What is the geometry of a gauge potential in the A-B experiment?

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In summary, the conversation discusses an article written in 1981 by Bernstein and Phillips on fiber bundles and quantum fields. The article is still considered a useful reference and is often used by lecturers at universities. The question is about how the authors determine the geometry of the magnetic vector potential in the original A-B experiment, which is described as a hemisphere, and outside the solenoid, which is described as a truncated cone. The conversation also includes a link to another article by T. T. Wu and C. N. Yang, which is a standard reference for the geometry and topology of the AB effect. The link to the original article mentioned is also provided.
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Anko
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Aharonov-Bohm effect
Hi, this is a question about an article in the Scientific American magazine.

In 1981 Bernstein and Phillips wrote an article about fiber bundles and quantum fields, and I believe it's still a useful reference, the kind of thing lecturers would use at university.

Anyway, my question is, how do the authors determine that the geometry of the magnetic vector potential, in the original A-B experiment, is topologically a hemisphere, and that outside the solenoid the potential is geometrically a truncated cone?
 
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Do you have a link to the article? Concerning the geometry/topology and the AB effect a standard reference is

T. T. Wu and C. N. Yang, Concept of nonintegrable phase factors and global for-
mulation of gauge fields, Phys. Rev. D 12, 3845 (1975),
http://link.aps.org/abstract/PRD/v12/i12/p3845
 
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FAQ: What is the geometry of a gauge potential in the A-B experiment?

What is the Aharonov-Bohm (A-B) experiment?

The Aharonov-Bohm experiment demonstrates that charged particles are affected by electromagnetic potentials, even in regions where the magnetic and electric fields are zero. This is achieved by showing a phase shift in the wave function of electrons that travel around a region with a confined magnetic field, without entering it.

What is a gauge potential?

A gauge potential is a mathematical function used in field theories, such as electromagnetism, to describe the electromagnetic field. In the context of the Aharonov-Bohm experiment, the vector potential \( \mathbf{A} \) is the gauge potential that influences the phase of the electron's wave function.

How does the gauge potential affect the electron in the A-B experiment?

In the A-B experiment, the gauge potential \( \mathbf{A} \) affects the electron by inducing a phase shift in its wave function. This phase shift is given by the line integral of the vector potential along the electron's path, which results in observable interference patterns even though the electrons travel through regions with zero magnetic field.

What is the geometric interpretation of the gauge potential in the A-B experiment?

The geometric interpretation of the gauge potential in the A-B experiment involves the concept of holonomy or parallel transport in a fiber bundle. The phase shift experienced by the electron can be understood as a consequence of the curvature of the connection (gauge potential) in this bundle, which is related to the magnetic flux enclosed by the electron's path.

Why is the A-B experiment significant in quantum mechanics?

The A-B experiment is significant because it provides direct evidence that potentials, rather than just fields, play a fundamental role in quantum mechanics. It challenges classical intuitions by showing that the physical effects of electromagnetic potentials can be observed even in regions where the fields are zero, highlighting the importance of the underlying geometry and topology in quantum theory.

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