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Gauge-Related States Are Physically Distinct: A Reductio ad Absurdum

Shan Gao

Published Aug 20, 2026• 0 citations
OpenAlex

Abstract

Gauge theories are conventionally interpreted by identifying gauge-related configurations as one and the same physical state. We argue that this identification is untenable. The key distinction is between dynamics and physical identity: a dynamics determines how configurations evolve, whereas physical identity, an ontological relation between configurations, is independent of that dynamics. We apply this principle to the local QED transformation $(A_\mu,\psi)\rightarrow(A_\mu-\partial_\mu\Lambda,e^{ie\Lambda/\hbar}\psi).$ Although this transformation is a symmetry of the massless electromagnetic dynamics, a dynamics on the same configuration variables containing a gauge-field mass term distinguishes the two configurations. If they were one and the same physical state, their identity could not depend on this change of dynamics; hence the gauge-related configurations cannot be identified as one physical state. The Higgs mechanism does not change the conclusion: enlarging the configuration space restores gauge covariance without establishing gauge redundancy. The result extends to non-Abelian gauge theories and, with diffeomorphisms replacing gauge transformations, to general relativity. Gauge covariance is thus a symmetry of the equations, not a principle of physical identity. Gauge-related configurations are physically distinct, and the electromagnetic potential $A_\mu$ should consequently be regarded as a real physical field rather than a mathematical redundancy.

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•Aug 23, 2026•0 citations
Gauge theories are conventionally interpreted by identifying gauge-related configurations as one and the same physical state. We argue that this identification is untenable. The key distinction is between dynamics and physical identity: a dynamics determines how configurations evolve, whereas physical identity, an ontological relation between configurations, is independent of that dynamics. We apply this principle to the local QED transformation (Aμ,ψ)→(Aμ−∂μΛ,eieΛ/ℏψ).(A_\mu,\psi)\rightarrow(A_\mu-\partial_\mu\Lambda,e^{ie\Lambda/\hbar}\psi).(Aμ​,ψ)→(Aμ​−∂μ​Λ,eieΛ/ℏψ). Although this transformation is a symmetry of the massless electromagnetic dynamics, a dynamics on the same configuration variables containing a gauge-field mass term distinguishes the two configurations. If they were one and the same physical state, their identity could not depend on this change of dynamics; hence the gauge-related configurations cannot be identified as one physical state. The Higgs mechanism does not change the conclusion: enlarging the configuration space restores gauge covariance without establishing gauge redundancy. The result extends to non-Abelian gauge theories and, with diffeomorphisms replacing gauge transformations, to general relativity. Gauge covariance is thus a symmetry of the equations, not a principle of physical identity. Gauge-related configurations are physically distinct, and the electromagnetic potential AμA_\muAμ​ should consequently be regarded as a real physical field rather than a mathematical redundancy.
Gauge invariance is conventionally interpreted as a fundamental redundancy of description, implying that gauge potentials represent the same physical situation and that only gauge-invariant quantities, such as the electromagnetic field strengths and closed-loop holonomies, are physically observable. Within this orthodox view, the gauge potential AμA_\muAμ​ in quantum electrodynamics possesses no independent ontological status. In this paper we re-examine this interpretation by treating the electromagnetic source in the Aharonov--Bohm (AB) effect as a fully quantized dynamical system whose backreaction on the test particle is explicitly included. We show that, once source dynamics are incorporated, the standard proof of gauge invariance no longer applies. The observable AB phase acquires an additional contribution that can be expressed as the sum of two open-path integrals of the gauge potential and is generally gauge dependent. Since this contribution affects measurable interference, it must correspond to a unique physical quantity, suggesting that the gauge potential itself is physically real rather than merely a redundant mathematical representation. We then argue that the Lorenz condition, ∂μAμ=0\partial_\mu A^\mu = 0∂μ​Aμ=0, should be interpreted not as a gauge choice but as a fundamental physical constraint which, together with the appropriate initial and boundary conditions, uniquely determines the gauge potential. Within this framework, the open-path relative phase of a charged particle, ∫γ1Aμdxμ−∫γ2Aμdxμ\int_{\gamma_1} A_\mu dx^\mu - \int_{\gamma_2} A_\mu dx^\mu∫γ1​​Aμ​dxμ−∫γ2​​Aμ​dxμ, becomes a well-defined observable, even though it is formally gauge dependent under transformations that do not preserve the Lorenz condition or boundary conditions. To establish the operational significance of this quantity, we propose a Spatial-to-Internal Phase Transfer (SIPT) protocol that coherently transfers the open-path phase of a single fermion into localized internal quantum memories, allowing the phase to be reconstructed through local correlation measurements without recombining the original spatial branches. We discuss possible implementations based on atomic electron capture and semiconductor--superconductor hybrid systems, and argue that such an experiment would constitute an \emph{experimentum crucis}, analogous to the Michelson--Morley experiment, capable of distinguishing experimentally between gauge redundancy and gauge-potential realism. Finally, we show that the conventional gauge-fixed formulation of quantum electrodynamics is mathematically equivalent to a formulation in which the Lorenz condition is imposed as a fundamental law. We further argue that local gauge invariance of the interaction follows from current conservation and the structure of the matter–field coupling rather than representing an independent physical symmetry, and that the gauge-parameter independence of physical amplitudes does not uniquely establish gauge redundancy. These results call for a reexamination of the conceptual foundations of quantum gauge theories and open the door to experimental tests that could transform our understanding of the ontological status of gauge potentials.
It is a foundational principle of quantum electrodynamics (QED) that local gauge transformations are redundancies of description, leaving all physical observables invariant. We show that this principle fails for finite-time, finite-space quantum interference processes. Using the exact path-integral treatment of the Aharonov–Bohm effect with a quantized source, we prove that the relative phase between the two interfering paths acquires a gauge-dependent contribution when the source backreacts and develops branch-dependent charge distributions. Its gauge variation reduces to a boundary term involving the gauge function evaluated at the source and detector points, weighted by the environment's branch density difference. Consequently, physical predictions for finite-time quantum interference are generically gauge dependent. The only way to obtain unique predictions is to fix a specific gauge; the causally retarded Lorenz gauge emerges as the unique physically preferred choice. Thus the gauge potential AμA_\muAμ​ in the preferred gauge must be regarded as a real, physical field.
The distinction between gauge independence and gauge covariance in electrodynamics is considered for quantities of physical interest. The gauge independence of directly observable quantities, such as the spectra of operators and energy conserving transition rates, is discussed and contrasted with the gauge covariance of other quantities such as scalar products. The quasigauge-invariance of the classical Lagrangian and the invariance of the Euler-Lagrange equations under the addition of a total time derivative to the Lagrangian are discussed. The consequent inherent nonuniqueness of the Lagrangian and Hamiltonian formulations of classical mechanics is pointed out. A physical interpretation of the explicit gauge dependence of classical canonical momenta and of the expectation values of the corresponding quantum mechanical operators is presented. The gauge independence of energy conserving transition rates calculated in the conventional finite-order time dependent perturbation theory is discussed and illustrated. The gauge dependence of the conventional time dependent transition amplitudes in the presence of electromagnetic fields is then discussed, and gauge independent transition amplitudes are constructed. An alternative formulation of the quantum mechanics of charged particles is obtained in terms of a new gauge independent Hamiltonian, which is, as might be expected, unique only within an arbitrary canonical transformation.
Quantum theory, gauge fields, and general relativity traditionally rest on independent foundations. This paper argues that they share a single structural origin when the Self-Consistent Reference Frame Principle (RCC) and causality are applied to a local Hilbert-space framework. The unifying object is the comparator U_{ij}, which translates states between sites. In gauge theory it becomes the parallel transporter yielding the Yang-Mills action; in quantum mechanics it forces the Born rule through a functional equation without Gleason's theorem; in gravity it becomes the Levi-Civita connection giving the Einstein-Hilbert action via Lovelock's theorem. Among the ten theorems and two lemmas, the Phase Unity Theorem establishes that the comparator Schur phase, the gauge transformation parameter, and the quantum global phase are structural correspondences of a single U(1) symmetry. The Coleman-Mandula separation of internal and spacetime symmetries follows from RCC as a consequence of distinct encoding schemes. The Higgs mechanism emerges as a corollary: RCC prohibits bare mass terms for gauge fields. The framework yields one parameter-free falsifiable prediction: a pattern of SMEFT Wilson coefficients testable at colliders.
Quantum theory, gauge fields, and general relativity traditionally rest on independent foundations. This paper argues that they can be understood within a unified logical framework when the Self-Consistent Reference Frame Principle (RCC) and causality are applied to a local Hilbert-space framework. The unifying object is the comparator U_{ij}, which translates states between sites. In gauge theory it becomes the parallel transporter yielding the Yang-Mills action; in quantum mechanics it forces the Born rule through a functional equation without Gleason's theorem; in gravity it becomes the Levi-Civita connection giving the Einstein-Hilbert action via Lovelock's theorem. Among the nine theorems and two lemmas, the Phase Unity Theorem establishes that the comparator Schur phase, the gauge transformation parameter, and the quantum global phase are structural correspondences of a single U(1) symmetry. The Coleman-Mandula separation of internal and spacetime symmetries follows from RCC as a consequence of distinct encoding schemes. The Higgs mechanism emerges as a corollary: RCC prohibits bare mass terms for gauge fields. The framework yields one parameter-free falsifiable prediction: a pattern of SMEFT Wilson coefficients that can be tested at colliders, albeit with current precision limitations.
Quantum state tomography aims to reconstruct the quantum state of a system, including its phase and coherence information. In electron tomography experiments, the reconstructed object is understood as the quantum state of the electron. However, for charged particles, the bare electron state is gauge dependent: ψ(x)→eieΛ(x)/ℏψ(x)\psi(\mathbf{x})\rightarrow e^{ie\Lambda(\mathbf{x})/\hbar}\psi(\mathbf{x})ψ(x)→eieΛ(x)/ℏψ(x). This raises a fundamental question: how can an experiment reconstruct a gauge-dependent quantum state? One possible resolution is that tomography does not reconstruct the bare electron state but instead a gauge-invariant dressed electron state, ΨD=W[A]ψ\Psi_D=W[A]\psiΨD​=W[A]ψ, where the electromagnetic dressing compensates the gauge transformation of the electron field. In this paper, we examine this alternative possibility and show that it is not viable. We prove a structural no-go theorem showing that measurement interactions generated by the QED current coupling cannot directly measure a gauge-invariant dressed electron state. We further show that although interpreting tomography as the reconstruction of the reduced electron state of a dressed electron is possible, the dressed-electron interpretation fails in the Aharonov--Bohm effect: the dressed formalism can reproduce the observed phase only as a mathematical redefinition, not as a physically dynamical mechanism. Consequently, the quantum state reconstructed in electron tomography is indeed the bare electron state, as assumed in the standard interpretation of these experiments. The experimental reconstruction of this gauge-dependent quantum state implies that the electromagnetic potential entering the quantum dynamics cannot be regarded as merely a conventional mathematical artifact. Rather, it must correspond to a real physical configuration uniquely determined by the complete physical laws governing the electromagnetic field, such as Maxwell's equations together with the Lorenz condition, and by the appropriate boundary and causal conditions. This provides empirical support for the reality of the electromagnetic potential AμA_\muAμ​. The same reasoning suggests an analogous conclusion for gravity, where the reconstruction of bare quantum states supports the reality of the gravitational field gμνg_{\mu\nu}gμν​.