Thermodynamics of ideal spin fluids and pseudo-gauge ambiguity

This paper resolves the violation of local thermodynamic relations in relativistic spin fluids by identifying a specific family of pseudo-gauges that restore standard thermodynamics, thereby deriving universal relations and explicit equations of state for free Dirac fermions and scalar fields.

The Gist

Imagine you are trying to describe a swirling, spinning fluid, like a giant, hot whirlpool of particles created in a particle collider. Physicists call this a "spin fluid." To understand how it moves and behaves, they use a set of rules called hydrodynamics.

However, there's a major problem. When physicists try to measure the "stuff" inside this fluid (like how much energy, pressure, or spin it has), they get different answers depending on how they choose to look at it. It's like trying to measure the volume of a spinning top: if you measure it while it's wobbling, the number looks different than if you measure it while it's perfectly still.

This paper, written by Jay Armas and Akash Jain, solves a long-standing mystery about these confusing measurements. Here is the breakdown in simple terms:

1. The Problem: The "Camera Angle" Issue

In physics, there are things called conserved currents. Think of these as the "receipts" of the universe. They tell you how much energy, momentum, or spin is in a system. These receipts must add up correctly no matter what.

But here's the catch: You can tweak the way you write these receipts without changing the total amount of stuff. In the world of spinning fluids, this is called a pseudo-gauge transformation.

• The Analogy: Imagine you are describing a spinning dancer. You can describe her motion relative to the stage, relative to the audience, or relative to her own spinning body. All these descriptions are "true," but they give you different numbers for her speed and position.
• The Conflict: When physicists compared the "receipts" from microscopic models (the tiny quantum rules) with the "receipts" from hydrodynamics (the big fluid rules), they didn't match. The numbers for temperature, pressure, and density were all wrong. It looked like the laws of thermodynamics (the rules of heat and energy) were broken.

2. The Solution: Finding the "True" Angle

The authors realized that the mismatch wasn't because the laws of physics were broken, but because they were comparing apples to oranges. They were looking at the fluid from the wrong "camera angle" (the wrong pseudo-gauge).

They went on a hunt to find a specific "camera angle" where the rules of thermodynamics work perfectly.

• The Metaphor: Imagine you are trying to tune a radio. You hear static and noise (the wrong thermodynamic relations). The authors found the exact frequency (the thermodynamic pseudo-gauge) where the static disappears, and you hear the music clearly (the standard thermodynamic relations).

They proved that if you adjust your measurements to this specific angle, the energy, pressure, and spin density suddenly make sense and obey the standard rules of physics.

3. The Twist: The "Ambiguity" of Spin

Even after finding the right angle, there's a catch. For most fluids, the "spin" part of the measurement is still a bit fuzzy.

• The Analogy: Imagine you are trying to weigh a spinning coin. You can weigh it, but because it's spinning, there's a tiny bit of wiggle room in the number. You can't be 100% sure of the exact weight unless you know something special about the coin.
• The Exception: However, the authors found a special case: Conformal fluids (fluids made of massless particles, like light or massless electrons). These fluids have a special symmetry (they look the same at any scale). Because of this symmetry, the "wiggle room" disappears completely. You can measure them with perfect precision, and there is only one correct answer.

4. The Application: Cracking the Code

The authors didn't just find the theory; they applied it to real-world examples:

• Dirac Fermions: These are the particles that make up electrons and quarks.
• Scalar Fields: These are simpler theoretical particles.

By using their new "tuning method," they calculated the exact thermodynamic properties (pressure, density, etc.) for these particles. They showed that previous studies had the wrong numbers because they were using the wrong "camera angle." Their new numbers fix the errors and align perfectly with what we expect from basic physics.

Summary

• The Issue: Spinning fluids were giving confusing answers about their temperature and pressure because physicists were measuring them from the wrong perspective.
• The Fix: The authors found the specific perspective (pseudo-gauge) where the laws of thermodynamics work correctly.
• The Result: They fixed the equations for spinning fluids, allowing us to accurately describe the "quark-gluon plasma" (the hot soup of the early universe) and other spinning systems.
• The Takeaway: Just like you need to stand in the right spot to see a 3D object clearly, you need to choose the right mathematical "viewpoint" to understand the thermodynamics of spinning fluids. Once you do, the chaos turns into order.

Technical Summary

1. Problem Statement

Relativistic hydrodynamics is the standard effective theory for describing the Quark-Gluon Plasma (QGP) produced in heavy-ion collisions. Recent experimental observations of global $\Lambda$-hyperon polarization have necessitated the development of spin hydrodynamics, a framework incorporating spin degrees of freedom.

However, a fundamental ambiguity exists in the formulation of spin hydrodynamics:

• Pseudo-gauge Ambiguity: Conserved currents (energy-momentum tensor $T^{\mu\nu}$, spin current $\Sigma^{\mu\nu\rho}$, and particle current $J^\mu$) are not unique. They can be improved by adding total derivative terms (pseudo-gauge transformations) that leave global charges invariant but alter the local constitutive relations.
• Thermodynamic Inconsistency: Microscopic calculations (e.g., from Quantum Field Theory for free Dirac fermions) often yield conserved currents that, in a generic pseudo-gauge, violate standard local thermodynamic relations (e.g., the Gibbs-Duhem relation or the definition of entropy density).
• The Core Question: How can one identify the correct thermodynamic variables (pressure, energy density, spin susceptibility) from microscopic models when the currents depend on an arbitrary pseudo-gauge choice?

2. Methodology

The authors perform a systematic analysis of ideal spin fluids using a perturbative expansion in the spin chemical potential ($\mu_{\mu\nu}$). Their approach involves:

1. Formalism Setup:

   • Defining the hydrodynamic variables: fluid velocity $u^\mu$, temperature $T$, spin chemical potential $\mu_{\mu\nu}$, and particle chemical potential $\mu$.
   • Constructing a basis of spacelike vectors derived from acceleration ($a^\mu$) and vorticity ($\omega^\mu$).
   • Defining the Free Energy Current $N^\mu$ and enforcing the local second law of thermodynamics ($\partial_\mu S^\mu \geq 0$).

2. Pseudo-Gauge Analysis:

   • Explicitly writing down the general form of pseudo-gauge transformations for $T^{\mu\nu}$, $\Sigma^{\mu\nu\rho}$, and $J^\mu$ involving arbitrary tensors $M$.
   • Deriving the Belinfante tensor ($T^{\mu\nu}_B$), which is invariant under certain spin-current improvements but still transforms under others.

3. Deriving Constraints:

   • They postulate the existence of a "Thermodynamic Pseudo-Gauge" where the constitutive relations align with standard thermodynamic definitions (e.g., $n = -u_\mu J^\mu$, $\epsilon = u_\mu u_\nu T^{\mu\nu}$).
   • By demanding that the transformed currents satisfy these standard definitions, they derive a set of Linear Partial Differential Equations (PDEs) for the pseudo-gauge parameters ($\gamma_i, \lambda_i$).

4. Solvability and Invariants:

   • They analyze the solvability of these PDEs. They find that solutions only exist if the original currents satisfy specific universal thermodynamic relations.
   • They construct pseudo-gauge invariants (quantities independent of the gauge choice) to test the validity of equations of state (EoS) derived from microscopic models.

3. Key Contributions

A. Identification of Thermodynamic Pseudo-Gauges

The authors identify a specific family of pseudo-gauges where the conserved currents satisfy standard thermodynamic relations. They provide the explicit conditions (Eq. 21 in the paper) that the pseudo-gauge parameters must satisfy to transform a generic set of currents into this thermodynamic frame.

B. Universal Thermodynamic Relations

A major theoretical result is the derivation of pseudo-gauge-independent thermodynamic relations that any equilibrium conserved current must satisfy to be compatible with thermodynamics.

• These relations link the components of the Belinfante tensor and charge current to thermodynamic derivatives (e.g., $\partial P / \partial \mu$).
• Crucially, these relations hold at any order in spin (quadratic, quartic, etc.) and are necessary conditions for the existence of a thermodynamic pseudo-gauge.

C. Quantification of Ambiguity in the Spin Equation of State (EoS)

The paper demonstrates that the spin Equation of State is inherently ambiguous in non-conformal theories.

• Even after fixing the currents to satisfy thermodynamic relations, there remains a residual freedom (parameterized by functions $\Lambda_i$) that shifts the pressure $p$ by terms involving $a^2$ and $\omega^2$.
• This implies that measuring equilibrium currents alone is insufficient to uniquely determine the spin-dependent EoS for general theories.

D. Resolution of Conformal Ambiguity

A significant exception is found for conformal (scale-invariant) theories (e.g., massless free fields).

• The stronger symmetry constraints of conformal invariance force the ambiguity parameters to vanish.
• Consequently, the thermodynamic variables and EoS for conformal spin fluids can be identified unambiguously.

4. Key Results and Applications

The authors apply their formalism to free Dirac fermions and scalar fields:

• Massless Dirac Fermions:

  • They derive the unique thermodynamic variables ($p, n, s, \epsilon$) and spin susceptibilities ($\chi_{\omega\omega}, \chi_{aa}$) that satisfy local thermodynamic relations.
  • Their results differ qualitatively from previous literature (e.g., Ref [35]), which violated standard thermodynamic relations.
  • They verify that their derived invariants ($I_2, I_4$) match the microscopic calculations.

• Massive Dirac Fermions:

  • Since the theory is not conformal, the EoS retains an ambiguity parameterized by $\Lambda_1$.
  • They provide the EoS up to quadratic order in spin, explicitly showing the dependence on the arbitrary function $\Lambda_1$.

• Scalar Fields:

  • Similar analysis is performed for free scalar fields, confirming the generalizability of their method.
  • They address the coupling to curved spacetime (parameter $\xi$) and show how pseudo-gauge transformations can remove certain dependencies.

5. Significance

• Bridging Micro and Macro: The work resolves the apparent contradiction between microscopic QFT calculations of spin polarization and the macroscopic framework of spin hydrodynamics. It provides a rigorous prescription to extract physical thermodynamic variables from microscopic data.
• Experimental Relevance: For heavy-ion collision experiments (like STAR and ALICE), this framework clarifies how to interpret polarization data. It suggests that while global observables are robust, local thermodynamic variables extracted from data depend on the chosen pseudo-gauge unless the system is conformal.
• Theoretical Foundation: The derivation of universal thermodynamic relations (Eq. 23) provides a new consistency check for any proposed spin hydrodynamic model or microscopic calculation.
• Future Directions: The authors highlight that for non-conformal theories, additional physical principles (e.g., stability, causality, or an action-based approach) may be required to fix the remaining EoS ambiguity.

In summary, this paper establishes a rigorous thermodynamic framework for spin hydrodynamics, proving that while pseudo-gauge ambiguities generally obscure the definition of thermodynamic variables, they can be systematically resolved or constrained, particularly in conformal systems.