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Global equilibrium at order $\mathcal{O}(\hbar^2)$ : pseudo-gauge ambiguity, Maxwell relations, and thermodynamic consistency

This paper investigates massive spin-1/2 particles in global thermodynamic equilibrium with rotation and acceleration up to order $\mathcal{O}(\hbar^2)$ , demonstrating that while global observables are pseudo-gauge independent, local quantities and thermodynamic consistency (specifically the validity of Maxwell relations) depend on the chosen pseudo-gauge, thereby favoring the kinetic-theory pseudo-gauge for applications like spin hydrodynamics.

Original authors: Asaad Daher, David Wagner, Masoud Shokri, Dirk H. Rischke

Published 2026-09-29

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CC BY 4.0

Original authors: Asaad Daher, David Wagner, Masoud Shokri, Dirk H. Rischke

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a fluid that does not just flow, but also spins and accelerates, carrying with it a hidden internal twist known as spin. This is the realm of relativistic spin hydrodynamics, a field of physics attempting to describe matter under the most extreme conditions imaginable, such as the super-hot, rapidly rotating soup of particles created when heavy atomic nuclei collide at nearly the speed of light. In these collisions, the resulting matter is not merely a hot gas; it is a system where the very act of spinning and accelerating influences how the particles move and interact. To understand this, physicists rely on thermodynamics, the branch of science that governs heat, energy, and how systems settle into balance. However, when a system is both spinning and accelerating, the standard rules of thermodynamics become insufficient. The fluid's velocity, usually just a measure of how fast something moves, becomes a fundamental property of the system itself, just like temperature or pressure. This creates a complex puzzle: how do we correctly describe the energy, the number of particles, and the internal spin of such a system without breaking the fundamental laws of physics?

A team of researchers has tackled this puzzle by studying a theoretical model of massive particles with spin, specifically looking at how quantum effects—tiny corrections that arise from the strange rules of the subatomic world—alter the behavior of this spinning, accelerating fluid. They focused on a specific level of precision, accounting for quantum corrections that are small but significant. Their work reveals that the standard way physicists have been calculating the properties of these systems is not always consistent. In physics, there are different mathematical "views" or frameworks, called pseudo-gauges, used to define quantities like energy and spin currents. While these different views all agree on the total amount of energy or the total number of particles in the entire system, they disagree on how these quantities are distributed locally. One view might say the energy density is high in a certain spot, while another says it is low, even though both describe the same physical reality.

The researchers discovered that for a system in global equilibrium, where everything is balanced despite the rotation and acceleration, the standard rule that defines a particle's mass and speed is actually modified by these quantum effects. This modification depends on the square of the thermal vorticity, a measure of how much the fluid is twisting and turning. By incorporating this new rule, the team calculated the particle flow, energy, and spin for three different mathematical frameworks. They found that in one specific framework, known as the kinetic-theory pseudo-gauge, all the thermodynamic rules hold together perfectly. The relationships between temperature, pressure, and spin remain consistent, and the energy can be calculated in different ways to yield the exact same result. This framework acts like a reliable map where every path leads to the same destination.

In contrast, the other two frameworks, which have been widely used in the past, failed this consistency check. When the researchers tried to calculate the thermodynamic pressure using these alternative views, they found that the result depended on the path taken through the mathematical landscape, a clear sign that the underlying thermodynamic relations were broken. To fix this, one would have to make arbitrary adjustments that feel forced and unsatisfactory. The study explicitly rules out the idea that these traditional frameworks are naturally consistent for systems with spin, acceleration, and rotation. Instead, the findings strongly suggest that the kinetic-theory approach is the correct choice for describing such systems. This conclusion is crucial for the future of spin hydrodynamics, as it provides a solid, consistent foundation for modeling the behavior of matter in the extreme environments found in heavy-ion collisions and potentially in other cosmic phenomena like rotating black holes. The work confirms that while the global properties of the universe remain unchanged by our choice of mathematical perspective, the local details of how energy and spin are distributed require a specific, consistent viewpoint to be understood correctly.

Technical Summary: Global Equilibrium at Order O(ℏ²): Pseudo-Gauge Ambiguity, Maxwell Relations, and Thermodynamic Consistency

Problem Statement
The paper addresses three unresolved issues in the theoretical framework of relativistic spin hydrodynamics: (i) the microscopic foundations of thermodynamic relations in systems with spin, acceleration, and rotation; (ii) the ambiguity arising from the freedom to choose different pseudo-gauges for defining hydrodynamic currents (particle-number, energy-momentum, and spin-current); and (iii) the numerical implementation of these theories. Specifically, the authors investigate whether the standard thermodynamic relations, particularly the Maxwell relations, hold consistently across different pseudo-gauge choices when quantum corrections up to order $O(\hbar^2)$ are included. Previous work had identified potential violations of thermodynamic consistency, but a systematic comparison of pseudo-gauges including $O(\hbar^2)$ mass-shell modifications was lacking.

Methodology
The authors study a system of massive spin-1/2 Boltzmann particles in global thermodynamic equilibrium (GTE) with non-vanishing acceleration and rotation. The analysis employs the Wigner-function formalism within quantum kinetic theory.

Wigner Function and Mass-Shell Modification: The authors derive the $O(\hbar^2)$ modification to the standard mass-shell condition ( $k^2 = m^2$ ). They show that in GTE, the mass-shell constraint acquires a correction quadratic in the thermal vorticity $\varpi_{\mu\nu}$ . This modified constraint is consistently applied throughout the calculations.
Pseudo-Gauge Transformations: The study explicitly computes the (net) particle-number current, energy-momentum tensor, and spin-current tensor in three distinct pseudo-gauges:
- The Canonical pseudo-gauge.
- The de Groot–van Leeuwen–van Weert (GLW) pseudo-gauge.
- A Kinetic-theory pseudo-gauge (a modification of GLW that yields the standard kinetic theory expressions for the energy-momentum and spin-current tensors).
Thermodynamic Consistency Checks: For each pseudo-gauge, the authors derive local thermodynamic quantities (density, energy density, pressure, entropy density, spin density). They then test the validity of the thermodynamic Maxwell relations, which are derived from the commutativity of second derivatives of the thermodynamic pressure. These relations ensure that the pressure is a valid thermodynamic potential and that thermodynamic quantities can be consistently computed as partial derivatives of the pressure.
Integral Identities: The authors utilize two alternative integral identities to compute the thermodynamic pressure: one involving the energy density, particle density, and spin tensor (Eq. 51), and a simpler one involving only the energy density (Eq. 61). Consistency requires that both methods yield identical results.

Key Contributions and Results

Modified Mass-Shell Condition: The paper demonstrates that at order $O(\hbar^2)$ , the mass-shell condition is modified by terms quadratic in thermal vorticity. This modification is essential for obtaining consistent thermodynamic quantities at this order.
Pseudo-Gauge Dependence of Local Quantities: The authors explicitly demonstrate that while global conserved charges (total particle number, total energy, total angular momentum) and the grand partition function are independent of the pseudo-gauge choice, local thermodynamic quantities (particle-number density, energy density, pressure, entropy density, and spin density) are pseudo-gauge dependent.
Thermodynamic Consistency and Maxwell Relations:
- Kinetic-Theory Pseudo-Gauge: The authors find that this gauge is thermodynamically consistent. All Maxwell relations are satisfied, and the thermodynamic pressure computed via the two different integral identities yields identical results. Furthermore, local thermodynamic quantities can be correctly recovered as partial derivatives of the pressure.
- Canonical and GLW Pseudo-Gauges: In contrast, these gauges generally violate the Maxwell relations. Consequently, computing the thermodynamic pressure via the two different integral identities yields different results, indicating a lack of thermodynamic consistency. While consistency can be artificially restored in the canonical gauge by imposing specific global equilibrium conditions ad hoc within the calculation, this is viewed as an unsatisfactory fix.
Anisotropy in Equilibrium: The study reveals that in the presence of acceleration and rotation, the equilibrium state is anisotropic. The energy-momentum tensor does not satisfy the standard eigenvalue equation $T^{\mu\nu}u_\nu = \varepsilon u^\mu$ ; instead, it includes a "Hall-like" energy current term orthogonal to the fluid velocity.

Significance and Claims
The paper claims that the choice of pseudo-gauge is not merely a mathematical redundancy but has physical implications for the formulation of thermodynamics in spin hydrodynamics. The primary conclusion is that the kinetic-theory pseudo-gauge is the preferred choice for applications such as spin hydrodynamics because it is the only one among those studied that maintains thermodynamic consistency (satisfying Maxwell relations and path-independent pressure integration) without requiring artificial adjustments.

The authors emphasize that their findings suggest that standard thermodynamic relations must be extended to include the fluid four-velocity as an independent thermodynamic variable when acceleration and rotation are present. They note that while global quantities are gauge-invariant, the local description of the fluid requires a gauge choice that respects thermodynamic consistency. The paper concludes that the kinetic-theory pseudo-gauge provides the most natural and consistent framework for these systems, whereas the canonical and GLW gauges lead to inconsistencies in the thermodynamic relations unless specific, somewhat arbitrary, conditions are imposed. Future work is noted to address the restoration of consistency in other gauges by properly accounting for surface-term ambiguities in the identification of thermodynamic currents.

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