On the non-uniqueness of the energy-momentum and spin currents

The Gist

Imagine you are trying to describe how a spinning top moves through space. You want to know two things: how much energy it has (its motion) and how it is spinning (its rotation). In the world of physics, specifically for tiny particles like electrons, scientists have been arguing for a long time about how to write down the exact "recipe" for these two things.

This paper, written by Rajeev Singh, tackles a confusing problem in physics: Why do we have so many different recipes for the same thing, and how do we pick the right one?

Here is the breakdown using simple analogies:

1. The Problem: The "Moving Map" Confusion

In physics, we use mathematical tools called tensors to map out energy and spin. Think of these tensors as maps.

• The Old Way (Noether's First Theorem): For decades, physicists used a standard method to draw these maps. But this method produced a map that was "crooked" or "asymmetric." It was like trying to draw a perfect circle but ending up with a squashed oval. To fix this, they had to use a special "correction tool" called a pseudogauge transformation (or the Belinfante improvement).
• The Ambiguity: The problem is that this "correction tool" isn't unique. It's like having a photo editor where you can choose from a hundred different filters. You can apply Filter A, Filter B, or Filter C. All of them make the photo look "okay," and they all preserve the total amount of light in the picture (the total energy). But they show the light distributed differently across the photo.
• The Dilemma: In the real world, we need to know exactly where the energy and spin are located, not just the total amount. If you use Filter A, the spin might look like it's in the center. If you use Filter B, the spin might look like it's on the edge. Which one is the "real" truth? The paper says that for a long time, we didn't have a rule to say which filter was the correct one.

2. The Solution: A New Compass (Noether's Second Theorem)

The author suggests we stop using the old method (First Theorem) and switch to a more powerful, but less commonly used, method called Noether's Second Theorem.

• The Analogy: Imagine you are trying to find the center of a storm.
  • The First Theorem is like looking at the wind from a distance and guessing where the eye is. You can guess, but you might be wrong, or you might have to add a "correction" later.
  • The Second Theorem is like having a GPS that is locked directly to the storm's internal structure. It doesn't guess; it calculates the center based on the storm's own rules.

By using this "GPS" (Noether's Second Theorem), the author shows that we don't need to guess or apply a filter. The math naturally forces the map to be perfect (symmetric) right from the start.

3. The Result: The "Perfect Map"

When the author applied this new method to free-moving particles (massive spin-half fermions, like electrons), two surprising things happened:

1. The Energy Map became Perfect: The resulting map of energy and momentum was naturally symmetric. It looked exactly like the "corrected" map that physicists had been trying to force into existence for years using the old filters.
2. The Spin Map Vanished: The map for the "spin" current turned out to be zero.

Wait, zero?
This sounds strange, but the author explains it this way: When you use the most fundamental, "local" rules of the universe (local symmetries) to define these currents, the spin doesn't need to be a separate, wandering entity. The energy and momentum are so perfectly organized that the "spin" part of the equation cancels itself out completely.

4. Why This Matters (According to the Paper)

The paper argues that this isn't just a mathematical trick. It solves the "which filter is right?" debate.

• No More Guessing: We don't need to choose between the "Belinfante" filter, the "GLW" filter, or the "HW" filter. Noether's Second Theorem gives us one unique, physically consistent answer.
• Consistency: This unique answer matches what we see in General Relativity (Einstein's theory of gravity), suggesting it is the "true" physical description.
• Stability: The paper notes that this specific version of the energy map behaves better when we look at tiny, hot systems (like the early universe or particle collisions), showing less chaotic "quantum noise" than the other versions.

Summary

Think of this paper as finding the master key to a lock that physicists have been trying to pick for years.

• Before: We had a pile of keys (different pseudogauges) that all opened the door, but we didn't know which one was the "real" key.
• Now: The author found a tool (Noether's Second Theorem) that forges the one true key. When you use this key, the door opens perfectly, the energy map is straight, and the confusing spin map disappears, leaving us with a single, clear picture of how these particles move and spin.

Technical Summary

Technical Summary: On the non-uniqueness of the energy-momentum and spin currents

Problem Statement
The paper addresses the long-standing ambiguity in defining the local distributions of energy-momentum and spin for relativistic systems, particularly within the framework of relativistic spin hydrodynamics. While the total conserved charges (total energy, momentum, and angular momentum) are invariant, the microscopic definitions of the energy-momentum tensor (EMT) and spin tensor derived via Noether's first theorem are not unique. Specifically, the canonical EMT derived from the Dirac Lagrangian is asymmetric, necessitating the use of "pseudogauge transformations" (or the Belinfante improvement procedure) to symmetrize it. These transformations introduce arbitrary super-potentials ($X^{\lambda,\mu\nu}$ and $Y^{\mu\nu,\lambda\rho}$), leading to multiple valid forms of the EMT and spin tensor (e.g., Belinfante-Rosenfeld, de Groot–van Leeuwen–van Weert, and Hilgevoord–Wouthuysen forms). The central issue is determining which specific decomposition of total angular momentum into orbital and spin components provides a physically meaningful local distribution, a question that remains unresolved in the context of gauge-invariant definitions and experimental interpretation.

Methodology
The author proposes resolving this ambiguity by applying Noether's second theorem rather than the widely used Noether's first theorem.

• Theoretical Framework: Noether's first theorem applies to global symmetries and yields conserved currents only up to the addition of divergence terms (super-potentials). In contrast, Noether's second theorem applies to local symmetries (local spacetime translations and local gauge transformations).
• Derivation Process: The paper considers a system of free massive Dirac fermions (spin-1/2). Instead of deriving the canonical currents and subsequently applying a pseudogauge transformation to symmetrize the EMT, the author derives the currents directly from the local symmetry of the action.
• Technical Implementation: The derivation involves calculating the total variations of the Dirac field ($\psi$) and its adjoint ($\bar{\psi}$) under local translations expressed as $\xi^\mu(x)$. By defining a quantity $D^{\mu\nu}$ based on the variation of the Lagrangian density with respect to these local parameters, the author constructs the EMT and spin tensor without introducing arbitrary super-potentials. The super-potential is effectively fixed by the local symmetry and the choice of boundary terms in the action.

Key Contributions and Results

1. Unique Derivation of EMT: Using Noether's second theorem, the paper derives an EMT for free Dirac fields that is inherently symmetric. The resulting expression, $T^{\mu\nu} = \frac{i}{4}\bar{\psi}(\gamma^\mu \overleftrightarrow{\partial}_\nu + \gamma^\nu \overleftrightarrow{\partial}_\mu)\psi$, matches the Belinfante-Rosenfeld (BR) form. Crucially, this result is obtained without performing a pseudogauge transformation, thereby eliminating the ambiguity associated with choosing specific super-potentials.
2. Vanishing Spin Tensor: The derivation yields a spin tensor $S^{\lambda,\mu\nu}$ that is identically zero ($S^{\lambda,\mu\nu} = 0$). This contrasts with the non-zero spin tensors obtained via pseudogauge transformations (such as the BR, GLW, or HW forms).
3. Resolution of Ambiguity: The paper argues that Noether's second theorem fixes the super-potential determined by local symmetry, selecting a unique, physically consistent pseudogauge. This approach suggests that the "ambiguity" in the local distribution of spin and energy is an artifact of using global symmetry methods (Noether's first theorem) where local symmetry methods are more appropriate.
4. Algebraic Consistency: The paper notes that the derived vanishing spin tensor trivially satisfies the $SO(3)$ angular momentum algebra, whereas spin tensors derived via pseudogauge transformations (GLW, HW) may not satisfy this algebra in all contexts.

Significance and Claims
The paper claims that utilizing Noether's second theorem provides a "gauge-invariant" and unique way to decompose total angular momentum, removing the need for ad-hoc pseudogauge transformations.

• Physical Consistency: The derived EMT is claimed to be the unique physically consistent form for free fermions, aligning with the Hilbert stress-energy tensor found in General Relativity.
• Hydrodynamic Implications: In the context of relativistic spin hydrodynamics, where macroscopic densities are ensemble averages of microscopic operators, this method offers a definitive prescription for the underlying quantum operators, potentially resolving debates regarding which pseudogauge describes experimental data (e.g., spin polarization in quark-gluon plasma).
• Novelty: While the symmetric EMT for free fermions was previously known, the author highlights that the derivation of the spin tensor using Noether's second theorem (resulting in a vanishing tensor) is presented here for the first time.
• Future Utility: The method is suggested as a tool for comparing different momentum and spin decompositions (such as Ji and Jaffe-Manohar decompositions) by providing a baseline derived strictly from local symmetries.

The paper maintains a modest tone regarding broader applications, focusing strictly on the theoretical derivation for free massive Dirac particles and the resolution of the super-potential ambiguity within this specific context.