This paper investigates a geometric framework for the gravitational Aharonov-Bohm effect in four-dimensional spacetime, demonstrating how spacetime curvature induces nonlocal quantum phase shifts within field-free regions. By constructing vector bundles on spacetime manifolds equipped with Levi-Civita connections, we derive the holonomy transformations for parallel-transported quantum states. Under the Newtonian approximation, metric decomposition into Minkowski background plus scalar potential perturbations reveals through the linearized Einstein field equations that the gravitationally induced phase shift is mathematically isomorphic to its electromagnetic counterpart. These results establish the quantum observability of gravitational gauge structures and provide theoretical support for experimental verification via atom interferometry.
This paper investigates a geometric framework for the gravitational Aharonov-Bohm effect in four-dimensional spacetime, demonstrating how spacetime curvature induces nonlocal quantum phase shifts within field-free regions. By constructing vector bundles on spacetime manifolds equipped with Levi-Civita connections, we derive the holonomy transformations for parallel-transported quantum states. Under the Newtonian approximation, metric decomposition into Minkowski background plus scalar potential perturbations reveals through the linearized Einstein field equations that the gravitationally induced phase shift is mathematically isomorphic to its electromagnetic counterpart. These results establish the quantum observability of gravitational gauge structures and provide theoretical support for experimental verification via atom interferometry.
The Aharonov-Bohm (AB) effect fundamentally demonstrates that electromagnetic potentials, rather than strictly classical fields, play a foundational, observable role in quantum mechanics. While standard quantum theory successfully accommodates this phenomenon via the axiomatic insertion of the minimal coupling principle, a purely geometric interpretation of the gauge potential has historically remained elusive. Building on a previously established complex spacetime framework—where conventional spacetime coordinates are extended into real and imaginary domains—this paper proposes a novel, localized geometric origin for the AB effect. We postulate that the imaginary spatial derivative of the complex wavefunction is physically equivalent to the electromagnetic gauge potential, thereby deriving the minimal coupling prescription directly from spacetime geometry rather than treating it as an independent axiom. Under this framework, the topological phase shift experienced by a charged particle in a field-free region arises naturally from its local evolution through the imaginary dimensions of complex spacetime. Furthermore, by treating the gauge potential as an affine connection or cross-metric term that links real and imaginary dimensions, we successfully redefine the complex metric tensor. This allows the classical electromagnetic field tensor to emerge naturally as the Riemann curvature of the complexified space. This approach provides a unified geometric foundation for both gauge potentials and quantum interference phenomena, resolving previous field-tensor metric limitations while offering a promising mathematical framework for future integrations into relativistic quantum mechanics and cosmic geometry.
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 Aμ, 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 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: This work develops a unified and gauge-invariant framework in which quantum interference is described as a measurement of relative holonomy between two coherent evolutions in a combined configuration space X = M × Λ. Both the Aharonov–Bohm (Abelian) and Wilczek–Zee (non-Abelian) effects appear as limits of a single Wilson-loop observable. A block connection is introduced on the direct-sum bundle E = H ⊕ (T X ⊗ ℂ), with structure group U(n) × O(p,q), thereby unifying Berry parallel transport and Levi-Civita geometry within one connection. The formalism ensures gauge and diffeomorphism covariance, analytic well-posedness under back-reaction, and establishes conditions for reconstructing the spacetime metric from holonomy data. The resulting theory provides a mathematically consistent and experimentally testable synthesis of geometric phase, interference, and spacetime connection. Potential physical implementations include mesoscopic rings, cold-atom systems, and superconducting circuits. Keywords: geometric phase; holonomy; Aharonov–Bohm effect; Berry phase; Wilczek–Zee connection; differential geometry; quantum interference; gauge theory.
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 Aharonov-Bohm (AB) effect highlights the fundamental role of electromagnetic potentials in quantum mechanics, manifesting as a phase shift for a charged particle in field-free regions. While well-established for static magnetic fluxes, the effect's behavior under time-varying fluxes remains an open and debated question. Employing the WKB method, we derive the AB phase shift for a time-dependent magnetic vector potential, demonstrating that for circular paths in the quasistatic regime, it is proportional to the time-averaged enclosed magnetic flux, ΔφAB=T1∫0TeΦ(t)dt, with the total phase shift, including kinetic contributions, equaling eΦ(0). For non-circular paths, the phase shift depends on both the flux history and path geometry, revealing the effect's hybrid nature involving gauge potentials and induced electric fields. We verify the consistency of our gauge choice with Maxwell's equations and discuss the implications for local versus nonlocal interpretations of the AB effect. We also generalize the results to scenarios with nonzero external magnetic fields, where the enclosed flux is through the actual electron paths, and for circular paths of radius R, the AB phase shift is also proportional to the time average of the enclosed flux Φenc(R,t), with the total phase shift depending only on the initial enclosed flux eΦenc(R,0); for general non-circular paths, the external magnetic field affects trajectories and phase accumulation through the Lorentz force, leading to additional path dependence. These findings clarify the role of gauge-dependent potentials and induced fields in the generalized AB effect, offering new theoretical insights and potential applications in quantum technologies.
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.