Aharonov–Bohm Effect
Author: Mark Higgins
Revised Date: 25th June 2026
Introduction
The Aharonov–Bohm (AB) Effect is one of the most profound demonstrations that quantum mechanics differs fundamentally from classical physics. First predicted by Yakir Aharonov and David Bohm in 1959, the effect shows that charged particles can experience measurable phase shifts even when travelling entirely through regions where both the electric field (E) and magnetic field (B) are exactly zero.
In conventional quantum theory, this phenomenon is interpreted as evidence that the electromagnetic vector potential possesses physical reality beyond being a mathematical convenience. While classical electromagnetism treats only electric and magnetic fields as observable, the AB Effect suggests that the potentials themselves influence quantum wavefunctions.
The experiment appears paradoxical because no classical force acts upon the particle. Electrons travel through field-free space, yet their interference pattern changes depending upon the magnetic flux enclosed by their path.
Within Holographic Space-Time (HST), this apparent paradox disappears. Rather than requiring action through empty space, the observed phase shift emerges naturally from changes in boundary coherence geometry. The electromagnetic potential becomes an effective description of deeper structural curvature encoded within the surrounding coherence field.
45.2 The Classical View
Classically, a charged particle experiences force according to the Lorentz equation:
\[ F = q \left( E + v \times B \right) \]
If
\[ E = 0 \]
and
\[ B = 0 \]
then
\[ F = 0 \]
Consequently,
Classical electrodynamics therefore predicts absolutely no measurable influence outside an ideal solenoid.
Yet experiments repeatedly demonstrate otherwise.
45.5 HST Reinterpretation
HST proposes a different ontology.
Rather than empty space carrying invisible potentials, the surrounding boundary possesses a structured coherence geometry.
The enclosed magnetic flux modifies this geometry globally.
Electrons travelling around different sides of the solenoid do not experience hidden forces.
Instead, they propagate through slightly different coherence landscapes.
Their internal coherence accumulates different structural phases because the surrounding boundary encoding differs.
The measured interference shift is therefore not produced by force.
It is produced by geometric coherence accumulation.
45.6 Boundary Coherence Encoding
Within HST every boundary carries a coherence density
\[ \rho_C \]
whose gradients define allowable propagation pathways.
The magnetic flux changes the coherence encoding around the enclosed region.
Although
\[ B = 0 \]
locally,
the global coherence map satisfies
\[ \rho_C \neq \mathrm{constant} \]
around the loop.
Thus two identical trajectories accumulate different recursive coherence histories.
When recombined,
their relative phase becomes
\[ \Delta \phi \propto \oint \nabla \rho_C \cdot d\mathbf{l} \]
The phase difference therefore reflects accumulated coherence curvature rather than hidden electromagnetic action.
Overview
This Flow Algebra implements the Aharonov–Bohm Effect as a process of boundary coherence encoding rather than one of force-mediated interaction. Instead of treating the electromagnetic vector potential as a fundamental physical entity that acts on particles at a distance, the flow reformulates the phenomenon within the Holographic Space-Time (HST) framework as the consequence of global coherence geometry encoded on the boundary surrounding a constrained magnetic region.
The implementation follows the complete physical evolution of the experiment—from the establishment of a coherent electron source to the emergence of an observable interference pattern—using recursive state transitions rather than differential field equations. Every stage of the experiment is represented as a coherent structural transformation, allowing the physical process to be described as an executable sequence of state updates.
Replacing Force with Structural Geometry
In conventional electrodynamics, the Aharonov–Bohm Effect presents a conceptual challenge because electrons exhibit measurable phase shifts despite travelling through regions where both the electric field
\[ E = 0 \]
and magnetic field
\[ B = 0 \]
throughout their entire trajectory.
Since the Lorentz force is
\[ F = q \left( E + v \times B \right) \]
classical mechanics predicts
\[ F = 0 \]
meaning that no measurable influence should occur.
The flow replaces this apparent contradiction by introducing a different causal mechanism.
Rather than allowing forces to act through empty space, the magnetic flux enclosed by the electron paths modifies the boundary coherence curvature, denoted by
\[ \Gamma_C \]
The observable phase shift is therefore generated by structural geometry instead of local force.
Interference as Structural Comparison
The flow then recombines the two coherent wavefunctions,
ψ1 + ψ2 → It
where
\[ I_t \]
represents the observable interference pattern.
The interference is therefore implemented as a comparison between two independently accumulated coherence histories.
The fringes shift because each path carries a different recursive structural memory rather than because one particle experienced a greater force than the other.
Interpreting the Vector Potential
A major theoretical implementation of the flow is the correspondence
\[ A \Longleftrightarrow \Gamma_C \]
Within HST, the electromagnetic vector potential is not regarded as a fundamental physical object.
Instead, it functions as an effective mathematical representation of the underlying boundary coherence curvature.
This correspondence allows the conventional quantum expression
\[ \Delta \phi = \frac{q}{\hbar} \oint \mathbf{A} \cdot d\mathbf{l} \]
to be reinterpreted as
\[ \Delta \phi \propto \oint \Gamma_C \cdot d\mathbf{l} \]
or equivalently,
\[ \Delta \phi \propto \oint \nabla \rho_C \cdot d\mathbf{l} \]
The measured phase shift therefore emerges from the geometry of coherence rather than from the action of an invisible potential.
Stage 5 — Propagation Along Independent Paths
The flow then separates the coherent electron wavefunction into two trajectories,
P_1and
P_2.These represent the two paths surrounding the solenoid.
A crucial point is that both paths satisfy the experimental conditions
E = 0and
B = 0.Consequently,
F = 0.No classical acceleration occurs anywhere along either trajectory.
When reading this section, the absence of force should not be viewed as an absence of structure.
Instead, it indicates that the particles move through different regions of an already encoded coherence geometry.
Stage 6 — Reading Phase Accumulation
Each propagation path accumulates structural phase independently.
The flow defines
φ_1and
φ_2as recursive integrals of the coherence density encountered along each trajectory.
These phases should not be interpreted as arbitrary wavefunction parameters.
Instead, they represent the cumulative structural memory acquired during propagation through the coherence field.
Although the particles travel through locally identical force-free regions, the global geometry differs, causing each path to accumulate a slightly different coherence history.
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Reading the Invariants
The invariants should be read as structural laws that must remain true throughout execution.
For example,
E(P_i)=0ensures that no electric field influences the electrons.
Similarly,
B(P_i)=0guarantees that the electrons never directly encounter magnetic fields.
The condition
|F(P_i)|≤ε_forceensures that the Lorentz force remains experimentally negligible.
Other invariants verify that boundary coherence has been successfully encoded and that sufficient phase difference exists to produce an observable interference shift.
If any invariant fails, the recursive model cannot generate the Aharonov–Bohm Effect.
Reading the Transition Rules
The transition block describes how the system moves from one complete physical state to the next.
Only when all structural conditions are simultaneously satisfied does the execution terminate with
OBSERVABLE_AB_PHASE_SHIFT.This should be interpreted as the successful completion of the physical experiment.
If any required condition is absent—for example, if there is no enclosed magnetic flux, insufficient coherence encoding, or loss of wavefunction coherence—the flow terminates in an alternative state such as
NO_OBSERVABLE_FRINGE_SHIFTor one of the defined failure modes.
Overview
This diagram presents the Holographic Space-Time (HST) interpretation of the Aharonov–Bohm Effect, reformulating one of quantum mechanics' most celebrated phenomena as an emergent consequence of boundary coherence geometry rather than action through electromagnetic potentials. Unlike the conventional interpretation, in which the vector potential influences electron phase despite vanishing local electric and magnetic fields, the HST framework proposes that the observable phase shift arises from the recursive encoding of magnetic flux into the surrounding boundary coherence.
The illustration follows the complete physical progression of the experiment, beginning with the formation of a coherent electron source and ending with the measurable displacement of the interference fringes. Each stage represents a structural transformation within the boundary rather than the transmission of a classical force.
1. Physical Setup
At the centre of the figure lies a long cylindrical solenoid, carrying an enclosed magnetic flux
\[ \Phi_B \]
The magnetic field is confined entirely within the solenoid interior.
Two coherent electron beams are emitted from a common source and travel around opposite sides of the solenoid before recombining at the detector.
The two trajectories are labelled
\[ P_1 \]
and
\[ P_2 \]
A large shaded toroidal surface surrounding the solenoid represents the HST boundary state
\[ B_t \]
within which coherence is organised.
Unlike conventional depictions that emphasise magnetic field lines, the boundary surface represents the geometric substrate upon which coherence evolves.
6. Interference Formation
After completing their respective paths, the electron wavefunctions recombine at the detector.
The interference intensity is represented by
\[ I_t = \left| \psi_1 + \psi_2 \right|^2 \]
The fringe pattern shown in the lower portion of the figure demonstrates that constructive and destructive interference arise naturally from the difference between the accumulated coherence histories of the two paths.
Nothing acts upon the electrons during recombination.
The detector merely reveals information that has already been encoded throughout propagation.
7. Observable Fringe Shift
The final panel illustrates the experimentally measurable displacement of the interference pattern.
Changing the enclosed magnetic flux
\[ \Phi_B \]
alters the boundary curvature
\[ \Gamma_C \]
which modifies the coherence density
\[ \rho_C \]
leading to a different accumulated phase difference
\[ \Delta \phi \]
The result is a lateral displacement of the interference fringes,
\[ \Delta x \]
without any classical force acting upon the electrons.
This visualises the central prediction of the Aharonov–Bohm experiment.
Diagram Logic Chain
Boundary
↓
Constraint Region
↓
Magnetic Flux
↓
Boundary Curvature
↓
Coherence Density
↓
Electron Paths
↓
Phase Accumulation
↓
Relative Phase
↓
Wavefunction Interference
↓
Observable Fringe ShiftComplete Structural Identity
| Diagram Layer | Flow Layer | Mathematical Layer |
|---|---|---|
| Boundary Geometry | B_t |
B_t = Boundary(t) |
| Constraint Region | S_t |
S_t = ConstraintRegion(B_t) |
| Magnetic Flux | Φ_B |
Φ_B = ∫ B · dA |
| Curvature Encoding | Γ_C |
Γ_C = Γ(B_t,Φ_B) |
| Coherence Density | ρ_C |
ρ_C = ρ(Γ_C) |
| Electron Paths | P_1, P_2 |
P_i ⊂ B_t |
| Phase Integration | φ_1, φ_2 |
φ_i = ∫Pᵢ f(ρ_C) ds |
| Phase Difference | Δφ |
Δφ = φ₁ − φ₂ |
| Interference | I_t |
I_t = |ψ₁ + ψ₂|² |
| Observation | S_{t+1} |
Observable Aharonov–Bohm fringe shift |
Execution Guarantee
The Flow Algebra presented in this chapter constitutes a deterministic structural execution model of the Aharonov–Bohm Effect within the Holographic Space-Time (HST) framework. Rather than describing the phenomenon through electromagnetic potentials acting in otherwise field-free space, the flow executes the experiment as a recursive sequence of boundary coherence transformations in which enclosed magnetic flux modifies the geometry of the surrounding coherence boundary.
Execution begins with the construction of the boundary state and the definition of the magnetic constraint region . The enclosed magnetic flux is recursively encoded into the boundary coherence curvature , which generates the coherence density field . Electron trajectories propagate through this structured coherence landscape while satisfying the invariant conditions
\[ E \left( P_i \right) = 0 \]
\[ B \left( P_i \right) = 0 \]
and
\[ F = q \left( E + v \times B \right) = 0 \]
Since no classical Lorentz force acts upon either electron trajectory, all observable phase evolution arises solely through recursive accumulation of coherence along the boundary.
Each propagation path independently integrates the local coherence density,
\[ \phi_i = \int_{P_i} f \left( \rho_C \right) \, ds \]
producing the relative phase
\[ \Delta \phi = \phi_1 - \phi_2 \]
The recombination of the coherent wavefunctions generates the measurable interference pattern
\[ I_t = \left| \psi_1 + \psi_2 \right|^2 \]
whose fringe displacement represents the final observable state
\[ S_{t+1} \]
The flow guarantees that the experimentally observed Aharonov–Bohm phase shift emerges if and only if the following structural conditions remain simultaneously satisfied throughout execution:
Violation of any one of these invariants causes execution to terminate in a defined failure state, including loss of interference, absence of observable phase shift, or invalid experimental configuration.
Consequently, the flow is both deterministic and structurally verifiable. Every execution path is traceable from the initial boundary configuration to the final interference pattern, with no hidden variables, undefined transitions, or non-recursive operations.
Formal Execution Guarantee
\[ B_t \rightarrow S_t \rightarrow \Phi_B \rightarrow \Gamma_C \rightarrow \rho_C \rightarrow \left( P_1, P_2 \right) \rightarrow \left( \phi_1, \phi_2 \right) \rightarrow \Delta \phi \rightarrow I_t \rightarrow S_{t+1} \]
subject to the invariants
\[ E \left( P_i \right) = 0 \]
\[ B \left( P_i \right) = 0 \]
\[ F \left( P_i \right) = 0 \]
\[ \Phi_B \neq 0 \]
\[ \Gamma_C \geq \Gamma_{\mathrm{min}} \]
\[ \rho_C \text{ continuous} \]
\[ \mathrm{Coherence} \left( P_1, P_2 \right) \geq C_{\mathrm{min}} \]
\[ \left| \Delta \phi \right| \geq \epsilon_{\mathrm{phase}} \]
HST Execution Identity
\[ \Phi_B \Longrightarrow \Gamma_C \Longrightarrow \rho_C \Longrightarrow \phi_i \Longrightarrow \Delta \phi \Longrightarrow I_t \]
Execution Guarantee: If an enclosed magnetic flux recursively encodes boundary coherence curvature while both electron trajectories remain locally field-free and mutually coherent, then the Flow Algebra guarantees the emergence of the Aharonov–Bohm interference shift as a deterministic consequence of accumulated boundary coherence.
Within the HST framework, the observed phase difference is therefore interpreted as the measurable signature of recursive boundary geometry rather than the action of electromagnetic forces or physically fundamental vector potentials.
