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Quantum State Tomography Implies the Reality of the Electromagnetic Potential

Shan Gao

Published Aug 23, 2026• 0 citations
OpenAlex

Abstract

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 \(g_{\mu\nu}\).

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Reality and Measurability of Gauge Potentials

Shan Gao

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.

•Jul 22, 2026•0 citations

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

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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

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

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

Is the Electron Dressed? The Aharonov-Bohm Effect Challenges Dirac's Proposal

Shan Gao

The dressing field method (DFM), introduced by Dirac in 1955, constructs gauge-invariant matter fields by attaching a Wilson line to the bare field. This dressed field is often interpreted as the physical electron. In this paper we provide a rigorous analysis of Dirac's DFM. We begin by defining the dressed field and the associated dressed electromagnetic potential, and we examine their path-dependence and multi-valuedness in multiply-connected spaces. We then derive the exact Dirac equation satisfied by the dressed field, as well as its non-relativistic limit, the Schrödinger equation. The probabilistic interpretation (Born rule) is stated in terms of the dressed field. Next, we apply the theory to the Aharonov-Bohm (AB) effect. Using explicit computations, we show that the interference pattern predicted by the dressed field is independent of the magnetic flux \(\Phi\), in direct contradiction with the experimentally observed AB phase shift \(e^{-ie\Phi}\). We trace the mathematical origin of this failure to the exact cancellation of topological phases between the dressing factor and the bare wave function, which follows from Stokes' theorem and the multi-valuedness of the Wilson line in a multiply-connected space. Finally, we discuss the implications: the dressed field cannot serve as a complete physical description of the electron. The Dirac DFM is therefore not equivalent to standard quantum mechanics in topologically non-trivial backgrounds.

•May 12, 2026•0 citations

Impossibility of Gauge-Invariant Explanations of the Aharonov–Bohm Effect

Shan Gao

We present a rigorous no-go theorem that excludes \emph{all} gauge-invariant dynamical explanations of the generalized, time-dependent Aharonov--Bohm (AB) effect. In this generalization, the AB phase is determined by the time-averaged magnetic flux enclosed by the particle's path, $\Delta \phi_{\mathrm{AB}} = \frac{e}{\hbar} \cdot \frac{1}{T} \int_0^T \Phi(t) \, dt$. Using a carefully designed protocol in which a constant flux $\Phi$ persists only for a finite interval $\Delta t$ after wavepacket splitting and is switched to zero before detection, we demonstrate that during the phase-generating period all gauge-invariant observables of the electron remain identical to those in a zero-flux experiment. Relativistic causality further prohibits backreaction from the source. Consequently, no gauge-invariant quantity---whether local or nonlocal, field-based or source-mediated---can account for the observed phase, which depends solely on flux history. We also critically examine Healey's holonomy interpretation and show that it, too, fails to explain the time-dependent AB effect, as its static formalism lacks the dynamical structure needed to capture temporal averaging. This result establishes the inescapable physical reality of gauge potentials in quantum theory and clarifies the foundational status of the AB effect as a genuine manifestation of potentials rather than fields.

•Feb 12, 2026•0 citations
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}).
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.
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.
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 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.
The dressing field method (DFM), introduced by Dirac in 1955, constructs gauge-invariant matter fields by attaching a Wilson line to the bare field. This dressed field is often interpreted as the physical electron. In this paper we provide a rigorous analysis of Dirac's DFM. We begin by defining the dressed field and the associated dressed electromagnetic potential, and we examine their path-dependence and multi-valuedness in multiply-connected spaces. We then derive the exact Dirac equation satisfied by the dressed field, as well as its non-relativistic limit, the Schrödinger equation. The probabilistic interpretation (Born rule) is stated in terms of the dressed field. Next, we apply the theory to the Aharonov-Bohm (AB) effect. Using explicit computations, we show that the interference pattern predicted by the dressed field is independent of the magnetic flux Φ\PhiΦ, in direct contradiction with the experimentally observed AB phase shift e−ieΦe^{-ie\Phi}e−ieΦ. We trace the mathematical origin of this failure to the exact cancellation of topological phases between the dressing factor and the bare wave function, which follows from Stokes' theorem and the multi-valuedness of the Wilson line in a multiply-connected space. Finally, we discuss the implications: the dressed field cannot serve as a complete physical description of the electron. The Dirac DFM is therefore not equivalent to standard quantum mechanics in topologically non-trivial backgrounds.
We present a rigorous no-go theorem that excludes \emph{all} gauge-invariant dynamical explanations of the generalized, time-dependent Aharonov--Bohm (AB) effect. In this generalization, the AB phase is determined by the time-averaged magnetic flux enclosed by the particle's path, ΔϕAB=eℏ⋅1T∫0TΦ(t) dt\Delta \phi_{\mathrm{AB}} = \frac{e}{\hbar} \cdot \frac{1}{T} \int_0^T \Phi(t) \, dtΔϕAB​=ℏe​⋅T1​∫0T​Φ(t)dt. Using a carefully designed protocol in which a constant flux Φ\PhiΦ persists only for a finite interval Δt\Delta tΔt after wavepacket splitting and is switched to zero before detection, we demonstrate that during the phase-generating period all gauge-invariant observables of the electron remain identical to those in a zero-flux experiment. Relativistic causality further prohibits backreaction from the source. Consequently, no gauge-invariant quantity---whether local or nonlocal, field-based or source-mediated---can account for the observed phase, which depends solely on flux history. We also critically examine Healey's holonomy interpretation and show that it, too, fails to explain the time-dependent AB effect, as its static formalism lacks the dynamical structure needed to capture temporal averaging. This result establishes the inescapable physical reality of gauge potentials in quantum theory and clarifies the foundational status of the AB effect as a genuine manifestation of potentials rather than fields.