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Comment on "Aharonov-Bohm Phase is Locally Generated Like All Other Quantum Phases"

This paper refutes Marletto and Vedral's claim that the Aharonov-Bohm phase is locally generated by entanglement, demonstrating instead that their model contains mathematical errors regarding gauge dependence and interaction signs, thereby reaffirming the conventional view that the phase arises from the vector potential rather than entanglement.

Original authors: Shan Gao

Published 2026-01-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shan Gao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Debate About "Invisible" Forces

Imagine you are walking through a park. There is a large, invisible fence around a secret garden. You can't see the fence, and you can't feel any wind or magnetic pull from it. However, as you walk around the garden, your internal compass (your "quantum phase") gets slightly twisted just because you walked around that specific area.

This is the Aharonov-Bohm (AB) effect. For decades, physicists have agreed that this twist happens because of an invisible "map" called the Vector Potential ( $A$ ). Even though the actual magnetic fields are zero where you are walking, this invisible map guides your path.

Recently, two researchers named Marletto and Vedral proposed a new idea. They suggested that the twist isn't caused by the invisible map at all. Instead, they claimed it happens because your walking path gets "entangled" (like two dancers holding hands) with the invisible quantum fields around you. They argued this happens locally (right where you are) and doesn't depend on how you draw your map.

Shan Gao, the author of this paper, says: "Hold on. That explanation is wrong."

Here is a breakdown of Gao's four main arguments against Marletto and Vedral's theory.

1. The Math Mistake: The "Half-Price" Error

Marletto and Vedral tried to calculate the energy of the interaction using a formula that looks like a standard physics equation for energy. However, Gao points out they made a simple but fatal math error.

The Analogy: Imagine you are buying a sandwich. The price is \ $10. Marletto and Vedral wrote a receipt that says, "The cost of interaction is half the price of the sandwich," so they charged you \$ 5.
The Reality: In physics, when two different things (like a moving charge and a solenoid) interact, you don't divide the energy by two. Gao shows that their formula has an extra "1/2" in it that shouldn't be there.
The Result: Because of this math error, their calculation of the energy is off by a factor of two. It's like trying to build a bridge with half the required steel; it just won't hold up.

2. The Sign Error: Walking the Wrong Way

Even if we fix the math, Gao found another problem: the direction is wrong.

The Analogy: Imagine you are pushing a swing. The correct physics says you should push it forward to make it go higher. Marletto and Vedral's formula says you should push it backward.
The Reality: In the standard theory (Quantum Electrodynamics or QED), the interaction energy is negative (let's call it -). Marletto and Vedral's formula gives a positive result (+).
The Result: This sign error means their model predicts the particle behaves in the exact opposite way it actually does. Gao shows that their formula only accidentally matches the real world under very specific, static conditions (like a still pond), but if the water starts moving (time-dependent fields), their formula breaks completely.

3. The "Local" Lie: The Map vs. The Field

Marletto and Vedral claimed their theory works for any path, even if the path isn't a complete circle, and that it doesn't matter how you draw your coordinate system (gauge independence). Gao says this is impossible.

The Analogy: Imagine you are navigating a city using a map. If you walk from Point A to Point B, the route you take on the map matters. If you change the map's projection (like switching from a flat map to a globe view), the distance and direction look different.
The Reality: Gao proves that for a path that isn't a closed loop, the "phase" (the twist in the wave) does depend on how you draw your map (the gauge).
The Result: Marletto and Vedral claimed their "field-based" energy was a universal truth that didn't care about the map. Gao shows that because their formula is actually just a specific version of the standard map (the Coulomb gauge), it fails when you change the map. Therefore, their claim that the effect is "gauge independent" is false.

4. The Entanglement Distraction: The Shadow, Not the Object

The most famous part of Marletto and Vedral's paper is the idea that entanglement (a spooky quantum connection) is the cause of the phase shift. Gao argues this is a misunderstanding of cause and effect.

The Analogy: Imagine a shadow puppet show. The puppet casts a shadow on the wall. Marletto and Vedral say, "The shadow is what makes the puppet move!" Gao says, "No. The puppet moves because of the light and the hand holding it. The shadow is just a side effect that happens to appear."
The Reality: In the quantum model, the particle and the field do get entangled. But Gao shows that this entanglement is just a byproduct of the interaction. The actual "twist" in the phase is determined entirely by the invisible Vector Potential ( $A$ ).
The Result: You can calculate the exact same phase shift using old-school, semi-classical physics without ever mentioning entanglement. Therefore, entanglement isn't the driver of the effect; it's just a passenger along for the ride.

The Conclusion

Shan Gao's paper is a "correction notice" for the physics community. He argues that:

Marletto and Vedral's math has a halving error and a sign error.
Their theory only works in one specific scenario (static fields in one specific gauge), not universally.
The entanglement they highlight is not the cause of the Aharonov-Bohm effect; the Vector Potential ( $A$ ) is.

The Bottom Line: The old explanation is still the right one. The invisible map (Vector Potential) guides the particle, and the "entanglement" is just a side effect, not the magic sauce.

Technical Summary: Comment on "Aharonov-Bohm Phase is Locally Generated Like All Other Quantum Phases"

Problem Statement
This Comment addresses a recent proposal by Marletto and Vedral [Phys. Rev. Lett. 125, 040401 (2020)] regarding the Aharonov-Bohm (AB) effect. Marletto and Vedral propose a quantum field theory (QFT) model asserting that the AB phase is locally mediated by entanglement between a charged particle and the quantized electromagnetic field. Their model claims that the interaction energy can be described by a field-based overlap integral, rendering the AB phase gauge-independent even for non-closed paths. The author, Shan Gao, challenges these assertions, arguing that they misinterpret the physical mechanism of the AB effect and contain fundamental mathematical and physical errors within the framework of Quantum Electrodynamics (QED).

Methodology
The paper employs a critical analysis of the QED framework to evaluate the Marletto-Vedral model. The methodology involves:

Hamiltonian Reconstruction: Re-examining the interaction Hamiltonian in the Coulomb gauge, correcting sign errors in the interaction terms, and deriving the transition amplitude using second-order time-ordered perturbation theory.
Energy Formulation Comparison: Comparing the standard QED interaction energy (derived from minimal coupling) with the field-based energy formulation ( $E_{field}$ ) proposed by Marletto and Vedral. This involves explicit integration of magnetic and electric field overlaps.
Gauge Analysis: Testing the equivalence of the proposed field-based energy against the vector potential interaction ( $qv \cdot A$ ) across different gauges (specifically Coulomb vs. non-Coulomb) and under both static and time-dependent conditions.
Entanglement Assessment: Analyzing the post-interaction state of the system to determine the causal role of entanglement between the charged particle and the photon field in generating the observable phase.

Key Contributions and Results

Mathematical Flaws in Field-Based Energy:
The analysis identifies a critical error in the prefactor of Marletto and Vedral's field-based energy equation ( $E_{field} = \frac{1}{2} \int ...$ ). The author demonstrates that the factor of $1/2$ is inappropriate for the interaction energy between two distinct sources (the solenoid and the charged particle), leading to an underestimation of the interaction strength by a factor of two. Furthermore, the derivation reveals a sign error: while standard QED yields an interaction energy of $-qv \cdot A_s$ (where $A_s$ is the solenoid's vector potential), the Marletto-Vedral formulation yields $+qv \cdot A_s$ in the Coulomb gauge.
Gauge Dependence and Validity:
The paper proves that the equivalence between the field-based energy and the interaction energy $qv \cdot A_s$ holds only approximately in the Coulomb gauge under static conditions. For time-dependent fields or other gauges, this equivalence breaks down. Consequently, the claim that the field-based energy provides a gauge-independent description of the AB phase for non-closed paths is refuted. The author confirms that, consistent with standard QED, the AB phase for non-closed paths is inherently gauge-dependent, transforming as $\phi'_{AB} = \phi_{AB} - \frac{q}{\hbar}[\chi(r_2) - \chi(r_1)]$ under a gauge transformation $A' = A + \nabla \chi$ .
The Role of Entanglement:
The analysis demonstrates that while entanglement between the charged particle and the photon field arises naturally within the QED framework, it is not the causal mechanism driving the AB phase. By tracing out the photon field, the author shows that the off-diagonal terms of the reduced density matrix retain the phase difference $\Delta \phi$ regardless of the photon overlap (which is approximately unity in the AB regime). The phase is fundamentally determined by the coupling to the vector potential $A$ , with entanglement being a byproduct rather than the generator of the effect.
Reaffirmation of the Vector Potential:
The paper concludes that the AB phase arises from the interaction with the vector potential $A$ via the photon propagator, which serves as an effective description of the coupling between the charge's current and the solenoid's current. The conventional semi-classical explanation, where the phase is driven by the local coupling to $A$ , remains the correct interpretation.

Significance
The paper claims to correct fundamental flaws in the Marletto-Vedral model that misrepresent the mechanism of the Aharonov-Bohm effect. By exposing the incorrect prefactor, the sign error in the interaction energy, and the invalid generalization of gauge independence, the author argues that the proposal of a "locally mediated" phase via entanglement is unsupported. The significance of this work lies in reaffirming the conventional QED interpretation: the AB phase is a manifestation of the physical significance of the vector potential $A$ , and entanglement plays no causal role in its generation. This analysis seeks to clarify the theoretical underpinnings of the AB effect, ensuring that interpretations remain consistent with established QED principles.