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Failure of Gauge Invariance or Necessity of Quantum Gravity?

Shan Gao

Published Jul 18, 2026• 0 citations
OpenAlex

Abstract

We analyse the gravitational Aharonov-Bohm (AB) phase within the semiclassical framework. We first review the traditional non-relativistic approximation, where a particle of rest mass \(m\) couples to the Newtonian potential via \(H_{\rm int}=m\Phi\), yielding a gauge-invariant phase analogous to the electromagnetic AB effect. We then show that this approximation fails when the particle's total energy varies between interferometer branches. A rigorous derivation from linearised general relativity reveals that in the weak-field, non-relativistic limit, gravity couples to the local energy density, not merely the rest mass. In a laser-driven atom interferometer, the two branches of the superposition differ by the location of an excitation energy \(\delta E\): either stored in the laser apparatus at \(\mathbf{x}_L\) or carried by the atom at \(\mathbf{x}_A(t)\). Under a gauge transformation $\delta h_{\mu\nu} = -\partial_\mu \xi_\nu - \partial_\nu \xi_\mu$, the gravitational AB phase acquires a residual \(\delta(\Delta \phi_{\rm grav}) = -\frac{\delta E}{\hbar c^2}[\xi_0(\mathbf{x}_L,t_f)-\xi_0(\mathbf{x}_A(t_f),t_f)]\) for a purely temporal generator, which is generically nonzero. Since experiments such as Overstreet et al. (2022) measure a stable, reproducible gravitational AB phase, we are compelled to choose between two mutually exclusive possibilities. If gauge invariance (diffeomorphism invariance) is a fundamental principle, then the semiclassical framework is incomplete and a quantum theory of gravity is necessary. If, on the other hand, gauge invariance fails, then semiclassical gravity can be a complete theory. In this case, the gravitational potentials must be understood as real physical fields, and the gauge-dependent phase is a genuine observable, consistent with experiment. These two resolutions are mutually exclusive, and their implications for the nature of gravity and gauge symmetry are profound.

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We analyse the gravitational Aharonov-Bohm (AB) phase within the semiclassical framework. We first review the traditional non-relativistic approximation, where a particle of rest mass (m) couples to the Newtonian potential via (H_{\rm int}=m\Phi), yielding a gauge-invariant phase analogous to the electromagnetic AB effect. We then show that this approximation fails when the particle's total energy varies between interferometer branches. A rigorous derivation from linearised general relativity reveals that in the weak-field, non-relativistic limit, gravity couples to the local energy density, not merely the rest mass. In a laser-driven atom interferometer, the two branches of the superposition differ by the location of an excitation energy (\delta E): either stored in the laser apparatus at (\mathbf{x}_L) or carried by the atom at (\mathbf{x}A(t)). Under a gauge transformation δhμν=−∂μξν−∂νξμ\delta h_{\mu\nu} = -\partial_\mu \xi_\nu - \partial_\nu \xi_\muδhμν​=−∂μ​ξν​−∂ν​ξμ​, the gravitational AB phase acquires a residual (\delta(\Delta \phi{\rm grav}) = -\frac{\delta E}{\hbar c^2}[\xi_0(\mathbf{x}_L,t_f)-\xi_0(\mathbf{x}_A(t_f),t_f)]) for a purely temporal generator, which is generically nonzero. Since experiments such as Overstreet et al. (2022) measure a stable, reproducible gravitational AB phase, we are compelled to choose between two mutually exclusive possibilities. If gauge invariance (diffeomorphism invariance) is a fundamental principle, then the semiclassical framework is incomplete and a quantum theory of gravity is necessary. If, on the other hand, gauge invariance fails, then semiclassical gravity can be a complete theory. In this case, the gravitational potentials must be understood as real physical fields, and the gauge-dependent phase is a genuine observable, consistent with experiment. These two resolutions are mutually exclusive, and their implications for the nature of gravity and gauge symmetry are profound.
Testing whether gravity is quantum or classical remains a cornerstone challenge in fundamental physics. We introduce a practical tabletop experiment to probe gravity's quantum nature via the gravitational Aharonov-Bohm (AB) effect in atom interferometry. Our approach hinges on two pivotal insights: First, unlike electromagnetic AB effects where constant charge ensures gauge-invariant phases for closed paths, semiclassical gravity couples to the particle's total energy, rendering the AB phase gauge-dependent and unobservable when energies differ between paths due to environmental noise or controlled perturbations. Second, in quantum gravity—treating source, particle, and field as quantum—the total phase splits into a gauge-dependent particle contribution and a gauge-invariant source backreaction, yielding precisely half the nominal AB phase as observable. This halved phase discriminates quantum from classical gravity without requiring massive superpositions, aligning with existing technology. Remarkably, Overstreet et al.'s (Science 375, 226 (2022)) data already reveal a nonzero measured AB phase, providing the first experimental evidence for gravitational quantization.
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.
Abstract In this paper, we study the phenomenon of quantum interference in the presence of external gravitational fields described by alternative theories of gravity. We analyze both non-relativistic and relativistic effects induced by the underlying curved background on a superposed quantum system. In the non-relativistic regime, it is possible to come across a gravitational counterpart of the Bohm–Aharonov effect, which results in a phase shift proportional to the derivative of the modified Newtonian potential. On the other hand, beyond the Newtonian approximation, the relativistic nature of gravity plays a crucial rôle. Indeed, the existence of a gravitational time dilation between the two arms of the interferometer causes a loss of coherence that is in principle observable in quantum interference patterns. We work in the context of generalized quadratic theories of gravity to compare their physical predictions with the analogous outcomes in general relativity. In so doing, we show that the decoherence rate strongly depends on the gravitational model under investigation, which means that this approach turns out to be a promising test bench to probe and discriminate among all the extensions of Einstein’s theory in future experiments.
The quantum nature of gravity is one of physics' greatest unsolved problems, with existing experimental proposals facing formidable challenges in maintaining spatial superpositions of massive objects against decoherence. We propose a fundamentally different approach that bypasses these difficulties. The central insight is that a quantum gravitational field can imprint a gauge-invariant phase on a particle's wavefunction even along an open trajectory, arising from the quantum backreaction of the source. This phase, which manifests prior to the final recombination stage of an interferometer and cannot be generated by a classical gravitational field, provides an operationally direct witness to the quantumness of gravity. Our experimental design uses a single photon whose accumulated gravitational phase due to the Earth is transferred via absorption to auxiliary atoms near the end of the interferometer sequence, preserving the open-path character essential for the test. We scale the interferometer to yield a detectable phase of order 10−310^{-3}10−3 rad and show how non-local correlation measurements on the atoms can access this gauge-invariant signature. Leveraging established quantum optical and atomic techniques, this proposal offers a feasible path to distinguish quantum from classical gravity. A positive result would provide the first experimental evidence for the quantum nature of gravitational fields, with profound implications for quantum gravity theories.
A consistent theory including gravitational effects in quantum mechanics has not been found. Although many different approaches have been proposed for theory of quantum gravity, several authors have suggested that gravity may be fundamentally classical. As there is no experimental evidence for a fundamentally quantum vs. fundamentally classical theory of gravity, it is hoped that investigating the consequences of semiclassical gravity, one may propose an experiment to distinguish between the quantum or classical nature of gravity. As opto- and electromechanical devices allow quantum limited measurement and control over massive, mechanical degrees of freedom, they are ideal systems to test violations of standard quantum mechanics arising from gravitational effects. After a brief introduction to the background theory and literature review, we begin by introducing a semiclassical model of gravity based on quantum measurement and control. We show under which conditions the theory behaves like a collapse model, and how a Newtonian potential is recovered in the multi particle case.We then go on to analyse our semiclassical model in two scenarios: firstly in a cosmological setting, and secondly by considering quantum clocks on the surface of the earth. In the cosmological description, we show that a violation of energy conservation, resulting from our model, can be reinterpreted as a dark energy fluid in the spacetime, suggesting that perhaps local violations of conservations of energy may be resolved in a cosmological framework. Secondly, we derive a minimum, gravitational dephasing rate for quantum clocks under the semiclassical model, which unsurprisingly, is is well below the resolution of current state of the art clocks. The next chapters diverge from analysing the gravitational model, and instead focus on novel methods of manipulating and measuring micromechanical devices, that may be used to experimentally probe the results of the semi-classical theory. Firstly we show that a sequence of coherently controlled pulses in an optomechanical system can be used to directly probe the momentum of an oscillator - opening up the possibility for direct detection of momentum diffusion, which is a consequence of the semiclassical gravity model. Secondly, we introduce an electro-mechanical device, and sequence of control operations, that may be understood as an interferometer for the wave function of a mechanical oscillator coupled to a qubit. We show how the resulting interference pattern can distinguish between unitary evolution, thermalisation/decoherence, and a purely classical description of the mechanical oscillator. We also show how the qubit-oscillator interaction can be used generate arbitrary wave functions of the mechanical oscillator, using only classical qubit driving.
The gauge invariance of the Aharonov-Bohm (AB) effect with a quantum treatment for the electromagnetic field is demonstrated. We provide an exact solution for the electromagnetic ground energy due to the interaction of the quantum electromagnetic field with the classical charges and currents that act as sources of the potentials in a classical description, in the Lorenz gauge. Then, we use first-order perturbation theory to compute an extra change on the electromagnetic ground energy due to the presence of a quantum charged particle with known wave function in the system. This energy in general depends on the quantum particle path in an interferometer, which results in an AB phase difference between the paths. The gauge invariance of this AB phase difference is then shown for the magnetic, electric, and the recently proposed electrodynamic versions of the AB effect. However, the AB phase difference could depend on the gauge for nonclosed paths, which reinforces the view that it only can be measured in closed paths.