Pseudo-Gauge Ambiguity in Field & Gauge Theories

• Pseudo-Gauge Ambiguity is the freedom to redefine local quantities, like energy–momentum and spin tensors, by adding total derivative terms while preserving global conservation laws.
• It manifests across gauge theories, quantum field theory, condensed matter, and cosmology, altering local distributions and the interpretation of observables.
• This ambiguity affects operator structures and impacts measurements of energy density, pressure, and spin, with implications for both microscopic and macroscopic analyses.

A pseudo-gauge ambiguity arises when the physical or mathematical description of a system possesses a freedom to redefine local quantities (such as symmetry generators, energy–momentum tensors, or conserved currents) in a way that preserves the global conservation laws, but alters the local structure and potentially the interpretation of physical observables. This concept manifests in diverse contexts, from gauge theories and quantum field theory (QFT) to condensed matter systems and cosmology. In the modern usage, "pseudo-gauge ambiguity" refers both to technical ambiguities driven by gauge fixing or local redefinitions and to operator redefinitions (often via superpotentials) that reshuffle the distribution of quantities such as energy, spin, or angular momentum while maintaining the relevant conservation laws.

1. Fundamental Principles of Pseudo-Gauge Ambiguity

Pseudo-gauge ambiguity is rooted in the mathematical freedom to redefine local objects in a theory without changing the integrated (global) physical quantities. In gauge theories, this originates from the redundancy in the field description due to gauge invariance. For example, in the specification of a gauge field $A_\mu$, certain degrees of freedom can be shifted by gradients or local transformations without altering gauge-invariant observables. More generally, in any theory with conserved quantities (e.g., energy–momentum, angular momentum), the basic tensors describing these quantities can be modified by adding total derivatives or divergences of "superpotential" tensors. These transformations form a group—the so-called "pseudo-gauge freedom"—by which the canonical Noether currents are not uniquely determined.

The archetypal example is the ambiguity in the energy–momentum tensor $T^{\mu\nu}$ arising from the addition of the divergence of an antisymmetric tensor $A^{\nu\mu\lambda}$: $$T'^{\mu\nu} = T^{\mu\nu} + \partial_\lambda A^{\nu\mu\lambda}, \qquad A^{\nu\mu\lambda} = -A^{\nu\lambda\mu}$$ Such ambiguity leaves all global conservation laws intact ($\int d^3x\, T^{0i}$ is invariant), but changes the local description of energy or momentum density.

2. Manifestations in Gauge Theory: Remnant Symmetries and Order Parameters

Pseudo-gauge ambiguity plays a central role in interpreting symmetry breaking in gauge theories, particularly in theories with local gauge invariance and spontaneous symmetry breaking. Elitzur’s theorem forbids the spontaneous breaking of local gauge symmetries; however, after gauge fixing (e.g., Landau or Coulomb gauge), a remnant global subgroup may remain unfixed. Order parameters sensitive to the breaking of these subgroups may signal "gauge symmetry breaking", but the location of this breaking in parameter space is ambiguous and depends on the particular subgroup selected by the gauge choice (0712.0999).

For example, in an SU(2) gauge–Higgs system:

• Landau gauge leaves a remnant global symmetry related to spacetime-constant and linear transformations, with an order parameter $$Q_L = \mathrm{Tr}[(1/V\sum_x\phi(x))(1/V\sum_x\phi(x))^{\dagger}]$$.
• Coulomb gauge leaves a larger remnant symmetry (space-independent, time-dependent), monitored by $$Q_C = \frac{1}{L_t}\sum_t \mathrm{Tr}[(1/V_3\sum_{\vec{x}}U_0(x,t))(1/V_3\sum_{\vec{x}}U_0(x,t))^{\dagger}]$$.

These order parameters turn nonzero at different points in coupling space, demonstrating that the actual phase boundary ("confinement" versus "Higgs") is not universally marked by remnant symmetry breaking. This constitutes a pseudo-gauge ambiguity: the physical significance of "gauge symmetry breaking" depends on the definition of the remnant subgroup.

3. Pseudo-Gauge Ambiguity in Field Theory: Energy-Momentum and Spin Tensors

The ambiguity in defining local conserved quantities is illustrated by the freedom in constructing the energy–momentum tensor (EMT) and associated spin tensors (Dey et al., 2023). The canonical EMT, derived from Noether’s theorem, is not invariant under pseudo-gauge transformations: $$T'^{\mu\nu} = T^{\mu\nu} + \frac{1}{2}\partial_\lambda[\Phi^{\lambda\mu\nu} + \Phi^{\mu\lambda\nu} + \Phi^{\nu\lambda\mu}]$$ Where $\Phi^{\lambda\mu\nu}$ is antisymmetric in the last two indices, and associated spin tensor is redefined as: $$S'^{\lambda,\mu\nu} = S^{\lambda,\mu\nu} - \Phi^{\lambda,\mu\nu} + \partial_\rho Z^{\mu\nu,\lambda\rho}$$ While global angular momentum and energy–momentum are conserved independently of the pseudo–gauge choice, the local densities and operator algebra (e.g., SO(3) for spin) do depend on the choice. Notably, the canonical spin operator satisfies the correct SO(3) algebra, whereas alternative choices may spoil this property, making the canonical definition preferable for connecting theory with experiment for spin-polarization observables.

4. Impact on Local Observables and Physical Interpretations

Pseudo-gauge ambiguity critically affects local observables such as energy density and pressure distributions, especially in composite systems. For example, in nucleon structure calculations within the Skyrme model with vector mesons, the spatial distributions of pressure $p(r)$, energy density $\varepsilon(r)$, and shear force $s(r)$ are found to differ between the canonical and Belinfante EMTs (Fukushima et al., 12 Sep 2025). The Belinfante form regularizes the central singularities present in the canonical distributions:

• $p_\text{can}(r) \sim r^{-2}$ as $r \to 0$ (singular),
• $p_\text{Bel}(r) \sim \text{finite constant}$.

Consequently, inferred quantities such as the confining force $f_\text{confining}(r) = -2 s(r)/r$ and the nucleon equation of state $p(\varepsilon)$ are pseudo–gauge dependent, complicating the physical interpretation derived from experimental data such as generalized parton distributions.

A plausible implication is that, without a unique principle for selecting the correct pseudo–gauge, the internal mechanical structure of composite objects cannot be uniquely determined from EMT form factors alone.

5. Quantum Fluctuations, Thermodynamics, and Coarse Graining

Pseudo-gauge ambiguity manifests in quantum statistical observables as well, specifically in fluctuations. For a hot relativistic gas of spin-$1/2$ fermions, the variance of energy in finite subsystems depends on the choice of pseudo–gauge for $T^{\mu\nu}$ (Das et al., 2021): $$T'^{\mu\nu} = T^{\mu\nu} + \partial_\lambda A^{\nu\mu\lambda}$$ The expectation value $\langle T^{00} \rangle$ is invariant, but variances such as $$\sigma^2_a = \langle:T^{00}_a::T^{00}_a:\rangle - \langle:T^{00}_a:\rangle^2$$ diverge for different pseudo–gauges. As the subsystem size $a$ increases (coarse graining), results converge to the canonical–ensemble limit and the ambiguity becomes irrelevant; in the quantum/small-$a$ regime, the ambiguity is manifest and must be considered in interpreting quantum fluctuations.

6. Dynamical Constraints and Hydrodynamic Models

Pseudo-gauge transformations impose constraints in hydrodynamics, particularly when connecting symmetric energy–momentum tensors. The superpotential $\Phi^{\lambda\mu\nu}$ must satisfy the conservation law (STS condition) (Drogosz et al., 9 Nov 2024): $\partial_\lambda \Phi^{\lambda\mu\nu} = 0$ In general, this is difficult to satisfy except in highly symmetric flows such as boost–invariant (Bjorken) flow, where a residual pseudo–gauge transformation survives and bulk/shear viscosity coefficients $\eta$, $\zeta$ become pseudo–gauge dependent. However, the specific linear combination $(4/3)\eta + \zeta$ entering the hydrodynamic equations of motion remains invariant. This suggests that pseudo–gauge ambiguity may affect microscopic interpretations of transport coefficients but not the macroscopic evolution.

7. Strategies for Addressing Pseudo-Gauge Ambiguity

Several theoretical strategies have been proposed to mitigate or resolve pseudo–gauge ambiguity:

• Formulation of pseudo–gauge invariant operators for local thermodynamic equilibrium, incorporating thermal shear and thermal vorticity (Becattini et al., 12 Jul 2025). The revised density operator eliminates ambiguity in mean values of observables in local equilibrium, with the Belinfante pseudo–gauge emerging as a natural choice.
• Imposing physically motivated boundary conditions (such as requiring vanishing potentials at spatial infinity (Gritsunov, 2013)) or compactifying spatial manifolds to achieve unique gauge fixing and avoid ambiguity in quantization (Raval, 2016, Zhou et al., 2016).
• Careful partitioning of light/matter degrees of freedom in cavity QED, through explicit dual (C-field) formalism, ensuring gauge–independence of canonical momenta and the Hamiltonian in truncated systems (Rouse et al., 2020).

8. Pseudo-Gauge Ambiguity in Condensed Matter and Cosmology

In Dirac/Weyl materials, pseudo–gauge fields arise from non–electromagnetic sources (strain, magnetization), mimicking gauge–like couplings at low energy. These fields have no independent gauge invariance, leading to pseudo–gauge ambiguity in observables, such as valley–dependent responses, anomalous currents, and piezoelectric effects (Yu et al., 2021). In graphene, optical (sublattice relative) displacements screen pseudogauge fields and pseudomagnetic fields, revising predictions for strain-induced effects (Beule et al., 3 Sep 2024).

In cosmology, large coordinate transformations, such as conformal Fermi coordinates, introduce ambiguity in the scalar–vector–tensor decomposition, but final cosmological observables remain fully invariant if all relativistic contributions are included (Mitsou et al., 2022).

---

Pseudo-gauge ambiguity encapsulates a pervasive freedom in local redefinitions of symmetry generators, conserved tensors, and operators, intrinsic across quantum field theory, condensed matter, hydrodynamics, and cosmological modeling. While global conservation laws and invariant observables are robust against these ambiguities, local densities, operator commutation relations, quantum fluctuations, and inferred mechanical properties may be contingent upon the pseudo–gauge choice. Understanding, constraining, and, where possible, removing pseudo–gauge ambiguity is essential for reliable physical interpretation and for connecting theory to experimental measurements.