Pseudo-Gauge Transformations in Field Theory

• Pseudo-gauge transformations are redefinitions of local energy–momentum and spin tensors using total divergences that preserve global conservation laws while altering local distributions.
• They play a critical role in distinguishing the split between orbital and spin angular momentum and ensuring thermodynamic consistency in relativistic spin hydrodynamics.
• Recent advances use pseudo-gauge invariant local equilibrium operators to remove ambiguities in spin-polarization observables, enabling more robust hydrodynamic modeling in high-energy collisions.

Pseudo-gauge transformations are redefinitions of the local energy–momentum and spin tensors by total divergences or superpotentials that preserve the conservation laws for energy, linear momentum, and angular momentum while changing the local redistribution of these quantities. In relativistic field theory this arbitrariness is described as pseudogauge freedom or symmetry, and it becomes operationally important whenever observables are computed from local densities rather than global charges. Recent work has shown that pseudo-gauge freedom controls the split between orbital and spin angular momentum, the form of local-equilibrium density operators, and the thermodynamic consistency of relativistic spin hydrodynamics (Dey et al., 2023, Becattini et al., 12 Jul 2025).

1. Formal definition and conservation structure

A commonly used formulation starts from a stress–energy tensor $T^{\mu\nu}$ and a spin tensor $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$ obeying

$$\partial_{\mu}T^{\mu\nu}=0, \qquad \partial_{\lambda}S^{\lambda,\mu\nu}=-2\,T^{[\mu\nu]}.$$

The associated total angular-momentum current is

$$J^{\lambda,\mu\nu} = x^\mu T^{\lambda\nu} - x^\nu T^{\lambda\mu} + S^{\lambda,\mu\nu},$$

whose conservation follows from the Ward identities. A pseudo-gauge transformation is generated by an arbitrary operator-valued superpotential $\Phi^{\alpha,\mu\nu}=\Phi^{\alpha,[\mu\nu]}$ through

$$S'^{\lambda,\mu\nu}=S^{\lambda,\mu\nu}-\Phi^{\lambda,\mu\nu},$$

$$T'^{\mu\nu} = T^{\mu\nu} +\frac12\partial_{\alpha} \Bigl( \Phi^{\alpha,\mu\nu} +\Phi^{\mu,\nu\alpha} +\Phi^{\nu,\mu\alpha} \Bigr).$$

These transformed tensors satisfy the same Ward identities, and the total charges

$$P^\nu=\int_\Sigma d\Sigma_\mu\,T^{\mu\nu}, \qquad J^{\mu\nu}=\int_\Sigma d\Sigma_\lambda\,J^{\lambda,\mu\nu}$$

remain invariant because the improvement contributes only surface terms (Becattini et al., 12 Jul 2025).

A more general formulation allows an additional rank-4 superpotential $Z^{\mu\nu,\lambda\rho}$ with

$$Z^{\mu\nu,\lambda\rho} = -\,Z^{\nu\mu,\lambda\rho} = -\,Z^{\mu\nu,\rho\lambda},$$

so that

$S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$0

Under standard boundary conditions, $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$1 and $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$2 vanish after reduction to surface integrals at spatial infinity (Dey et al., 2023).

The physical content of the transformation is therefore local rather than global. What changes are the local densities and currents; what does not change are the exactly conserved global generators.

2. Canonical, Belinfante, GLW, HW, and related realizations

For microscopic field theories, the canonical Noether currents provide the starting point. In flat $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$3 spacetime, a Lagrangian density $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$4 yields the canonical energy–momentum tensor

$S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$5

and canonical spin current

$S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$6

while the Belinfante tensor is

$S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$7

which is symmetric on the equations of motion. For free Dirac and Proca fields, explicit expressions for canonical, Belinfante, Hilgevoord–Wouthuysen, de Groot–van Leeuwen–van Weert, and Klein–Gordon-type pseudo-gauges are available, and the same framework extends to interacting theories and electromagnetic backgrounds (Armas et al., 20 Jan 2026, Weickgenannt et al., 2022).

Pseudogauge	Defining potentials	Resulting property
Canonical (Noether)	$S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$8	Original Noether $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$9 and $$\partial_{\mu}T^{\mu\nu}=0, \qquad \partial_{\lambda}S^{\lambda,\mu\nu}=-2\,T^{[\mu\nu]}.$$0
Belinfante–Rosenfeld	$$\partial_{\mu}T^{\mu\nu}=0, \qquad \partial_{\lambda}S^{\lambda,\mu\nu}=-2\,T^{[\mu\nu]}.$$1	$$\partial_{\mu}T^{\mu\nu}=0, \qquad \partial_{\lambda}S^{\lambda,\mu\nu}=-2\,T^{[\mu\nu]}.$$2, $$\partial_{\mu}T^{\mu\nu}=0, \qquad \partial_{\lambda}S^{\lambda,\mu\nu}=-2\,T^{[\mu\nu]}.$$3
GLW	$$\partial_{\mu}T^{\mu\nu}=0, \qquad \partial_{\lambda}S^{\lambda,\mu\nu}=-2\,T^{[\mu\nu]}.$$4	Symmetric $$\partial_{\mu}T^{\mu\nu}=0, \qquad \partial_{\lambda}S^{\lambda,\mu\nu}=-2\,T^{[\mu\nu]}.$$5, conserved $$\partial_{\mu}T^{\mu\nu}=0, \qquad \partial_{\lambda}S^{\lambda,\mu\nu}=-2\,T^{[\mu\nu]}.$$6
HW	$$\partial_{\mu}T^{\mu\nu}=0, \qquad \partial_{\lambda}S^{\lambda,\mu\nu}=-2\,T^{[\mu\nu]}.$$7 begins with the same bilinear as GLW plus an extra total-derivative term; $$\partial_{\mu}T^{\mu\nu}=0, \qquad \partial_{\lambda}S^{\lambda,\mu\nu}=-2\,T^{[\mu\nu]}.$$8	Conserved $$\partial_{\mu}T^{\mu\nu}=0, \qquad \partial_{\lambda}S^{\lambda,\mu\nu}=-2\,T^{[\mu\nu]}.$$9
KG-type	Obtained by adding a suitable $$J^{\lambda,\mu\nu} = x^\mu T^{\lambda\nu} - x^\nu T^{\lambda\mu} + S^{\lambda,\mu\nu},$$0-term	$$J^{\lambda,\mu\nu} = x^\mu T^{\lambda\nu} - x^\nu T^{\lambda\mu} + S^{\lambda,\mu\nu},$$1, $$J^{\lambda,\mu\nu} = x^\mu T^{\lambda\nu} - x^\nu T^{\lambda\mu} + S^{\lambda,\mu\nu},$$2

For free fields, GLW and HW provide symmetric energy–momentum tensors and conserved spin tensors. In interacting or out-of-equilibrium settings, this simplification generally fails: interactions can spoil exact spin conservation, and one has $$J^{\lambda,\mu\nu} = x^\mu T^{\lambda\nu} - x^\nu T^{\lambda\mu} + S^{\lambda,\mu\nu},$$3 (Weickgenannt et al., 2022). In electromagnetic backgrounds, a mixed KG pseudo-gauge for matter combined with a Belinfante shift for the electromagnetic field yields a manifestly gauge-invariant splitting

$$J^{\lambda,\mu\nu} = x^\mu T^{\lambda\nu} - x^\nu T^{\lambda\mu} + S^{\lambda,\mu\nu},$$4

with Lorentz-force and spin-precession equations following for the matter sector (Weickgenannt et al., 2022).

These constructions exhibit the main structural feature of pseudo-gauge freedom: several locally inequivalent tensor decompositions can encode the same total $$J^{\lambda,\mu\nu} = x^\mu T^{\lambda\nu} - x^\nu T^{\lambda\mu} + S^{\lambda,\mu\nu},$$5 and $$J^{\lambda,\mu\nu} = x^\mu T^{\lambda\nu} - x^\nu T^{\lambda\mu} + S^{\lambda,\mu\nu},$$6.

3. Pseudo-gauge-invariant local equilibrium

A central recent development is the construction of a local thermodynamic equilibrium density operator that is itself pseudo-gauge invariant. Seeking a density operator of the form

$$J^{\lambda,\mu\nu} = x^\mu T^{\lambda\nu} - x^\nu T^{\lambda\mu} + S^{\lambda,\mu\nu},$$7

linear in $$J^{\lambda,\mu\nu} = x^\mu T^{\lambda\nu} - x^\nu T^{\lambda\mu} + S^{\lambda,\mu\nu},$$8 and $$J^{\lambda,\mu\nu} = x^\mu T^{\lambda\nu} - x^\nu T^{\lambda\mu} + S^{\lambda,\mu\nu},$$9, one considers the ansatz

$\Phi^{\alpha,\mu\nu}=\Phi^{\alpha,[\mu\nu]}$0

Requiring invariance under arbitrary $\Phi^{\alpha,\mu\nu}=\Phi^{\alpha,[\mu\nu]}$1 fixes

$\Phi^{\alpha,\mu\nu}=\Phi^{\alpha,[\mu\nu]}$2

Identifying $\Phi^{\alpha,\mu\nu}=\Phi^{\alpha,[\mu\nu]}$3, with $\Phi^{\alpha,\mu\nu}=\Phi^{\alpha,[\mu\nu]}$4 the inverse-temperature four-vector, and defining

$\Phi^{\alpha,\mu\nu}=\Phi^{\alpha,[\mu\nu]}$5

one obtains the unique pseudo-gauge-invariant local-equilibrium operator

$\Phi^{\alpha,\mu\nu}=\Phi^{\alpha,[\mu\nu]}$6

At global equilibrium, $\Phi^{\alpha,\mu\nu}=\Phi^{\alpha,[\mu\nu]}$7 is a Killing field, $\Phi^{\alpha,\mu\nu}=\Phi^{\alpha,[\mu\nu]}$8, and the operator reduces to

$\Phi^{\alpha,\mu\nu}=\Phi^{\alpha,[\mu\nu]}$9

A naïve addition of a conserved charge term,

$$S'^{\lambda,\mu\nu}=S^{\lambda,\mu\nu}-\Phi^{\lambda,\mu\nu},$$0

is not in general pseudo-gauge invariant unless $$S'^{\lambda,\mu\nu}=S^{\lambda,\mu\nu}-\Phi^{\lambda,\mu\nu},$$1 is constant or $$S'^{\lambda,\mu\nu}=S^{\lambda,\mu\nu}-\Phi^{\lambda,\mu\nu},$$2 has a suitable improvement that drops out of the hypersurface integral (Becattini et al., 12 Jul 2025).

The principal consequence is that expectation values

$$S'^{\lambda,\mu\nu}=S^{\lambda,\mu\nu}-\Phi^{\lambda,\mu\nu},$$3

become independent of the chosen split between orbital and spin angular momentum. Ambiguities in local energy flow and spin density drop out, and for all local operators built from fields and their derivatives one obtains exactly the same result as in the Belinfante pseudo-gauge. In the Belinfante frame, where $$S'^{\lambda,\mu\nu}=S^{\lambda,\mu\nu}-\Phi^{\lambda,\mu\nu},$$4 and $$S'^{\lambda,\mu\nu}=S^{\lambda,\mu\nu}-\Phi^{\lambda,\mu\nu},$$5 is symmetric,

$$S'^{\lambda,\mu\nu}=S^{\lambda,\mu\nu}-\Phi^{\lambda,\mu\nu},$$6

yet the correlators coincide with those computed from the full pseudo-gauge-invariant operator (Becattini et al., 12 Jul 2025).

This construction is directly motivated by high-energy nuclear collisions. After a short time $$S'^{\lambda,\mu\nu}=S^{\lambda,\mu\nu}-\Phi^{\lambda,\mu\nu},$$7, the system is often assumed to have locally thermalized on the hyperboloid $$S'^{\lambda,\mu\nu}=S^{\lambda,\mu\nu}-\Phi^{\lambda,\mu\nu},$$8. Replacing the true pseudo-gauge-independent quantum state by $$S'^{\lambda,\mu\nu}=S^{\lambda,\mu\nu}-\Phi^{\lambda,\mu\nu},$$9 on $$T'^{\mu\nu} = T^{\mu\nu} +\frac12\partial_{\alpha} \Bigl( \Phi^{\alpha,\mu\nu} +\Phi^{\mu,\nu\alpha} +\Phi^{\nu,\mu\alpha} \Bigr).$$0 removes the unphysical dependence of spin-polarization observables on the choice of spin tensor. As a corollary, mean spin-polarization calculations performed in the Belinfante pseudo-gauge are identified as the unique physically correct result (Becattini et al., 12 Jul 2025).

4. Thermodynamic pseudo-gauges and second-order hydrodynamics

In relativistic hydrodynamics with spin, one introduces the spin chemical potential $$T'^{\mu\nu} = T^{\mu\nu} +\frac12\partial_{\alpha} \Bigl( \Phi^{\alpha,\mu\nu} +\Phi^{\mu,\nu\alpha} +\Phi^{\nu,\mu\alpha} \Bigr).$$1 and the free-energy current

$$T'^{\mu\nu} = T^{\mu\nu} +\frac12\partial_{\alpha} \Bigl( \Phi^{\alpha,\mu\nu} +\Phi^{\mu,\nu\alpha} +\Phi^{\nu,\mu\alpha} \Bigr).$$2

Ideal spin hydrodynamics posits

$$T'^{\mu\nu} = T^{\mu\nu} +\frac12\partial_{\alpha} \Bigl( \Phi^{\alpha,\mu\nu} +\Phi^{\mu,\nu\alpha} +\Phi^{\nu,\mu\alpha} \Bigr).$$3

with constitutive relations

$$T'^{\mu\nu} = T^{\mu\nu} +\frac12\partial_{\alpha} \Bigl( \Phi^{\alpha,\mu\nu} +\Phi^{\mu,\nu\alpha} +\Phi^{\nu,\mu\alpha} \Bigr).$$4

and thermodynamic identities

$$T'^{\mu\nu} = T^{\mu\nu} +\frac12\partial_{\alpha} \Bigl( \Phi^{\alpha,\mu\nu} +\Phi^{\mu,\nu\alpha} +\Phi^{\nu,\mu\alpha} \Bigr).$$5

However, if one computes improved currents $$T'^{\mu\nu} = T^{\mu\nu} +\frac12\partial_{\alpha} \Bigl( \Phi^{\alpha,\mu\nu} +\Phi^{\mu,\nu\alpha} +\Phi^{\nu,\mu\alpha} \Bigr).$$6 in an arbitrary pseudo-gauge and then defines the rest-frame densities from $$T'^{\mu\nu} = T^{\mu\nu} +\frac12\partial_{\alpha} \Bigl( \Phi^{\alpha,\mu\nu} +\Phi^{\mu,\nu\alpha} +\Phi^{\nu,\mu\alpha} \Bigr).$$7, the standard first-law and Gibbs–Duhem relations need not hold. This leads to the notion of a smaller family of “thermodynamic” pseudo-gauges in which the ideal-fluid form is retained (Armas et al., 20 Jan 2026).

At quadratic order in spin, these thermodynamic pseudo-gauges are parameterized by an arbitrary function $$T'^{\mu\nu} = T^{\mu\nu} +\frac12\partial_{\alpha} \Bigl( \Phi^{\alpha,\mu\nu} +\Phi^{\mu,\nu\alpha} +\Phi^{\nu,\mu\alpha} \Bigr).$$8. Under such a shift, the pressure transforms as

$$T'^{\mu\nu} = T^{\mu\nu} +\frac12\partial_{\alpha} \Bigl( \Phi^{\alpha,\mu\nu} +\Phi^{\mu,\nu\alpha} +\Phi^{\nu,\mu\alpha} \Bigr).$$9

Thus the spin equation of state retains a residual one-parameter ambiguity even within the thermodynamic class; in conformal theories this ambiguity is fixed by scale invariance. By contrast, combinations such as

$$P^\nu=\int_\Sigma d\Sigma_\mu\,T^{\mu\nu}, \qquad J^{\mu\nu}=\int_\Sigma d\Sigma_\lambda\,J^{\lambda,\mu\nu}$$0

are pseudo-gauge invariant. Explicit free-field examples were worked out for free massless Dirac fermions and scalar fields (Armas et al., 20 Jan 2026).

A complementary hydrodynamic result is that extending hydrodynamics by a spin variable is equivalent to modifying conventional symmetric hydrodynamics by a set of non-dissipative, second-order terms. In a parity-even sector one may write

$$P^\nu=\int_\Sigma d\Sigma_\mu\,T^{\mu\nu}, \qquad J^{\mu\nu}=\int_\Sigma d\Sigma_\lambda\,J^{\lambda,\mu\nu}$$1

$$P^\nu=\int_\Sigma d\Sigma_\mu\,T^{\mu\nu}, \qquad J^{\mu\nu}=\int_\Sigma d\Sigma_\lambda\,J^{\lambda,\mu\nu}$$2

$$P^\nu=\int_\Sigma d\Sigma_\mu\,T^{\mu\nu}, \qquad J^{\mu\nu}=\int_\Sigma d\Sigma_\lambda\,J^{\lambda,\mu\nu}$$3

The second law requires five entropy-production coefficients $$P^\nu=\int_\Sigma d\Sigma_\mu\,T^{\mu\nu}, \qquad J^{\mu\nu}=\int_\Sigma d\Sigma_\lambda\,J^{\lambda,\mu\nu}$$4 to vanish, which reduces the seven transport functions $$P^\nu=\int_\Sigma d\Sigma_\mu\,T^{\mu\nu}, \qquad J^{\mu\nu}=\int_\Sigma d\Sigma_\lambda\,J^{\lambda,\mu\nu}$$5 to two independent ones. These may be taken as $$P^\nu=\int_\Sigma d\Sigma_\mu\,T^{\mu\nu}, \qquad J^{\mu\nu}=\int_\Sigma d\Sigma_\lambda\,J^{\lambda,\mu\nu}$$6 and $$P^\nu=\int_\Sigma d\Sigma_\mu\,T^{\mu\nu}, \qquad J^{\mu\nu}=\int_\Sigma d\Sigma_\lambda\,J^{\lambda,\mu\nu}$$7, equivalently $$P^\nu=\int_\Sigma d\Sigma_\mu\,T^{\mu\nu}, \qquad J^{\mu\nu}=\int_\Sigma d\Sigma_\lambda\,J^{\lambda,\mu\nu}$$8 and the spin susceptibility $$P^\nu=\int_\Sigma d\Sigma_\mu\,T^{\mu\nu}, \qquad J^{\mu\nu}=\int_\Sigma d\Sigma_\lambda\,J^{\lambda,\mu\nu}$$9. In the same framework one obtains the nondissipative heat-current relation

$Z^{\mu\nu,\lambda\rho}$0

which is described as a vorticity-driven thermal Hall effect in the no-drag frame (Li et al., 2020).

5. Dynamical constraints and symmetry-restricted residual freedom

Classical pseudo-gauge transformations in hydrodynamic models can be analyzed by decomposing the superpotential $Z^{\mu\nu,\lambda\rho}$1 into Lorentz-invariant pieces relative to the fluid velocity $Z^{\mu\nu,\lambda\rho}$2. Writing

$Z^{\mu\nu,\lambda\rho}$3

and separating a fully transverse part

$Z^{\mu\nu,\lambda\rho}$4

one arrives at a decomposition with $Z^{\mu\nu,\lambda\rho}$5 independent components, matching a general $Z^{\mu\nu,\lambda\rho}$6 (Drogosz et al., 2024).

If both the original and transformed energy–momentum tensors are required to be symmetric, then the superpotential must satisfy

$Z^{\mu\nu,\lambda\rho}$7

the STS condition. When $Z^{\mu\nu,\lambda\rho}$8 is constructed only from hydrodynamic fields $Z^{\mu\nu,\lambda\rho}$9 and without higher gradients, the analysis implies that essentially only

$$Z^{\mu\nu,\lambda\rho} = -\,Z^{\nu\mu,\lambda\rho} = -\,Z^{\mu\nu,\rho\lambda},$$0

survives. Even then, the six independent components of the STS condition become six equations for one scalar function $$Z^{\mu\nu,\lambda\rho} = -\,Z^{\nu\mu,\lambda\rho} = -\,Z^{\mu\nu,\rho\lambda},$$1, which generically has no solution unless further symmetries are present (Drogosz et al., 2024).

A special case is $$Z^{\mu\nu,\lambda\rho} = -\,Z^{\nu\mu,\lambda\rho} = -\,Z^{\mu\nu,\rho\lambda},$$2-dimensional boost-invariant Bjorken flow, where $$Z^{\mu\nu,\lambda\rho} = -\,Z^{\nu\mu,\lambda\rho} = -\,Z^{\mu\nu,\rho\lambda},$$3 holds identically for any function of proper time. The residual pseudo-gauge transformation is

$$Z^{\mu\nu,\lambda\rho} = -\,Z^{\nu\mu,\lambda\rho} = -\,Z^{\mu\nu,\rho\lambda},$$4

For the hydrodynamic tensor

$$Z^{\mu\nu,\lambda\rho} = -\,Z^{\nu\mu,\lambda\rho} = -\,Z^{\mu\nu,\rho\lambda},$$5

this residual transformation shifts the viscosities according to

$$Z^{\mu\nu,\lambda\rho} = -\,Z^{\nu\mu,\lambda\rho} = -\,Z^{\mu\nu,\rho\lambda},$$6

while preserving the combination

$$Z^{\mu\nu,\lambda\rho} = -\,Z^{\nu\mu,\lambda\rho} = -\,Z^{\mu\nu,\rho\lambda},$$7

Hence the bulk and shear viscosity coefficients are pseudo-gauge dependent in this symmetry class, whereas the linear combination entering the Bjorken equation of motion is invariant (Drogosz et al., 2024).

This suggests that pseudo-gauge transformations can alter local transport parametrizations without modifying the hydrodynamic structure that actually governs the evolution.

6. Operator algebra, kinetic theory, and broader usage

Pseudo-gauge freedom is not innocuous at the level of local spin operators. For equal-time spin generators

$$Z^{\mu\nu,\lambda\rho} = -\,Z^{\nu\mu,\lambda\rho} = -\,Z^{\mu\nu,\rho\lambda},$$8

the canonical spin density yields

$$Z^{\mu\nu,\lambda\rho} = -\,Z^{\nu\mu,\lambda\rho} = -\,Z^{\mu\nu,\rho\lambda},$$9

By contrast, in the GLW and HW gauges the transformed spin operators acquire improvement terms whose commutators do not vanish in general, so the resulting operators fail to satisfy the $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$00 algebra. This is the basis for the conclusion that only the canonical spin tensor defines bona fide spin-angular-momentum operators. At the same time, polarization vectors constructed through the Pauli–Lubanski vector can still be pseudogauge-independent, and GLW/HW remain useful in classical or semiclassical settings or whenever one prefers a conserved spin tensor (Dey et al., 2023).

In Wigner-function kinetic theory, different pseudo-gauges correspond to different reorganizations of the microscopic Clifford components $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$01, $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$02, $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$03, and related quantities into energy–momentum and spin densities. For interacting Dirac and Proca fields, the gradient-expanded Boltzmann equation

$S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$04

shows that nonlocal collisions mix orbital and spin degrees of freedom and spoil naïve spin conservation in the canonical pseudo-gauge, while HW, GLW, or KG-type choices absorb the same transfer differently. In global equilibrium one recovers a thermal-vorticity-driven polarization with

$S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$05

leading to the familiar equilibrium spin tensor $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$06 (Weickgenannt et al., 2022).

The term “pseudo-gauge” also appears in a mathematically distinct setting: time-dependent diffeomorphisms $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$07 acting on autonomous ordinary differential equations. In that usage, the transformed system is

$S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$08

and the linear gauge transformation $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$09 is the special case $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$10. Solution correspondence, symmetry transfer, and transformation of first integrals persist, but the identification problem becomes a PDE for $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$11 rather than the matrix ODE $S^{\lambda,\mu\nu}=S^{\lambda,[\mu\nu]}$12 of the linear case (Gaeta et al., 8 Jun 2025).

Across these contexts, pseudo-gauge transformations encode the same abstract pattern: they preserve the underlying dynamical or conserved content while reshuffling its local representation. In relativistic spin physics, the contemporary emphasis has shifted from the mere existence of this freedom to the identification of invariant quantities, admissible thermodynamic gauges, and operator constructions that remain physically meaningful under it.