Thermodynamics of ideal spin fluids and pseudo-gauge ambiguity

Abstract: Conserved currents of relativistic spin fluids derived from microscopic models are known to violate local thermodynamic relations. We present a systematic analysis of pseudo-gauge improvements in ideal spin hydrodynamics and identify a family of pseudo-gauges where standard thermodynamic relations are satisfied. We quantify pseudo-gauge ambiguities in the spin equation of state and derive universal thermodynamic relations that apply to conserved currents in any pseudo-gauge. As an application, we extract the thermodynamic variables and equations of state for free Dirac fermions and scalar fields.

• The paper introduces a comprehensive formalism to resolve pseudo-gauge ambiguities in the thermodynamics of spin fluids.
• It derives universal constraints and invariants through pseudo-gauge transformations applied to free Dirac fermion and scalar field models.
• The study clarifies conditions for equilibrium consistency and highlights differences between conformal and non-conformal fluids impacting experimental analyses.

Thermodynamics of Ideal Spin Fluids and Pseudo-Gauge Ambiguity: A Technical Overview

Introduction

The paper "Thermodynamics of ideal spin fluids and pseudo-gauge ambiguity" (2601.14421) systematically investigates the role of pseudo-gauge transformations in the constitutive relations and thermodynamic structure of relativistic spin hydrodynamics. Building upon discrepancies observed between microscopic model calculations and hydrodynamic predictions for conserved currents, the authors develop a comprehensive framework for pseudo-gauge improvements, derive universal thermodynamic constraints, and elucidate the equation-of-state (EoS) ambiguities induced by these transformations. They apply their formalism to free Dirac fermion and scalar field systems, resolving open contradictions and identifying the conditions for thermodynamic consistency.

Spin Hydrodynamics and Conserved Currents

Spin hydrodynamics extends conventional hydrodynamics by incorporating non-symmetric energy-momentum tensors $T^{\mu\nu}$, spin currents $\Sigma^{\mu\nu\rho}$, and potentially an additional conserved U(1) current $J^\mu$. The conservation laws intertwine these objects and the fluid variables: velocity $u^\mu$, temperature $T$, chemical potentials $\mu$, and spin chemical potential $\mu_{\mu\nu}$. Unlike conventional fluids, spin hydrodynamics admits nontrivial spacetime profiles even in equilibrium, leading to ambiguity in the identification of local thermodynamic densities.

Pseudo-Gauge Transformations and Ambiguity

A central insight of the paper is the formal non-uniqueness of the definition of conserved currents due to pseudo-gauge transformations. These constitute local improvement terms (parameterized by antisymmetric tensors) added to the energy-momentum, spin, and charge currents without affecting the global conservation laws. However, in equilibrium, these terms can shift the local densities (e.g., particle density, energy density) by derivatives of arbitrary functions of the hydrodynamic variables, potentially violating familiar thermodynamic relations that underpin the identification of entropy, energy, and charge.

The authors provide a detailed algebraic characterization of these transformations and their effect, particularly in the ideal spin fluid context. They introduce the Belinfante tensor to formalize a pseudo-gauge-invariant combination, yet show that ambiguity persists unless additional symmetry (such as conformal invariance) intervenes.

Constitutive Relations and Thermodynamic Constraints

The entropy and free energy currents are generalised to incorporate spin, with explicit formulae for their divergence ensuring compatibility with the local second law of thermodynamics. Constitutive relations for energy-momentum, spin, and charge are then developed, including their decomposition into rest-frame components and susceptibilities to acceleration and vorticity.

A key result is the identification of thermodynamic frames (specific pseudo-gauges) wherein all currents, in equilibrium, satisfy standard thermodynamic relations. The authors derive linear partial differential equations for the parameters governing pseudo-gauge improvements, yielding explicit criteria for compatibility. Importantly, they prove that only a subset of microscopic currents derived in arbitrary pseudo-gauges admit such a thermodynamic interpretation.

Universal Thermodynamic Relations and Invariants

A major technical achievement is the derivation, order-by-order in spin chemical potential, of pseudo-gauge-independent thermodynamic relations. These impose stringent constraints on the functional form of the conserved currents: for instance, at quadratic order, the deviation of the particle current from the derivative of the pressure must satisfy a universal relation involving derivatives with respect to acceleration and vorticity. At higher orders, analogous relations are explicitly presented.

Additionally, the authors construct pseudo-gauge invariants—combinations of susceptibilities and derivatives of the EoS—which serve as robust diagnostics for thermodynamic consistency, independent of the pseudo-gauge choice.

Equation of State Ambiguity and Conformal Fluids

The framework uncovers intrinsic ambiguities in the spin-dependent EoS: given any solution, homogeneous solutions of the associated PDEs can be superposed, shifting the coefficients of acceleration and vorticity terms. This ambiguity is parameterized by arbitrary functions (e.g., $\Lambda_i$), and is generic except in the conformal case, where scaling symmetry uniquely fixes all relevant structures. Thus, for theories such as massless free fields, the EoS and thermodynamic variables are unambiguously defined, aligning universally with microscopic calculations.

Application to Microscopic Models

Explicit calculations for free Dirac fermion systems (both massless, conformal, and massive, non-conformal) demonstrate the procedure and highlight the distinction between thermodynamic and pseudo-gauge-invariant quantities. The derived thermodynamic densities and EoS satisfy all imposed relations and match the values of the defined invariants. For massless Dirac fermions, the pressure, entropy, and susceptibilities are given in closed analytic form in terms of temperature, chemical potential, and vorticity, with the latter scaling as expected in conformal hydrodynamics.

Similarly, for scalar fields, the framework produces correct thermodynamic observables in the conformal limit, again resolving discrepancies found in prior work. For non-conformal cases (massive scalars), the ambiguity is parameterized explicitly and linked to the aforementioned pseudo-gauge freedom.

Practical and Theoretical Implications

The results have significant implications for the interpretation of experimental measurements (e.g., in heavy-ion collisions where the QGP displays macroscopic vorticity and polarization) and for the matching of hydrodynamic effective theories with underlying QFT calculations. The clarified structure of pseudo-gauge ambiguity ensures that only observables invariant under allowed transformations can be meaningfully compared; local identification of densities and thermodynamic variables must account for the precise pseudo-gauge employed.

From a theoretical perspective, the work paves the way for future studies of stability, causality, and non-equilibrium corrections in spin hydrodynamics, and suggests that action-based approaches may further restrict the ambiguity or prefer certain pseudo-gauge choices. The explicit formulation of universal thermodynamic relations provides a foundation for such investigations.

Conclusions

The systematic analysis and construction of thermodynamic pseudo-gauges for ideal spin fluids presented in this work resolve longstanding contradictions between microscopic and hydrodynamic descriptions. The identification of universal, pseudo-gauge-independent constraints and invariants, as well as the explicit parametrization of EoS ambiguity, establish a robust framework for spin hydrodynamics compatible with both theory and experiment. The unique status of conformal fluids, wherein ambiguity is fixed, highlights the role of symmetry in hydrodynamic theory. Prospective developments include exploration of broader physical principles (beyond symmetry) to constraint pseudo-gauge freedom, non-perturbative extensions, and action-based formulations.