User:Nonabelian/Aharonov-Bohm Effect



The Aharonov–Bohm (AB) effect is a quantum mechanical phenomenon where an electrically charged particle, an electron, is affected by an electromagnetic potential despite there being no magnetic or electric force (i.e. the magnetic field and electric field are zero). This effect was long seen as defying conventional thinking in physics: usually forces are described by vector fields, which in turn are described by changes in potential energy, given by the gradient of a scalar field. If the force is zero, their should no change in potential energy either. A satisfactory explanation of why this is the case was not established until the 1970's with the use of gauge theory. The AB effect was the first experimental proof that gauge fields are fundamental fields in the universe, rather than either potential or force fields.

In the most common case describing the AB effect, a double slit experiment is set up with a long solenoid in the middle. A charged quantum particle, an electron, travels around the solenoid and experiences a change in its phase that can be observed in its diffraction pattern. This phase shift happens despite the magnetic field in the solenoid being fully enclosed and having negligible strength in the outer region where the particle's wave-function passes; similarly the particle also has negligible chance of its wave-function passing through the inside the solenoid.

The AB effect has prompted extensive discussion and debate since David Bohm and Yakir Aharonov published their paper in 1959. One reason for this widespread interest is that the effect challenges conventional thinking. Moreover, it touches on crucial elements of quantum mechanics, such as the tangible reality of potential fields, which previously could be argued as having now physical significance but a mere mathematical convenience. The AB effect had had renewed interest with the theory of gauge fields. In a famous paper by Wu and Yang in 1975, they extend vector potentials in the AB experiment to be non-abelian gauge fields. This theory suggests that gauge fields represent the most basic and essential components of physical reality, and that both the vector and potential fields lead to an incomplete view of reality. Notably, the AB effect stands out as the only experimental proof supporting the idea of gauge theory.

In 1985 Michael Berry showed that the Aharonov-Bohm effect can be generalised in the geometrical concept of the Berry Phase.

Overview
In traditional electromagnetism as described by Maxwell's equations, all electromagnetic phenomena shown by charged particles can be described by the electric $\mathbf{E}$ and magnetic $\mathbf{B}$ fields. These forces arise form the differences in the potential energy fields. However, since the absolute value of these potential fields cannot be measured, and only the differences, the argument can be made that the potentials are mere mathematical conveniences with no physical meaning or that can be described fully in other terms of the electric $\mathbf{E}$ and magnetic $\mathbf{B}$ fields. The AB effect ultimately shows that this is an incomplete understanding of electromagnetism and the potentials can have physical consequences. While it might be tempting to think that the AB effect shows that potentials are more fundamental, the AB effect was ultimately used to show that it is neither the vector or potential fields are which can describe all electromagnetic phenomena but a novel aspect called the non-intergrable phase factor.

Maxwell's equations can be generalised into a gauge fields and the properties of electromagnetism can be shown to be a gauge field with U(1) symmetry.

quantum-mechanical phenomenon in which an electrically charged particle is affected by a The underlying mechanism is the coupling of the electromagnetic potential with the complex phase of a charged particle's wave function, and the Aharonov–Bohm effect is accordingly illustrated by interference experiments.

The most commonly described case, sometimes called the Aharonov–Bohm solenoid effect, takes place when the wave function of a charged particle passing around a long solenoid experiences a phase shift as a result of the enclosed magnetic field, despite the magnetic field being negligible in the region through which the particle passes and the particle's wavefunction being negligible inside the solenoid. This phase shift has been observed experimentally.

here are also magnetic Aharonov–Bohm effects on bound energies and scattering cross sections, but these cases have not been experimentally tested. An electric Aharonov–Bohm phenomenon was also predicted, in which a charged particle is affected by regions with different electrical potentials but zero electric field, but this has no experimental confirmation yet.

History
Walter Franz was arguably the first to notice that an enclosed magnetic flux could have an effect on an electon quantum interference pattern in his 1939 paper but it is not clear if he envisaged an electron still being affected with no interaction with the magnetic vector field. Werner Ehrenberg (1901–1975) and Raymond E. Siday predicted the effect in 1949 but the full significance of the physicality of the potential field only became apparent after the detailed description by Yakir Aharonov and David Bohm, published in 1959. After publication of the 1959 paper, Bohm was informed of Ehrenberg and Siday's work, which was acknowledged and credited in Bohm and Aharonov's subsequent 1961 paper.

However, the debate raged as to whether the observation was due to experimental error or could be explained by other processes that would mean the AB effect did not really exist. By the time the debate was settled in 1986 with an experiment, Bohm had died.

The Aharonov-Bohm effect is an effect first discussed by physicists Aharonov and Bohm in 1959 that was observed experimentally shortly after.

Aharanov and Berry jointly received the Wolt Prize in 1998 for their for the discovery of quantum topological and geometrical phases.

It demonstrates the influence of a magnetic field on the phase of the wavefunction $Ψ$ of a charged particle, even when the wave function is zero in the region where the magnetic field is nonvanishing.

Topology
The AB effect gained significant recognition during the late 1970s. due to the discovery of its connection to the theory of gauge fields.

Gauge fields were first introduced by Yang and Mills back in 1954, and then expanded upon by Utiyama in 1956. As the years went on, this theory started playing a pivotal role in understanding the fundamental forces of nature. An iconic milestone was achieved when scientists Weinberg and Salam successfully combined the theories of electromagnetism (the force behind electric and magnetic effects) and the weak force (one of the forces responsible for nuclear reactions) into one unified framework. This merger further elevated the importance of gauge fields, making them a strong contender for a single theory that could describe all known interactions in the universe.

In 1975, Wu and Yang added another layer to this understanding of gauge fields. They presented a new and fundamental explanation of electromagnetism. Instead of focusing on conventional terms like field strength or vector potential as being fundamental, a novel aspect called the non-integrable (i.e. path dependant) phase factor is shown to be fundamental and explain all electromagnetic effects.

The non-integrable phase factor is a mathematical term that helps describe the behavior of all particles in an electromagnetic field. Via a mathematical tool known as fiber bundles, Wu and Yang were able to exend this key insight to more complex, non-abelian gauge fields. In this framework, the non-integrable phase factor can be visualised as a sort of "guiding path" for charged particles.

Wu and Yan explained that the Aharonov-Bohm effect, when seen through this fibre bundle lens, can be be explained as a geometric consequence of this fibre bundle "guiding path". The AB effect is the only experimental proof that electromagnetism is fundamentally described by a gauge field.

Relation to Geometric (Berry) Phase
The Aharonov-Bohm effect is generalised in the geometrical concept of the Berry Phase.

In his original paper (Berry, 1984) Berry argues that the AB e�ect is a speci�c instance of his geometrical phase and hence an instance of anholon

A geometric phase is what mathematicians would call a U(1) holonomy. The mathematics of holonomies are described by the theory of fiber bundles and their related mathematical concepts of connections and parallel transport.

Relation to Magnetic Monopoles
For H= U(l), the electromagnetic gauge group, Wu and Yang (1975) showed that the single valuedness of 7 leads to the Dirac quantisation condition and that the homotopy class of 7 is labelled by the magnetic charge. The relationship between the work of Wu and Yang and Dirac has been discussed by Brandt and Primack (1977).

Aharaonov-Casher effect
The Aharaonov-Casher effect is an analogous effect for the electric fields.

Gravitational Aharonov-Bohm effect
The Gravitational Aharonov-Bohm (AB) effect is a quantum phenomenon where the wave function of a particle undergoes a phase shift in the presence of a gravitational potential, even when there is no classical force acting on the particle. This is analogous to the electromagnetic AB effect, where the wave function of a particle is shifted by an electrostatic (scalar) potential in the absence of any classical force field. The gravitational effect shares the features of its electromagnetic cousin in being nondispersive, non-local, and topological in nature: i.e. no number of local measurements at any location (e.g. by gravimeters) in which the particle is allowed to exist can predict the gravitostatic AB effect.

In 2022 the gravitational Aharonov-Bohm effect effect was observed experimentally for the first time. The observation was based on a test design first proposed in 2012. In the experiment, ultra-cold rubidium atoms in superposition were launched vertically inside a vacuum tube and split with a laser so that one part would go higher than the other and then recombined back. Outside of the chamber at the top sits a mass that changes the gravitational potential. Thus, the part that goes higher should experience a greater change which manifests as an interference pattern when the wave packets recombine resulting in a measurable phase shift.

Multiple other tests have been proposed.