The Aharonov–Bohm (AB) Effect in the Theory of Entropicity (ToE): Entropic Holonomy, Thresholded Distinguishability, and Quantum Phase

The Aharonov–Bohm effect is one of the cleanest demonstrations in modern physics that global structure can have physical consequences even when local classical fields vanish along the trajectory of a particle. In the standard magnetic Aharonov–Bohm configuration, an electron beam is split into two coherent paths that encircle a confined magnetic flux. Along each accessible path the magnetic field is zero, yet the recombined beams exhibit a measurable phase shift. In conventional notation, the phase difference is

Δφ_AB = (q/ħ) ∮_C A·dl = (qΦ/ħ),

where q is the particle charge, A is the vector potential, C is the closed loop obtained by concatenating the two paths, and Φ is the enclosed magnetic flux. This formula is gauge-invariant because the observable phase depends only on the loop holonomy, or equivalently on the enclosed flux, not on the particular gauge choice for A. Standard discussions therefore interpret the AB effect either as evidence for the physical significance of gauge potentials, or more carefully as evidence that the physically relevant object is the global holonomy of the gauge connection rather than the local field strength alone. That interpretation is well established in the literature.

The Theory of Entropicity (ToE), as first formulated and further developed by John Onimisi Obidi, does not deny this structure. On the contrary, if ToE is to be viable, it must reproduce it. The question is not whether ToE can ignore the standard AB phase, but whether it can explain it at a deeper level without losing gauge invariance or topological exactness. The answer is that ToE can give a coherent explanation by treating the AB phase as an instance of entropic holonomy. The novelty is not that the phase exists, which standard quantum theory already explains, but that the phase is interpreted as arising from transport through a globally nontrivial entropic manifold whose local classical field strength may vanish while its connection remains topologically nontrivial.

To formulate this rigorously, we begin from the single axiom of the Theory of Entropicity (ToE): entropy is a universal physical field S(x). Matter, geometry, and phase structure are emergent from this field. In a low-energy sector describing a charged matter excitation transported along a path γ, the phase accumulated by the excitation must be derived from an effective one-form connection on the entropic manifold. Let this effective phase connection be denoted by Ω. The transport phase is then

θ[γ] = ∫_γ Ω.

The only way for ToE to recover the standard electromagnetic limit is for Ω to contain, at minimum, the usual electromagnetic contribution. Thus one writes

Ω = (q/ħ) A_i dx^i + Ω_S,

where Ω_S is the entropic contribution induced by the entropy field and its geometric data. The relative phase between two paths is therefore

Δφ_ToE = ∮_C Ω = (q/ħ) ∮_C A·dl + ∮_C Ω_S.

Using Stokes’ theorem in the ordinary electromagnetic sector gives

(q/ħ) ∮_C A·dl = (q/ħ) Φ.

Thus

Δφ_ToE = (qΦ/ħ) + H_S(C),

where

H_S(C) = ∮_C Ω_S

is the entropic holonomy around the loop C.

This formula is the precise ToE generalization of the Aharonov–Bohm phase. It immediately yields the correct standard result in the experimentally established regime if the entropic holonomy term vanishes, or is exact and therefore integrates to zero, in ordinary AB interferometry. In that case

H_S(C) = 0 and Δφ_ToE = qΦ/ħ.

So the first requirement of rigor is satisfied: ToE reproduces the standard AB phase as a low-energy or ordinary-sector limit.

The second requirement is gauge invariance. In the standard AB effect, gauge transformations act as

A → A + ∇χ,

and the closed-loop phase remains invariant because

∮_C ∇χ · dl = 0.

The ToE extension must preserve this. That means the entropic connection Ω_S cannot be introduced arbitrarily. It must be defined so that either it is itself gauge-invariant, or it transforms only by an exact differential,

Ω_S → Ω_S + dη,

with η a single-valued scalar on the accessible domain. Then

∮_C dη = 0,

and the total observable phase remains gauge-invariant. So ToE can remain consistent with standard AB gauge structure, but only if the entropic connection is built as a genuine connection one-form on the relevant configuration bundle, not as an arbitrary added phase term.

The third requirement is topological consistency. In the standard AB setup, the accessible region is multiply connected because the flux tube is excluded from the electron’s path domain. The phase is therefore controlled by the topology of the punctured region, not by the local field strength along the path. ToE interprets this exactly in terms of entropic geometry: the region is not “empty” in the ontologically trivial sense. It is entropically structured. The local classical field may vanish along the electron path, but the entropic connection around the excluded region remains globally nontrivial. The particle responds not to a local force field but to the holonomy class of the entropic connection.

This is not hand waving. It is a straightforward translation of the standard gauge-holonomy statement into the ontology of ToE. The conventional claim is that the connection matters globally even when F = dA vanishes locally on the path. The ToE claim is that this global connection is an emergent manifestation of the deeper entropic manifold. Thus the AB effect becomes a direct example of a general ToE principle: global entropic geometry can have physically measurable consequences even in regions where the local classical field strength vanishes.

The place where ToE potentially contributes something genuinely new is in the relation between continuous phase accumulation and physically realized measurement. In standard quantum mechanics, the phase difference evolves continuously and interference visibility follows from the superposition of path amplitudes. In ToE, that continuous phase evolution is accepted, but physical distinguishability is not assumed to be automatic. The Theory of Entropicity requires that physically distinguishable events be realized only when the relevant entropic separation crosses the Obidi Curvature Invariant threshold. This does not alter the phase formula itself. Rather, it alters the ontology of when the phase becomes a physically realized observation.

To state this properly, let the two path amplitudes after transport be

ψ_1 = a e^{iθ_1}, ψ_2 = a e^{iθ_2},

with equal magnitude a for simplicity, and let the relative phase be

Δφ = θ_1 - θ_2.

The recombined intensity is

I(Δφ) ∝ |ψ_1 + ψ_2|^2 = 2a^2 [1 + cos(Δφ)].

Standard theory stops here. ToE goes further and asks when two interference configurations are physically distinguishable as realized events. Let P(Δφ) denote the detector probability distribution generated by the phase difference Δφ, and let P_0 denote a reference distribution. Then the entropic distinguishability between the two output configurations is measured by an invariant functional D[P(Δφ), P_0]. According to the Theory of Entropicity, a physically realized distinction in the measurement apparatus requires

D[P(Δφ), P_0] ≥ ln 2.

This is the point at which the AB effect in ToE becomes more than a reinterpretation. The phase may accumulate continuously, but the realized interference event is thresholded by the OCI through the measuring arrangement. The phase is therefore not denied, nor discretized at the level of wave transport. Rather, its physical registration is governed by the same thresholded distinguishability law that governs all measurement and observation in ToE.

This also clarifies the role of the No-Rush Theorem. The No-Rush Theorem is not amended here. It remains a theorem about the impossibility of zero-time dynamical realization. What produces the stronger spatiotemporal statement is the conjunction of the No-Rush Theorem with the OCI threshold. In the AB setting, that means the interference pattern cannot become physically realized as a distinguishable observation in zero time, nor without the relevant spatiotemporal and apparatus-integrated entropic deformation crossing the OCI threshold. The underlying phase transport remains continuous. The observed physical event remains thresholded. This is completely consistent with the general ToE architecture developed earlier.

One may now state the strongest rigorous ToE claim about the AB effect. The standard Aharonov–Bohm phase is reproduced as the electromagnetic limit of an entropic holonomy law,

Δφ_ToE = (qΦ/ħ) + H_S(C),

with H_S(C) vanishing or becoming exact in ordinary AB experiments. The new interpretive content is that the measured phase difference is not merely a formal property of a gauge bundle but the observable manifestation of global entropic connection structure. The new structural content is that the physical realization of AB interference is subject to the same thresholded distinguishability law as all other observations. And the new potential predictive content, if developed further, is that nonzero entropic holonomy corrections H_S(C) might appear in topological quantum phase experiments beyond the ordinary regime.

We state that the last point above must be handled with this caveat. At present, without an explicit derivation of Ω_S from the Obidi Action, one cannot honestly claim that ToE has produced a new AB prediction already confirmed or numerically precise. What ToE does provide now is a rigorous framework in which such a prediction could be formulated. The immediate task for a fully developed ToE treatment would be to derive Ω_S explicitly from the entropic action, show under what conditions H_S(C) vanishes, and identify any regime in which it does not. Only then would ToE move from interpretive depth to independently testable novelty in the AB domain.

The ToE conclusion is therefore as follows. The Theory of Entropicity offers a conceptually strong and mathematically consistent explanation of the Aharonov–Bohm effect as an entropic holonomy phenomenon. It preserves the standard phase formula and gauge invariance in the ordinary limit, while providing a deeper ontological account of why a globally nontrivial connection can matter when local classical fields vanish. Its most useful present contribution is explanatory and unificatory: it places the AB effect, geometric phase, and thresholded measurement within one common entropic-geometric framework. Its future usefulness depends on deriving the entropic connection term explicitly from the Obidi Action and determining whether nontrivial entropic holonomy corrections survive in experimentally accessible regimes.

References

Aharonov, Y., & Bohm, D. (1959). Significance of electromagnetic potentials in the quantum theory. Physical Review, 115(3), 485–491.

Anandan, J., & Aharonov, Y. (1988). Geometric quantum phase and angles. Physical Review D, 38, 1863–1870.

Cohen, E., Larocque, H., Bouchard, F., et al. (2019). Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond. Nature Reviews Physics.

Kasunic, K. J. (2019). Magnetic Aharonov-Bohm effects and the quantum phase shift: A heuristic interpretation. American Journal of Physics, 87, 745 ff.

Li, X., Hansson, T. H., & Ku, W. (2022). Gauge-independent description of the Aharonov-Bohm effect. Physical Review A, 106, 032217.

Saldanha, P. L. (2024). Gauge invariance of the Aharonov-Bohm effect in a quantum electrodynamics framework. Physical Review A, 109, 062205.

Vaidman, L. (2012). Role of potentials in the Aharonov-Bohm effect. Physical Review A, 86, 040101.