Perfect spin hydrodynamics at all orders in spin polarization

This paper demonstrates that two distinct frameworks for perfect spin hydrodynamics—based on classical kinetic theory and the Wigner function—yield conserved currents with identical forms at every order of spin polarization expansion, differing only by a monotonically increasing multiplicative factor.

The Gist

Imagine you are trying to describe how a swarm of tiny, spinning tops (particles) moves and flows together in a hot, chaotic soup. Physicists have been arguing for a long time about the best way to do this.

One group says, "Let's treat these tops like tiny, spinning gyroscopes we can see and touch." This is the Classical approach.
The other group says, "No, these tops are quantum objects; they follow weird, fuzzy rules that only exist in the quantum world." This is the Quantum approach.

Usually, we expect these two descriptions to only match when the tops are spinning so fast and so wildly that their quantum weirdness averages out and looks "classical." But this paper asks: What happens when the tops are spinning slowly? Do the two descriptions still agree?

The Big Discovery

The author, Zbigniew Drogosz, set up a mathematical "taste test" to compare these two recipes for describing spinning particles. He looked at the formulas used to calculate three main things:

1. How many particles are there? (Baryon current)
2. How much energy and momentum are they carrying? (Energy-momentum tensor)
3. How are they spinning? (Spin tensor)

He expanded the formulas like a recipe, adding ingredients step-by-step. The first step is the simplest (low spin), the second step adds more detail, the third adds even more, and so on.

The "Cookie Cutter" Analogy

Here is the surprising result:

Imagine both the Classical and Quantum chefs are baking cookies.

• The Shape: When they cut out the cookies (the mathematical structure of the formulas), they cut out the exact same shape at every single step of the process. Whether they are making the first cookie or the hundredth, the shape is identical.
• The Size: The only difference is the size of the cookie.
  • At the very first step (low spin), both chefs cut out cookies of the exact same size. The two theories are perfect twins here.
  • At the second step, the Quantum chef's cookie is slightly smaller than the Classical chef's.
  • At the third step, the difference gets bigger.
  • At the tenth step, the Classical chef is baking a giant cookie, while the Quantum chef is baking a tiny crumb.

The paper proves that the "size difference" follows a strict rule. As you add more complex steps (higher orders of spin), the Classical recipe predicts values that get exponentially larger than the Quantum recipe.

Why Does This Matter?

This explains a mystery in the field. Scientists had noticed that for heavy-ion collisions (where they smash atoms together to create a "soup" of particles), the Classical and Quantum theories seemed to work in the same range of conditions.

This paper explains why:

• In the real world, the "spin" of the particles is usually quite low.
• Because the spin is low, we only need the first few steps of the recipe.
• In those first few steps, the two theories are almost identical (the cookies are the same size).
• The theories only start to disagree wildly if you try to describe a situation with extremely high spin, which is a condition that rarely happens in these experiments.

The "Magic Number" Twist

The author also found a clever trick. If you could magically change the "size setting" on the Classical chef's machine (a parameter called the spin normalization constant) for every single step of the recipe, you could make the Classical cookies match the Quantum ones perfectly forever.

However, in reality, that setting is a fixed number. Because it's fixed, the two theories naturally drift apart as the spin gets stronger.

The Bottom Line

The paper concludes that for the "perfect" spinning fluids we see in nature (where friction is ignored), the Classical and Quantum descriptions are structurally identical. They are built on the same blueprint. They only differ in a scaling factor that grows as the spin gets more intense.

So, for the low-spin situations we actually observe in heavy-ion collisions, you can safely use the simpler Classical picture, knowing it will give you the right answer because it matches the complex Quantum picture almost perfectly.

Technical Summary

Problem Statement
The paper addresses the theoretical validity and correspondence between classical and quantum descriptions of spin hydrodynamics for spin-1/2 particles. While the correspondence principle generally suggests that classical physics emerges from quantum physics at large quantum numbers, the spin of fundamental particles (order of $\hbar$) does not naturally satisfy this condition. Despite this, classical spin pictures (based on Mathisson's framework) have been successfully used to describe polarization phenomena in heavy-ion collisions, often yielding results consistent with quantum approaches at low spin polarizations. The central problem investigated is whether this agreement persists at higher orders of spin polarization or if the two frameworks fundamentally diverge, and if so, how precisely they differ.

Methodology
The author compares two recently developed frameworks for perfect spin hydrodynamics:

1. Classical Spin Description: Based on classical kinetic theory extended to include a spin four-vector $s^\mu$. The system is described using equilibrium distribution functions for particles and antiparticles in an extended phase space. The analysis utilizes the GLW (Gorbar-Liu-Wang) pseudogauge, where the spin tensor is non-zero and conserved. The scalar density $n_{cl}$ serves as a generating function for conserved currents (baryon current $N^\mu$, energy-momentum tensor $T^{\mu\nu}$, and spin tensor $S^{\lambda,\mu\nu}$).
2. Quantum Spin Description: Based on a novel Wigner function approach. Similar to the classical case, conserved currents are derived by differentiating a quantum scalar density $n_{qt}$.

The core of the methodology involves expanding both the classical and quantum scalar densities in powers of the dimensionless spin polarization tensor $\omega_{\alpha\beta} \equiv \Omega_{\alpha\beta}/T$. The author performs explicit calculations of integrals over the spin space for the classical case, utilizing the properties of the spin four-vector integration measure and the symmetrized products of projectors. These results are then compared term-by-term with the expansion of the quantum scalar density, which is expanded in powers of the dual vector $a^\mu$ (derived from $\omega_{\alpha\beta}$).

Key Contributions and Results

• Structural Equivalence: The paper demonstrates that the conserved current tensors in both the classical and quantum approaches possess fully analogous structures at every order of the expansion in $\omega_{\alpha\beta}$.
• Multiplicative Factor Divergence: While the functional forms are identical, the two descriptions differ by a relative multiplicative factor at each order of expansion.
  • At the lowest non-trivial order, the factor is 1, meaning the descriptions are exactly equivalent.
  • At higher orders, the factor increases monotonically. Specifically, for the $2n$-th order contribution to the baryon current and energy-momentum tensor (and the $(2n-1)$-th order for the spin tensor), the classical result is larger than the quantum result by a factor of $\frac{3^n}{2n+1}$, assuming the classical spin normalization constant is fixed at the SU(2) Casimir eigenvalue ($\ell^2 = 3/4$).
• Normalization Analysis: The study shows that for the classical and quantum descriptions to be exactly equivalent at all orders, the classical spin normalization constant $\ell$ would not be a constant but would need to vary with the expansion order, specifically $\ell^{2n} = \frac{2n+1}{2^{2n}}$.
• Limit Behavior: In the limit of infinite order ($n \to \infty$), the required normalization constant approaches $1/2$. This explains the observed correspondence in application ranges found in previous literature, suggesting that the quantum approach effectively reduces the classical normalization from $\sqrt{3}/2$ (valid for low polarizations) to $1/2$ as polarization increases.

Significance and Claims
The paper claims to provide a precise quantification of the separation between classical and quantum perfect spin hydrodynamics at higher orders of spin polarization. The primary significance lies in establishing that the classical description is valid specifically for sufficiently low values of the spin polarization tensor components, where higher-order contributions are negligible.

The author posits that the correspondence at low orders is conceptually possible because the classical framework considers a system of many particles where polarization averages out to a low value. The work clarifies that the agreement between the two frameworks is not universal but breaks down systematically as the expansion order increases, with the classical description overestimating the magnitude of conserved currents by a calculable factor. The paper concludes by noting the practical value of implementing these dynamic equations in 3D numerical simulations to further explore these hydrodynamic systems.