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

This paper presents an exact analytical and numerical study of a boost-invariant, out-of-equilibrium fluid of non-interacting Dirac fermions with finite spin potential, deriving explicit expressions for thermodynamic quantities and demonstrating that non-vanishing spin polarization in such free systems arises solely from the spin potential while shear-induced polarization and the spin Hall effect are absent.

The Gist

The Big Picture: A Quantum Dance Floor

Imagine a massive, high-energy dance party. This isn't a normal party; it's the Quark-Gluon Plasma (QGP) created when heavy atomic nuclei smash into each other at nearly the speed of light (like in the Large Hadron Collider).

In this chaotic ballroom, trillions of tiny particles (quarks and gluons) are swirling around. Usually, physicists treat these particles like a hot gas or a fluid. But these particles have a secret quantum superpower: Spin. Think of spin not as the particle spinning like a top, but as an internal compass needle pointing in a specific direction.

The big question this paper asks is: If we force these particles to align their compass needles (spin) in a specific way, how does the whole "fluid" react?

The Setup: The "Bjorken" Slide

To make the math possible, the authors imagine a very specific type of party. They use a coordinate system called Milne coordinates, which is like watching the party from a special perspective where everything expands perfectly symmetrically.

• The Analogy: Imagine a loaf of bread rising in the oven. As it expands, every crumb moves away from every other crumb, but the pattern looks the same from the center. This is "boost-invariant." The fluid is stretching out, but it looks the same at every angle.
• The Problem: In this specific stretching scenario, the fluid is not in perfect equilibrium. It's like a balloon being blown up; the air inside is moving and changing.

The Mystery: The "Spin Potential"

In standard physics, particles align their spins because of "vorticity"—basically, the fluid is swirling like a whirlpool. But in this specific "stretching bread" scenario, there is no whirlpool. So, why would the particles spin?

The authors introduce a new character: The Spin Potential ($\Omega$).

• The Analogy: Imagine the dance floor has a magnetic field that forces everyone to face North. Even if the music (the fluid flow) isn't swirling, the magnetic field (the spin potential) makes everyone align.
• The paper asks: What happens to the energy, pressure, and movement of the fluid if we crank up this "magnetic alignment" knob?

The Method: Solving the Puzzle Exactly

Most physicists use approximations (guessing the answer based on small changes). This paper is special because they solved the problem exactly.

• The Analogy: Instead of estimating how many people are on the dance floor by taking a quick photo, they counted every single person, tracked their exact steps, and calculated the total energy of the room down to the last decimal point.
• They used a mathematical tool called Milne modes (a specific way of describing the particles in this expanding universe) and a technique called Bogoliubov transformation (a fancy way of rearranging the math to make the messy equations clean and solvable).

The Key Discoveries

Here is what they found, translated from "Physics Speak" to "Human Speak":

1. The "No-Swirl" Rule

They found that in this specific stretching fluid, you cannot get spin polarization just from the fluid moving or shearing.

• The Metaphor: Imagine a conveyor belt moving straight. Even if the belt is vibrating (shear), it won't make the boxes on it turn to face a specific direction unless you force them to.
• The Result: If you turn off the "Spin Potential" knob, the particles remain randomly oriented, even if the fluid is hot and moving fast. This solves a long-standing puzzle about why some formulas predicted spin where there was none.

2. The "Magnetic Knob" Works

When they turned on the Spin Potential, the particles did align.

• The Result: They calculated exactly how much the particles align based on how strong the "magnetic knob" is.
• The Surprise: The alignment isn't perfect (100%). It depends on the particle's mass and how fast it's moving sideways. Heavy particles align better than light ones. Particles moving straight forward align better than those moving sideways.

3. The Fluid Doesn't Care (Much)

One of the most surprising findings is about the Pressure (how hard the fluid pushes against the walls).

• The Metaphor: You might think that forcing everyone to face North would make the crowd push harder in that direction, creating a lopsided pressure.
• The Result: Surprisingly, the fluid pushes equally in all directions. The "Spin Potential" adds a little bit of extra energy to the system (like adding more dancers), but it doesn't make the fluid squishy in one direction and hard in another. The pressure remains perfectly round (isotropic).

4. Thermodynamics Still Holds

There was a debate in the physics community: "If we add this new 'Spin Potential' rule, do the standard laws of thermodynamics (like how energy and temperature relate) break?"

• The Result: The authors proved that no, the laws still work. They showed that the math connecting the "Spin Potential" to the energy and pressure works perfectly, just like a standard recipe works even if you add a new spice.

Why Does This Matter?

This paper is a benchmark.

• The Analogy: Imagine you are building a new video game engine. Before you release it, you need a "test level" where you know the exact answer to every question. This paper provides that test level.
• Real World Application: Physicists studying the Quark-Gluon Plasma (the soup of the early universe) can now compare their complex computer simulations against the "Exact Answer" provided by this paper. If their simulations don't match this paper, they know their simulation is wrong.

Summary in One Sentence

The authors solved a complex quantum puzzle to show that in an expanding, stretching fluid, particles won't align their spins unless you force them to with a "spin potential," and when you do, the fluid gains energy but stays perfectly balanced in its pressure.

Technical Summary

1. Problem Statement

The paper addresses the theoretical challenges in describing the Quark-Gluon Plasma (QGP) produced in heavy-ion collisions, specifically focusing on spin polarization.

• The Puzzle: While thermal vorticity (related to angular velocity and acceleration) successfully explains some aspects of spin polarization, it fails to account for the observed momentum dependence, leading to the "polarization sign puzzle."
• The Framework: Recent developments in spin hydrodynamics introduce a finite spin potential ($S_{\mu\nu}$) as an independent dynamical variable to describe non-equilibrium spin degrees of freedom.
• The Gap: Constitutive relations for expectation values in the presence of a finite spin potential are largely unknown. Furthermore, the "pseudogauge ambiguity" (the choice between Belinfante and canonical energy-momentum/spin tensors) significantly affects out-of-equilibrium results. There is a lack of exact, non-perturbative benchmarks to test linear response theories and thermodynamic relations in this regime.

Goal: To compute exact expectation values for a system of non-interacting Dirac fermions in a longitudinally boost-invariant (Bjorken flow) background with a finite spin potential, without relying on linear response approximations.

2. Methodology

The authors employ a rigorous quantum field theory approach in curved spacetime coordinates (Milne coordinates) to handle the boost invariance.

• Setup:

  • System: Non-interacting Dirac fermions.
  • Geometry: Milne coordinates $(\tau, x, y, \eta)$, where $\tau$ is proper time and $\eta$ is space-time rapidity. This naturally incorporates longitudinal boost invariance.
  • Statistical Operator: The Zubarev operator for a local equilibrium state is constructed using the density operator $\hat{\rho} \propto \exp(-\hat{\Upsilon})$, where $\hat{\Upsilon}$ includes the stress-energy tensor, charge current, and the spin tensor coupled to the spin potential $S_{\mu\nu}$.
  • Spin Potential: Imposed to be purely transverse ($S_{xy}$) to satisfy boost invariance, defined via a reduced spin potential $S = \Omega/T$.

• Technical Steps:

  1. Solving the Dirac Equation: The Dirac equation is solved exactly in Milne coordinates. The solutions are expressed in terms of Hankel functions with matrix-valued orders, derived from the spin connection in the Milne metric.
  2. Mode Expansion: The field operator is expanded in terms of Milne creation and annihilation operators ($\hat{A}, \hat{B}^\dagger$). The authors demonstrate that these modes do not mix positive and negative frequencies relative to the Minkowski vacuum, confirming no spontaneous particle production occurs (the effect is a coordinate artifact).
  3. Diagonalization:
     • The statistical operator's exponent (effective Hamiltonian $\hat{\Pi}_\Omega$) is non-diagonal in the Milne basis.
     • A Bogoliubov transformation is performed to diagonalize the Hamiltonian.
     • Belinfante Pseudogauge: Analytic diagonalization is straightforward ($\Omega=0$).
     • Canonical Pseudogauge: With finite $\Omega$, the Hamiltonian becomes a dense $4\times4$ matrix. The authors derive an analytic expression for the eigenvalues and eigenvectors (though the explicit eigenvectors are complex, they are provided in the appendices).
  4. Renormalization: A specific renormalization procedure is applied by subtracting the vacuum contribution at a fixed decoupling time $\tau_0$ (instantaneous freeze-out), ensuring finite, physically meaningful expectation values.
  5. Calculation of Observables: Exact formulas are derived for the partition function, energy density, pressures, spin density, and spin polarization vector.

3. Key Contributions

• First Exact Analytic Partition Function: The paper provides the first exact analytic expression for the partition function of a relativistic fluid with a finite spin potential in a boost-invariant setting.
• Exact Spin Polarization: Derivation of exact expressions for the spin polarization vector as a function of the spin potential, valid for arbitrary values of the potential (not limited to linear response).
• Pseudogauge Comparison: A systematic comparison between the Belinfante and Canonical pseudogauges, highlighting how the choice of gauge affects the presence of spin torque and the generation of polarization.
• Thermodynamic Validation: Numerical verification that standard thermodynamic relations (connecting the partition function to energy, pressure, and spin density) hold true in this non-equilibrium system with a finite spin potential, at least within the canonical pseudogauge.

4. Key Results

A. Belinfante Pseudogauge ($\Omega = 0$)

• No Polarization: In a boost-invariant system with zero spin potential, the spin polarization vector vanishes identically, even in the presence of finite thermal shear and non-constant chemical potential.
• Implication: This proves that thermal shear alone is insufficient to generate net polarization in this symmetry class, ruling out certain linear-response predictions that suggest otherwise.
• Stress-Energy Tensor: The local equilibrium expectation values coincide with classical kinetic theory results (no quantum corrections).

B. Canonical Pseudogauge (Finite $\Omega$)

• Spin-Induced Polarization: A non-vanishing spin potential is the sole source of polarization in this free theory.
• Analytic Limits:
  • Purely Longitudinal Momentum ($p_T=0$): An exact analytic formula for polarization $P_z$ is derived (Eq. 5.4). It saturates to 1 for large spin potentials ($\Omega \gg T$).
  • Large Spin Potential Limit: For $\Omega \gg T$, the polarization depends on the transverse momentum: $P_z \approx \frac{m}{\sqrt{m^2+p_T^2}} \tanh(S/4)$.
• Numerical Results:
  • Energy & Pressure: The spin potential induces a modest enhancement (6–15%) in energy density and pressure but does not induce pressure anisotropy ($P_L = P_T$).
  • Thermodynamics: The numerically computed expectation values match those derived from the partition function via thermodynamic derivatives, confirming the consistency of the thermodynamic framework with spin potentials.
  • Torque: The spin torque (source of non-conservation) is found to be numerically vanishing in this specific configuration.

C. General Observations

• Symmetry Constraints: Longitudinal boost invariance is a powerful constraint that forbids shear-induced polarization. Polarization can only arise from the explicit spin potential or interactions (which are absent here).
• Quantum Corrections: While the local equilibrium stress-energy tensor matches classical theory, the free-streaming (out-of-equilibrium) evolution from an initial state $\tau_0$ shows quantum corrections due to the non-equivalence of the Milne vacuum and the Minkowski vacuum at the decoupling time. However, these corrections are suppressed for realistic heavy-ion parameters ($m_T \tau_0 \gg 1$).

5. Significance

• Benchmark for Spin Hydrodynamics: The results provide a rigorous, exact benchmark against which approximate linear-response formulas and spin hydrodynamic models can be tested.
• Resolution of the Sign Puzzle: The finding that shear alone cannot generate polarization in boost-invariant flows suggests that the polarization sign puzzle in heavy-ion collisions must be resolved by mechanisms beyond simple thermal gradients, such as the presence of a finite spin potential or interaction effects.
• Thermodynamic Consistency: The work demonstrates that thermodynamic relations remain valid even when extending the ensemble to include a spin potential, supporting the theoretical consistency of spin hydrodynamics.
• Phenomenological Impact: The derived polarization formulas suggest that if a spin potential exists at freeze-out, it must be small ($S \sim 10^{-2}$) to match experimental data, placing it well within the regime where linear response theory is applicable, yet the exact formulas allow for exploration of strong-coupling regimes.

In summary, this paper establishes a mathematically exact framework for studying spin in relativistic fluids, clarifying the role of symmetries and the necessity of a spin potential for generating polarization in the absence of interactions.