spin potential is set to zero, or a nonzero shear-induced contribution to polarization in the same free boost-invariant setup.

read the original abstract

We study a boost-invariant, out-of-equilibrium fluid of non-interacting Dirac fermions with a finite canonical spin potential. After solving the Dirac equation in Milne coordinates, we exactly diagonalize the non-equilibrium density operator and compute the partition function and expectation values of relevant observables, including spin polarization, energy density, longitudinal and transverse pressures, spin density, and \emph{spin torque}, i.e. the source of spin non-conservation. We find an analytic expression for the partition function at finite spin potential, and show numerically that thermodynamic relations connecting it to thermodynamic functions hold in the system under consideration. We show that, in a boost-invariant system, both shear-induced polarization and the spin Hall effect are absent, and that a non-vanishing polarization can only arise from a finite spin potential in a free theory. We obtain an analytic expression for the spin polarization as a function of the spin potential in some particular cases, and otherwise compute numerically its exact expectation value at finite spin potential. Our results are discussed in the context of relativistic spin hydrodynamics and quark--gluon plasma phenomenology.

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.

Referee Report

0 major / 2 minor

Summary. The paper claims to derive exact expectation values for observables in a boost-invariant, out-of-equilibrium fluid of non-interacting Dirac fermions with finite canonical spin potential. After solving the Dirac equation in Milne coordinates and exactly diagonalizing the density operator, an analytic partition function is obtained, from which expectation values of spin polarization, energy density, longitudinal and transverse pressures, spin density, and spin torque are computed. The results show that shear-induced polarization and the spin Hall effect are absent, with polarization arising only from the spin potential, and thermodynamic relations are verified numerically.

Significance. If the results hold, this work is significant for providing a parameter-free, exact solution in a free theory that serves as a benchmark for relativistic spin hydrodynamics. The analytic partition function and the demonstration of the absence of shear-induced effects in boost-invariant systems offer clear insights into the origins of spin polarization. The numerical confirmation of thermodynamic relations adds to the robustness, making it useful for phenomenology in quark-gluon plasma where spin effects are studied.

minor comments (2)
  1. The abstract highlights an analytic expression for spin polarization 'in some particular cases'; explicitly identifying these cases (e.g., specific limits of temperature or chemical potential) would improve clarity without altering the central results.
  2. The term 'spin torque' is introduced as the source of spin non-conservation; including its explicit definition or derivation from the divergence of the spin current in the main text would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, including the recognition of its significance as a parameter-free benchmark for relativistic spin hydrodynamics. The referee's summary accurately captures our main results on the absence of shear-induced polarization and the spin Hall effect in boost-invariant systems, as well as the analytic partition function and numerical verification of thermodynamic relations. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper begins from the free Dirac equation solved exactly in Milne coordinates for a boost-invariant system, incorporates a constant spin potential into the density operator, and performs an exact diagonalization in the mode basis to obtain the partition function and all expectation values (polarization, energy density, pressures, spin density, torque). The reported absence of shear-induced polarization and spin Hall effect, plus the polarization arising only from finite spin potential, are direct consequences of this diagonalization and the resulting traces; no parameters are fitted to data and then relabeled as predictions, no uniqueness theorems or ansatze are imported via self-citation, and thermodynamic relations are verified numerically after the analytic partition function is derived. The setup is parameter-free once the spin potential is given and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central results rest on the free Dirac equation in curved Milne coordinates, the assumption that the density operator is diagonalizable in the chosen basis, and the definition of the canonical spin potential as an external parameter. No new particles or forces are introduced.

free parameters (1)
  • spin potential
    Introduced as a finite external parameter controlling the spin density; its value is not derived from first principles but treated as given.
axioms (2)
  • domain assumption The Dirac equation can be solved exactly in Milne coordinates for boost-invariant flow.
    Invoked to obtain the mode functions used for diagonalization.
  • domain assumption The non-equilibrium density operator for free fermions with finite spin potential is exactly diagonalizable.
    Required to compute the partition function and all expectation values analytically or numerically.

pith-pipeline@v0.9.0 · 5484 in / 1537 out tokens · 41759 ms · 2026-05-10T07:53:37.642799+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Boost-invariant and cylindrically symmetric perfect spin hydrodynamics

    hep-ph 2026-05 unverdicted novelty 5.0

    In boost-invariant cylindrical spin hydrodynamics, azimuthal-longitudinal coupling in the spin tensor produces nonzero total polarization only via the longitudinal magnetic component coupled to the azimuthal electric ...

Reference graph

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