Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Reality and Measurability of Gauge Potentials

Shan Gao

Published Jul 22, 2026• 0 citations
OpenAlex

Abstract

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 \(A_\mu\) 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, $\partial_\mu A^\mu = 0$, 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, \(\int_{\gamma_1} A_\mu dx^\mu - \int_{\gamma_2} A_\mu dx^\mu\), 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.

Related papers

Does the Aharonov-Bohm Effect Imply the Reality of Electromagnetic Potentials?

Shan Gao

The Aharonov--Bohm (AB) phase for a closed interferometric loop is widely regarded as gauge invariant. This conclusion is usually obtained by treating the electromagnetic source as a fixed background that is unaffected by the test charge. We investigate whether this conclusion remains valid when backreaction is included within the standard gauge-fixed formulation of quantum electrodynamics (QED). Our analysis indicates that, when the interfering branches become entangled with distinct electromagnetic field configurations, changes of the gauge-fixing condition can induce a non-vanishing shift of the total phase. This result gives rise to an apparent paradox: the total phase predicted by the QED formalism appears gauge dependent, whereas experiments robustly exhibit a definite interference shift. We examine two possible resolutions of this paradox. The standard approach attributes the observed interference only to a gauge-invariant component of the total phase and treats the remaining gauge-dependent contribution as physically irrelevant. We argue, however, that the QED formalism itself predicts the total phase governing the interference outcome and does not provide a dynamical prescription that extracts a gauge-invariant component while discarding the remaining terms. We then explore an alternative perspective in which the electromagnetic potential $A_\mu$ is regarded as a physical field whose configuration is determined by the Lorenz condition together with appropriate causal and boundary conditions. From this viewpoint, the total phase can be interpreted as a well-defined physical quantity within the QED formalism. We conclude by discussing a possible experimental test based on direct measurements of open-path relative phases, which may help clarify the ontological status of electromagnetic potentials.

•Aug 6, 2026•0 citations

Quantum Evidence for the Reality of Electromagnetic Potentials and the Gravitational Field

Shan Gao

The ontological status of the fundamental fields appearing in gauge theories and general relativity has been one of the deepest conceptual questions in modern physics. In electromagnetism, the central issue concerns whether the electromagnetic potential \(A_\mu\), rather than only the gauge-invariant field strength \(F_{\mu\nu}\), represents a physically real structure. In general relativity, the analogous question concerns whether the gravitational field \(g_{\mu\nu}\), despite its dependence on coordinate choices, should be regarded as a physically real field or merely as a mathematical representation of invariant spacetime relations. In this paper, we present a comprehensive argument for the reality of both \(A_\mu\) and \(g_{\mu\nu}\). For the electromagnetic case, we show that when the source is treated as a fully quantized dynamical system with backreaction, the closed-loop Aharonov--Bohm phase acquires a gauge-dependent contribution that is the sum of two open-path integrals of \(A_\mu\). Since this phase contributes to observed interference patterns, it must correspond to a unique physical quantity, forcing the conclusion that \(A_\mu\) is a real physical field. We further argue that the Lorenz condition is the additonal physical law that, together with boundary conditions, determines \(A_\mu\) uniquely. For the gravitational case, we extend this analysis to both closed-loop and open-path phases. First, we show that closed-loop gravitational phases, such as those measured in the Colella-Overhauser-Werner (COW) experiment and the Overstreet et al. experiment, become gauge dependent under diffeomorphisms when source backreaction is properly accounted for. Second, we show that existing quantum optical experiments using local homodyne tomography already measure open-path relative phases of photons in spatially separated modes, which contain a gravitational contribution—the open-path gravitational phase. Since both closed-loop and open-path gravitational phases are gauge dependent under diffeomorphisms, their measurability entails the reality of the gravitational field \(g_{\mu\nu}\). We further argue that the harmonic (de Donder) gauge is the additional physical law that, together with boundary conditions, determines \(g_{\mu\nu}\) uniquely. These results establish that both electromagnetic potentials and the gravitational field are real physical fields, not mere mathematical auxiliaries. They challenge the conventional identification of gauge invariance with physical reality and have profound implications for non-Abelian gauge theories and quantum gravity.

•Jul 29, 2026•0 citations

Is Gauge Invariance a Universal Physical Principle? A Critical Test from Finite-Time QED

Shan Gao

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 \(A_\mu\) in the preferred gauge must be regarded as a real, physical field.

•Jul 7, 2026•0 citations

The Aharonov-Bohm Effect Explained: Reality of Gauge Potentials and Its Implications

Gao, Shan

The Aharonov-Bohm (AB) effect, wherein an electron acquires a phase shift in a field-free region due to electromagnetic potentials, poses a profound challenge to the ontology of quantum mechanics and gauge theories. This paper demonstrates that gauge-invariant explanations, which attribute the phase to measurable quantities, rely on nonlocal and discontinuous mechanism and fail to account for its continuous accumulation along the electron's path—a process evident in the generalized AB effect, where a time-varying magnetic flux induces a phase that builds gradually over time, as predicted by quantum mechanics. Through a new analysis spanning quantum mechanics and quantum electrodynamics, I argue that the electromagnetic potentials $A_\mu$, fixed in the Lorenz gauge, emerge as the fundamental physical reality, offering a local, relativistically consistent account of the phase’s generation. This exclusion of gauge-invariant paradigms reverberates across gauge theories: it redefines the Higgs mechanism, favoring dynamic potentials over static invariants, and extends to general relativity, where gravitational potentials $g_{\mu\nu}$ may anchor spacetime’s substantival reality via a gravitational AB effect. By unraveling the AB phase’s continuous generation—locally mediated by potentials—this study not only addresses a long-standing conundrum but also bridges electromagnetic, gravitational, and Yang-Mills frameworks, offering a unified potential-centric perspective with novel implications for physics and philosophy.

•Mar 16, 2025•0 citations

The Aharonov-Bohm Phase is Gauge Dependent, But Its Gauge-Invariant Core is Observable

Shan Gao

Contrary to long-standing textbook doctrine, we demonstrate that the Aharonov-Bohm (AB) phase for a closed interferometric loop — the relative phase governing interference — becomes gauge dependent when backreaction of the test charge on the electromagnetic source is included in a fully quantum treatment. The standard Stokes-theorem argument fails because different interfering paths become entangled with distinct field configurations, permitting branch-dependent gauge transformations. We prove that the total phase decomposes into (1) a gauge-invariant core tied to the fixed background flux, which fully accounts for the observed interference shift, and (2) a purely gauge-dependent backreaction term that can be eliminated by an appropriate gauge choice and carries no observable consequences. This decomposition follows from a general framework for observables in gauge theories, which we establish from a foundational axiom that purely gauge-dependent quantities have no observational consequences. From this axiom, we derive theorems showing that any gauge-dependent quantity contains an extractable gauge-invariant core that constitutes its measurable content, while the remaining gauge-dependent component is physically irrelevant. This framework resolves the apparent paradox between formal gauge dependence and robust experimental observations, and offers a conceptually sharper understanding of observables in quantum electrodynamics, non-abelian gauge theories, general relativity, and other gauge-based theories.

•Feb 5, 2026•0 citations

Quantum State Tomography Implies the Reality of the Electromagnetic Potential

Shan Gao

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: \(\psi(\mathbf{x})\rightarrow e^{ie\Lambda(\mathbf{x})/\hbar}\psi(\mathbf{x})\). 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, \(\Psi_D=W[A]\psi\), 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 \(A_\mu\). The same reasoning suggests an analogous conclusion for gravity, where the reconstruction of bare quantum states supports the reality of the gravitational field \(g_{\mu\nu}\).

•Aug 23, 2026•0 citations
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 (A_\mu) 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, (\int_{\gamma_1} A_\mu dx^\mu - \int_{\gamma_2} A_\mu dx^\mu), 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.
The Aharonov--Bohm (AB) phase for a closed interferometric loop is widely regarded as gauge invariant. This conclusion is usually obtained by treating the electromagnetic source as a fixed background that is unaffected by the test charge. We investigate whether this conclusion remains valid when backreaction is included within the standard gauge-fixed formulation of quantum electrodynamics (QED). Our analysis indicates that, when the interfering branches become entangled with distinct electromagnetic field configurations, changes of the gauge-fixing condition can induce a non-vanishing shift of the total phase. This result gives rise to an apparent paradox: the total phase predicted by the QED formalism appears gauge dependent, whereas experiments robustly exhibit a definite interference shift. We examine two possible resolutions of this paradox. The standard approach attributes the observed interference only to a gauge-invariant component of the total phase and treats the remaining gauge-dependent contribution as physically irrelevant. We argue, however, that the QED formalism itself predicts the total phase governing the interference outcome and does not provide a dynamical prescription that extracts a gauge-invariant component while discarding the remaining terms. We then explore an alternative perspective in which the electromagnetic potential AμA_\muAμ​ is regarded as a physical field whose configuration is determined by the Lorenz condition together with appropriate causal and boundary conditions. From this viewpoint, the total phase can be interpreted as a well-defined physical quantity within the QED formalism. We conclude by discussing a possible experimental test based on direct measurements of open-path relative phases, which may help clarify the ontological status of electromagnetic potentials.
The ontological status of the fundamental fields appearing in gauge theories and general relativity has been one of the deepest conceptual questions in modern physics. In electromagnetism, the central issue concerns whether the electromagnetic potential AμA_\muAμ​, rather than only the gauge-invariant field strength FμνF_{\mu\nu}Fμν​, represents a physically real structure. In general relativity, the analogous question concerns whether the gravitational field gμνg_{\mu\nu}gμν​, despite its dependence on coordinate choices, should be regarded as a physically real field or merely as a mathematical representation of invariant spacetime relations. In this paper, we present a comprehensive argument for the reality of both AμA_\muAμ​ and gμνg_{\mu\nu}gμν​. For the electromagnetic case, we show that when the source is treated as a fully quantized dynamical system with backreaction, the closed-loop Aharonov--Bohm phase acquires a gauge-dependent contribution that is the sum of two open-path integrals of AμA_\muAμ​. Since this phase contributes to observed interference patterns, it must correspond to a unique physical quantity, forcing the conclusion that AμA_\muAμ​ is a real physical field. We further argue that the Lorenz condition is the additonal physical law that, together with boundary conditions, determines AμA_\muAμ​ uniquely. For the gravitational case, we extend this analysis to both closed-loop and open-path phases. First, we show that closed-loop gravitational phases, such as those measured in the Colella-Overhauser-Werner (COW) experiment and the Overstreet et al. experiment, become gauge dependent under diffeomorphisms when source backreaction is properly accounted for. Second, we show that existing quantum optical experiments using local homodyne tomography already measure open-path relative phases of photons in spatially separated modes, which contain a gravitational contribution—the open-path gravitational phase. Since both closed-loop and open-path gravitational phases are gauge dependent under diffeomorphisms, their measurability entails the reality of the gravitational field gμνg_{\mu\nu}gμν​. We further argue that the harmonic (de Donder) gauge is the additional physical law that, together with boundary conditions, determines gμνg_{\mu\nu}gμν​ uniquely. These results establish that both electromagnetic potentials and the gravitational field are real physical fields, not mere mathematical auxiliaries. They challenge the conventional identification of gauge invariance with physical reality and have profound implications for non-Abelian gauge theories and quantum gravity.
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 Aharonov-Bohm (AB) effect, wherein an electron acquires a phase shift in a field-free region due to electromagnetic potentials, poses a profound challenge to the ontology of quantum mechanics and gauge theories. This paper demonstrates that gauge-invariant explanations, which attribute the phase to measurable quantities, rely on nonlocal and discontinuous mechanism and fail to account for its continuous accumulation along the electron's path—a process evident in the generalized AB effect, where a time-varying magnetic flux induces a phase that builds gradually over time, as predicted by quantum mechanics. Through a new analysis spanning quantum mechanics and quantum electrodynamics, I argue that the electromagnetic potentials AμA_\muAμ​, fixed in the Lorenz gauge, emerge as the fundamental physical reality, offering a local, relativistically consistent account of the phase’s generation. This exclusion of gauge-invariant paradigms reverberates across gauge theories: it redefines the Higgs mechanism, favoring dynamic potentials over static invariants, and extends to general relativity, where gravitational potentials gμνg_{\mu\nu}gμν​ may anchor spacetime’s substantival reality via a gravitational AB effect. By unraveling the AB phase’s continuous generation—locally mediated by potentials—this study not only addresses a long-standing conundrum but also bridges electromagnetic, gravitational, and Yang-Mills frameworks, offering a unified potential-centric perspective with novel implications for physics and philosophy.
Contrary to long-standing textbook doctrine, we demonstrate that the Aharonov-Bohm (AB) phase for a closed interferometric loop — the relative phase governing interference — becomes gauge dependent when backreaction of the test charge on the electromagnetic source is included in a fully quantum treatment. The standard Stokes-theorem argument fails because different interfering paths become entangled with distinct field configurations, permitting branch-dependent gauge transformations. We prove that the total phase decomposes into (1) a gauge-invariant core tied to the fixed background flux, which fully accounts for the observed interference shift, and (2) a purely gauge-dependent backreaction term that can be eliminated by an appropriate gauge choice and carries no observable consequences. This decomposition follows from a general framework for observables in gauge theories, which we establish from a foundational axiom that purely gauge-dependent quantities have no observational consequences. From this axiom, we derive theorems showing that any gauge-dependent quantity contains an extractable gauge-invariant core that constitutes its measurable content, while the remaining gauge-dependent component is physically irrelevant. This framework resolves the apparent paradox between formal gauge dependence and robust experimental observations, and offers a conceptually sharper understanding of observables in quantum electrodynamics, non-abelian gauge theories, general relativity, and other gauge-based theories.
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​.