Gravitational Aharonov-Bohm Effect

• Gravitational Aharonov-Bohm effect is a quantum phenomenon where a particle acquires a phase shift solely due to gravitational potential differences, even in the absence of classical forces.
• It is rooted in the topological and gauge-invariant properties of spacetime, demonstrating nonlocal interference that challenges conventional gravitational dynamics.
• Experimental proposals use precise interferometric setups in controlled gravitational environments to detect these subtle, potential-induced phase shifts.

The electric Aharonov–Bohm (AB) effect is a paradigmatic manifestation of quantum nonlocality and gauge structure, wherein a charged particle acquires a physically observable phase shift purely due to the spacetime dependence of the electromagnetic scalar potential, in the complete absence of any local electromagnetic fields along its trajectory. This phenomenon stands as the electric analogue of the better-known magnetic AB effect and exemplifies the fundamental role played by the electromagnetic 4-potential $A_\mu$ (or equivalently, multivalued complex potentials) in quantum theory, as opposed to the local fields $\mathbf{E}$ and $\mathbf{B}$ that govern classical dynamics (Friedman et al., 2010, Weder, 2010).

1. Theoretical Foundation and Mathematical Structure

The electric AB effect rests on the quantum mechanical prediction that a charged particle of charge $q$ traversing a region where the scalar potential $V(x, t)$ may vary in time but is spatially constant, and where the electric field $\mathbf{E} = -\nabla V - \partial_t \mathbf{A}$ strictly vanishes, acquires a phase

$$\Delta\phi = \frac{q}{\hbar} \int_{t_i}^{t_f} [V_2(t) - V_1(t)]\, dt$$

if the wavefunction components are subjected to potentials $V_1(t)$ and $V_2(t)$ on two different spacetime branches (Friedman et al., 2010, Weder, 2010, Eskin, 2014). This phase is observable as a shift in the interference pattern when the branches are recombined, even though no classical force acts and no electric field is present along the particle's path.

Formally, this phase follows from the time-dependent Schrödinger equation: $$i\hbar\,\partial_t\psi(x,t) = -\frac{\hbar^2}{2m} \nabla^2\psi(x,t) + q\,V(x,t)\,\psi(x,t)$$ with $\mathbf{E}$0 constant in each spatially disjoint region traversed by the $\mathbf{E}$1th packet (Weder, 2010, Eskin, 2014). The general solution factorizes as

$\mathbf{E}$2

where $\mathbf{E}$3 is the field-free evolution. The net measurable phase shift between two arms is thus set exclusively by the difference of the time-integrals of their respective potentials (Friedman et al., 2010, Weder, 2010, Eskin, 2014).

2. Topological Character and Gauge Invariance

The electric AB effect is fundamentally topological, dependent on the global (non-contractible) structure of the particle's configuration space in spacetime and the holonomy of the $\mathbf{E}$4 connection defined by $\mathbf{E}$5 (Saldanha et al., 2024, Eskin, 2015, Eskin, 2014). The gauge invariance of the phase shift is guaranteed because under $\mathbf{E}$6, the wavefunction acquires a compensating local phase that precisely cancels the gauge ambiguity, so only differences or closed integrals of the 4-potential around nontrivial spacetime loops are physically observable (Friedman et al., 2010, Saldanha et al., 2024). Thus,

$\mathbf{E}$7

becomes nontrivial only when the spacetime domain is multiply connected—for instance, when the system's configuration space is transiently disconnected, as is necessary for the robust appearance of the electric AB effect (Eskin, 2014, Eskin, 2015).

3. Rigorous Proofs, Required Topology, and Physical Constraints

A mathematically rigorous proof requires that the following conditions be met (Weder, 2010, Eskin, 2014):

• The regions through which the wavepackets propagate are strictly field-free ($\mathbf{E}$8), which classically forbids any phase shift.
• The scalar potential $\mathbf{E}$9 is not globally gauge-trivial due to time-dependent connectivity of the spatial domain, realized via moving boundaries, inhomogeneous gating, or equivalent methods.
• The wavepackets are held in spatially disjoint, perfectly screened domains (e.g., Faraday cages), and the switching of electrostatic potentials is done nonadiabatically with respect to the transit time.

The central theorem (Eskin, (Eskin, 2014)) states: if $\mathbf{B}$0 in a multiply disconnected (in time) spacetime domain and the scalar potential is not gauge equivalent to zero, then observable interference effects occur that cannot be described by any local field theory. Conversely, when the spacetime domain remains simply connected (no topological obstruction), the potential can be gauged away and no phase persists.

The presence of "obstacles"—regions dynamically excluded from the configuration space—enables non-contractible spacetime loops, which are the necessary locus for nonzero AB phases (Eskin, 2015, Eskin, 2014).

4. Quantum Locality, Induced Charges, and Physical Realism

A crucial consideration is the role of induced charges and the completeness of the physical model. In a real metallic environment, the passage of a charge through a conducting tube induces a screening charge of $\mathbf{B}$1 on the tube, such that the combination ($\mathbf{B}$2, $\mathbf{B}$3) is locally neutral. A fully quantum treatment, as well as a careful classical analysis, shows that when induced charges are included, the total energy of the system (moving charge plus induced charge) is independent of the applied scalar potential (Wang, 2014). The phase shift cancels exactly: $\mathbf{B}$4 for each branch, so the interference pattern remains unchanged, and the naive electric AB phase is not physically realized. This suggests that any real observation of the electric AB effect demands strict control of electromagnetic boundary conditions to ensure that the induced charge contribution does not cancel the phase.

A fully quantum mechanical treatment with both the source and the particle quantized (e.g., quantized capacitor plates and electron) yields the same net shift as the standard classical analysis, but shows flexibility in how the accumulated phase can be distributed among the subsystems—ultimately, however, only gauge-invariant, global phase differences are observable (Pearle et al., 2015).

5. Topological Extensions: Electrodynamic and Time-Dependent Variants

Variants involving time-dependent magnetic fields (time-dependent AB effect) or "electrodynamic" (vector-potential-driven) AB phases extend the standard effect. For example, varying the current in a solenoid produces an external electric field, but with appropriate Faraday shielding, one can arrange for both $\mathbf{B}$5 and $\mathbf{B}$6 along the particle's paths, yet a nonlocal, spacetime holonomy remains, producing an AB phase proportional to the change in solenoidal flux: $\mathbf{B}$7 where $\mathbf{B}$8 is the solenoid flux (Saldanha et al., 2024, Saldanha, 2023). In such setups, the observable phase is purely potential-based and not associated to any local field action. The phenomenon persists as a consequence of the nontrivial spacetime topology, and the phase is robustly gauge invariant (Saldanha et al., 2024, Saldanha, 2023, Choudhury et al., 2019).

A related line of inquiry analyzes the electric AB effect in the context of quantum electrodynamics, showing that the phase can be understood as arising from local virtual photon exchange (longitudinal or scalar modes), and that the end result for the phase shift remains unchanged for arbitrary conductor geometries, provided the relevant boundary conditions are realized (Saldanha, 2019).

6. Modern Observables and Experimental Proposals

Contemporary implementations and proposed experimental designs extend beyond spatially separated beams. In superconducting circuit platforms, a time-dependent scalar potential difference between two isolated superconductors (e.g., across a capacitor in a DC Josephson junction) yields a phase shift between their order parameters of

$\mathbf{B}$9

manifested as a measurable change in the Josephson current, even after the potential difference is turned off—an electromagnetic "memory" effect (Bachlechner et al., 2019). The phase shift is acquired without any dissipative current or local field acting on the condensate (Bachlechner et al., 2019).

In yet another variant, the electric AB effect may be observed spectroscopically as a shift in energy levels for atoms confined within a Faraday cage with spatially uniform, time-varying scalar potential,

$q$0

corresponding to a ladder of sidebands reminiscent of the Floquet or AC Stark effect, but in strict absence of any local electric field (Chiao et al., 2022). This formulation establishes the electric AB effect as a quantum coherent energy-level shift.

Superconducting qubits (e.g., Cooper pair boxes) can realize a "loop-free" AB effect: a qubit in a superposition of different charge states experiences a relative phase proportional to an externally engineered "geometric potential," even though no closed spatial path is traversed and the electric field is localized away from the device (Kim et al., 2018). This approach circumvents the constraints of ultra-fast voltage switching and direct spatial interference, allowing the AB phase to be read out as a shift in Ramsey fringes or spectroscopic lines.

7. Experimental Constraints, Controversies, and Open Questions

Despite a firm theoretical foundation and a rigorous mathematical formalism, direct unambiguous experimental observation of the electric AB effect in elementary particle interferometry remains a significant challenge (Weder, 2010). Major constraints include:

• The need for ultra-fast switching of potentials relative to particle transit times.
• Maintaining strict field-free conditions along both arms, ensuring sufficiently long coherence lengths.
• Accounting for any induced charge contributions or residual field leakage that may cancel or mask the expected phase shift (Wang, 2014).

Multiple theoretical works emphasize that observing a robust electric AB shift is only feasible when the scalar potential is not globally gauge-equivalent to zero due to spacetime topology (e.g., via disconnected regions or moving boundaries) (Eskin, 2014, Eskin, 2015). Conversely, in static, connected geometries, the effect is rigorously absent.

Finally, the conceptual status of the electric AB effect—whether the phase is purely a manifestation of gauge structure, nonlocality, virtual boson exchange, or a memory effect—remains an active subject of foundational interest and debate (Saldanha, 2019, Pearle et al., 2015, Bachlechner et al., 2019). The topological and gauge-theoretic nature of the scalar potential in quantum theory is unambiguously revealed by the AB effect and continues to motivate diverse inquiries in both condensed matter and fundamental physics.

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Key References:

• Friedman & Ostapenko, “The scalar complex potential and the Aharonov-Bohm effect” (Friedman et al., 2010)
• Weder et al., “The Electric Aharonov-Bohm Effect” (Weder, 2010)
• Eskin, “A Simple Proof of Magnetic and Electric Aharonov-Bohm Effects” (Eskin, 2014)
• Gritsevets, “Influence of Induced Charges in the Electric Aharonov-Bohm Effect” (Wang, 2014)
• Bachlechner & Kleban, “Testing the electric Aharonov-Bohm effect with superconductors” (Bachlechner et al., 2019)
• Chiao, Hu, et al., “Energy level shift of quantum systems via the electric Aharonov-Bohm effect” (Chiao et al., 2022)
• Saldanha, “Electrodynamic Aharonov-Bohm effect” (Saldanha, 2023)
• Saldanha, “Local description of the Aharonov-Bohm effect with a quantum electromagnetic field” (Saldanha, 2019)
• Vaidman et al., “Quantum Mechanical Inclusion of the Source in the Aharonov-Bohm Effects” (Pearle et al., 2015)
• Saldanha et al., “Topological schemes for the electrodynamic Aharonov-Bohm effect” (Saldanha et al., 2024)
• Shim et al., “Electric Aharonov-Bohm effect without a loop in a Cooper pair box” (Kim et al., 2018)