The **Aharonov–Bohm effect**, sometimes called the **Ehrenberg–Siday–Aharonov–Bohm** effect, is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential (φ, **A**), despite being confined to a region in which both the magnetic field **B** and electric field **E** are zero.^{ [1] } 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.

- Significance
- Potentials vs. fields
- Global action vs. local forces
- Locality of electromagnetic effects
- Magnetic solenoid effect
- Monopoles and Dirac strings
- Electric effect
- Aharonov–Bohm nano rings
- Mathematical interpretation
- See also
- References
- Further reading
- External links

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.^{ [2] } There 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.^{ [2] } A separate "molecular" Aharonov–Bohm effect was proposed for nuclear motion in multiply connected regions, but this has been argued to be a different kind of geometric phase as it is "neither nonlocal nor topological", depending only on local quantities along the nuclear path.^{ [3] }

Werner Ehrenberg (1901–1975) and Raymond E. Siday first predicted the effect in 1949.^{ [4] } Yakir Aharonov and David Bohm published their analysis in 1959.^{ [1] } 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.^{ [5] }^{ [6] } The effect was confirmed experimentally, with a very large error, while Bohm was still alive. By the time the error was down to a respectable value, Bohm had died.^{ [7] }

In the 18th and 19th centuries, physics was dominated by Newtonian dynamics, with its emphasis on forces. Electromagnetic phenomena were elucidated by a series of experiments involving the measurement of forces between charges, currents and magnets in various configurations. Eventually, a description arose according to which charges, currents and magnets acted as local sources of propagating force fields, which then acted on other charges and currents locally through the Lorentz force law. In this framework, because one of the observed properties of the electric field was that it was irrotational, and one of the observed properties of the magnetic field was that it was divergenceless, it was possible to express an electrostatic field as the gradient of a scalar potential (e.g. Coulomb's electrostatic potential, which is mathematically analogous to the classical gravitational potential) and a stationary magnetic field as the curl of a vector potential (then a new concept – the idea of a scalar potential was already well accepted by analogy with gravitational potential). The language of potentials generalised seamlessly to the fully dynamic case but, since all physical effects were describable in terms of the fields which were the derivatives of the potentials, potentials (unlike fields) were not uniquely determined by physical effects: potentials were only defined up to an arbitrary additive constant electrostatic potential and an irrotational stationary magnetic vector potential.

The Aharonov–Bohm effect is important conceptually because it bears on three issues apparent in the recasting of (Maxwell's) classical electromagnetic theory as a gauge theory, which before the advent of quantum mechanics could be argued to be a mathematical reformulation with no physical consequences. The Aharonov–Bohm thought experiments and their experimental realization imply that the issues were not just philosophical.

The three issues are:

- whether potentials are "physical" or just a convenient tool for calculating force fields;
- whether action principles are fundamental;
- the principle of locality.

Because of reasons like these, the Aharonov–Bohm effect was chosen by the * New Scientist * magazine as one of the "seven wonders of the quantum world".^{ [8] }

It is generally argued that the Aharonov–Bohm effect illustrates the physicality of electromagnetic potentials, *Φ* and **A**, in quantum mechanics. Classically it was possible to argue that only the electromagnetic fields are physical, while the electromagnetic potentials are purely mathematical constructs, that due to gauge freedom aren't even unique for a given electromagnetic field.

However, Vaidman has challenged this interpretation by showing that the Aharonov–Bohm effect can be explained without the use of potentials so long as one gives a full quantum mechanical treatment to the source charges that produce the electromagnetic field.^{ [9] } According to this view, the potential in quantum mechanics is just as physical (or non-physical) as it was classically. Aharonov, Cohen, and Rohrlich responded that the effect may be due to a local gauge potential or due to non-local gauge-invariant fields.^{ [10] }

Two papers published in the journal * Physical Review A * in 2017 have demonstrated a quantum mechanical solution for the system. Their analysis shows that the phase shift can be viewed as generated by a solenoid's vector potential acting on the electron or the electron's vector potential acting on the solenoid or the electron and solenoid currents acting on the quantized vector potential.^{ [11] }^{ [12] }

Similarly, the Aharonov–Bohm effect illustrates that the Lagrangian approach to dynamics, based on energies, is not just a computational aid to the Newtonian approach, based on forces. Thus the Aharonov–Bohm effect validates the view that forces are an incomplete way to formulate physics, and potential energies must be used instead. In fact Richard Feynman complained^{[ citation needed ]} that he had been taught electromagnetism from the perspective of electromagnetic fields, and he wished later in life he had been taught to think in terms of the electromagnetic potential instead, as this would be more fundamental. In Feynman's path-integral view of dynamics, the potential field directly changes the phase of an electron wave function, and it is these changes in phase that lead to measurable quantities.

The Aharonov–Bohm effect shows that the local **E** and **B** fields do not contain full information about the electromagnetic field, and the electromagnetic four-potential, (*Φ*, **A**), must be used instead. By Stokes' theorem, the magnitude of the Aharonov–Bohm effect can be calculated using the electromagnetic fields alone, *or* using the four-potential alone. But when using just the electromagnetic fields, the effect depends on the field values in a region from which the test particle is excluded. In contrast, when using just the electromagnetic four-potential, the effect only depends on the potential in the region where the test particle is allowed. Therefore, one must either abandon the principle of locality, which most physicists are reluctant to do, or accept that the electromagnetic four-potential offers a more complete description of electromagnetism than the electric and magnetic fields can. On the other hand, the Aharonov–Bohm effect is crucially quantum mechanical; quantum mechanics is well known to feature non-local effects (albeit still disallowing superluminal communication), and Vaidman has argued that this is just a non-local quantum effect in a different form.^{ [9] }

In classical electromagnetism the two descriptions were equivalent. With the addition of quantum theory, though, the electromagnetic potentials *Φ* and **A** are seen as being more fundamental. ^{ [13] } Despite this, all observable effects end up being expressible in terms of the electromagnetic fields, **E** and **B**. This is interesting because, while you can calculate the electromagnetic field from the four-potential, due to gauge freedom the reverse is not true.

The magnetic Aharonov–Bohm effect can be seen as a result of the requirement that quantum physics be invariant with respect to the gauge choice for the electromagnetic potential, of which the magnetic vector potential forms part.

Electromagnetic theory implies that a particle with electric charge travelling along some path in a region with zero magnetic field , but non-zero (by ), acquires a phase shift , given in SI units by

Therefore, particles, with the same start and end points, but travelling along two different routes will acquire a phase difference determined by the magnetic flux through the area between the paths (via Stokes' theorem and ), and given by:

In quantum mechanics the same particle can travel between two points by a variety of paths. Therefore, this phase difference can be observed by placing a solenoid between the slits of a double-slit experiment (or equivalent). An ideal solenoid (i.e. infinitely long and with a perfectly uniform current distribution) encloses a magnetic field , but does not produce any magnetic field outside of its cylinder, and thus the charged particle (e.g. an electron) passing outside experiences no magnetic field . However, there is a (curl-free) vector potential outside the solenoid with an enclosed flux, and so the relative phase of particles passing through one slit or the other is altered by whether the solenoid current is turned on or off. This corresponds to an observable shift of the interference fringes on the observation plane.

The same phase effect is responsible for the quantized-flux requirement in superconducting loops. This quantization occurs because the superconducting wave function must be single valued: its phase difference around a closed loop must be an integer multiple of (with the charge for the electron Cooper pairs), and thus the flux must be a multiple of . The superconducting flux quantum was actually predicted prior to Aharonov and Bohm, by F. London in 1948 using a phenomenological model.^{ [14] }

The first claimed experimental confirmation was by Robert G. Chambers in 1960,^{ [15] }^{ [16] } in an electron interferometer with a magnetic field produced by a thin iron whisker, and other early work is summarized in Olariu and Popèscu (1984).^{ [17] } However, subsequent authors questioned the validity of several of these early results because the electrons may not have been completely shielded from the magnetic fields.^{ [18] }^{ [19] } An early experiment in which an unambiguous Aharonov–Bohm effect was observed by completely excluding the magnetic field from the electron path (with the help of a superconducting film) was performed by Tonomura et al. in 1986.^{ [20] }^{ [21] } The effect's scope and application continues to expand. Webb *et al.* (1985)^{ [22] } demonstrated Aharonov–Bohm oscillations in ordinary, non-superconducting metallic rings; for a discussion, see Schwarzschild (1986)^{ [23] } and Imry & Webb (1989).^{ [24] } Bachtold *et al.* (1999)^{ [25] } detected the effect in carbon nanotubes; for a discussion, see Kong *et al.* (2004).^{ [26] }

The magnetic Aharonov–Bohm effect is also closely related to Dirac's argument that the existence of a magnetic monopole can be accommodated by the existing magnetic source-free Maxwell's equations if both electric and magnetic charges are quantized.

A magnetic monopole implies a mathematical singularity in the vector potential, which can be expressed as a Dirac string of infinitesimal diameter that contains the equivalent of all of the 4π*g* flux from a monopole "charge" *g*. The Dirac string starts from, and terminates on, a magnetic monopole. Thus, assuming the absence of an infinite-range scattering effect by this arbitrary choice of singularity, the requirement of single-valued wave functions (as above) necessitates charge-quantization. That is, must be an integer (in cgs units) for any electric charge *q*_{e} and magnetic charge *q*_{m}.

Like the electromagnetic potential **A** the Dirac string is not gauge invariant (it moves around with fixed endpoints under a gauge transformation) and so is also not directly measurable.

Just as the phase of the wave function depends upon the magnetic vector potential, it also depends upon the scalar electric potential. By constructing a situation in which the electrostatic potential varies for two paths of a particle, through regions of zero electric field, an observable Aharonov–Bohm interference phenomenon from the phase shift has been predicted; again, the absence of an electric field means that, classically, there would be no effect.

From the Schrödinger equation, the phase of an eigenfunction with energy *E* goes as . The energy, however, will depend upon the electrostatic potential *V* for a particle with charge *q*. In particular, for a region with constant potential *V* (zero field), the electric potential energy *qV* is simply added to *E*, resulting in a phase shift:

where *t* is the time spent in the potential.

The initial theoretical proposal for this effect suggested an experiment where charges pass through conducting cylinders along two paths, which shield the particles from external electric fields in the regions where they travel, but still allow a time dependent potential to be applied by charging the cylinders. This proved difficult to realize, however. Instead, a different experiment was proposed involving a ring geometry interrupted by tunnel barriers, with a constant bias voltage *V* relating the potentials of the two halves of the ring. This situation results in an Aharonov–Bohm phase shift as above, and was observed experimentally in 1998, albeit in a setup where the charges do traverse the electric field generated by the bias voltage. The original time dependent electric Aharonov–Bohm effect has not yet found experimental verification.^{ [27] }

Nano rings were created by accident^{ [28] } while intending to make quantum dots. They have interesting optical properties associated with excitons and the Aharonov–Bohm effect.^{ [28] } Application of these rings used as light capacitors or buffers includes photonic computing and communications technology. Analysis and measurement of geometric phases in mesoscopic rings is ongoing.^{ [29] }^{ [30] }^{ [31] } It is even suggested they could be used to make a form of slow glass.^{ [32] }

Several experiments, including some reported in 2012,^{ [33] } show Aharonov–Bohm oscillations in charge density wave (CDW) current versus magnetic flux, of dominant period *h*/2*e* through CDW rings up to 85 µm in circumference above 77 K. This behavior is similar to that of the superconducting quantum interference devices (see SQUID).

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The Aharonov–Bohm effect can be understood from the fact that one can only measure absolute values of the wave function. While this allows for measurement of phase differences through quantum interference experiments, there is no way to specify a wavefunction with constant absolute phase. In the absence of an electromagnetic field one can come close by declaring the eigenfunction of the momentum operator with zero momentum to be the function "1" (ignoring normalization problems) and specifying wave functions relative to this eigenfunction "1". In this representation the i-momentum operator is (up to a factor ) the differential operator . However, by gauge invariance, it is equally valid to declare the zero momentum eigenfunction to be at the cost of representing the i-momentum operator (up to a factor) as i.e. with a pure gauge vector potential . There is no real asymmetry because representing the former in terms of the latter is just as messy as representing the latter in terms of the former. This means that it is physically more natural to describe wave "functions", in the language of differential geometry, as sections in a complex line bundle with a hermitian metric and a U(1)-connection . The curvature form of the connection, , is, up to the factor i, the Faraday tensor of the electromagnetic field strength. The Aharonov–Bohm effect is then a manifestation of the fact that a connection with zero curvature (i.e. flat), need not be trivial since it can have monodromy along a topologically nontrivial path fully contained in the zero curvature (i.e. field free) region. By definition this means that sections that are parallelly translated along a topologically non trivial path pick up a phase, so that covariant constant sections cannot be defined over the whole field free region.

Given a trivialization of the line-bundle, a non-vanishing section, the U(1)-connection is given by the 1-form corresponding to the electromagnetic four-potential *A* as where *d* means exterior derivation on the Minkowski space. The monodromy is the holonomy of the flat connection. The holonomy of a connection, flat or non flat, around a closed loop is (one can show this does not depend on the trivialization but only on the connection). For a flat connection one can find a gauge transformation in any simply connected field free region(acting on wave functions and connections) that gauges away the vector potential. However, if the monodromy is nontrivial, there is no such gauge transformation for the whole outside region. In fact as a consequence of Stokes' theorem, the holonomy is determined by the magnetic flux through a surface bounding the loop , but such a surface may exist only if passes through a region of non trivial field:

The monodromy of the flat connection only depends on the topological type of the loop in the field free region (in fact on the loops homology class). The holonomy description is general, however, and works inside as well as outside the superconductor. Outside of the conducting tube containing the magnetic field, the field strength . In other words, outside the tube the connection is flat, and the monodromy of the loop contained in the field-free region depends only on the winding number around the tube. The monodromy of the connection for a loop going round once (winding number 1) is the phase difference of a particle interfering by propagating left and right of the superconducting tube containing the magnetic field. If one wants to ignore the physics inside the superconductor and only describe the physics in the outside region, it becomes natural and mathematically convenient to describe the quantum electron by a section in a complex line bundle with an "external" flat connection with monodromy

- magnetic flux through the tube /

rather than an external EM field . The Schrödinger equation readily generalizes to this situation by using the Laplacian of the connection for the (free) Hamiltonian

- .

Equivalently, one can work in two simply connected regions with cuts that pass from the tube towards or away from the detection screen. In each of these regions the ordinary free Schrödinger equations would have to be solved, but in passing from one region to the other, in only one of the two connected components of the intersection (effectively in only one of the slits) a monodromy factor is picked up, which results in the shift in the interference pattern as one changes the flux.

Effects with similar mathematical interpretation can be found in other fields. For example, in classical statistical physics, quantization of a molecular motor motion in a stochastic environment can be interpreted as an Aharonov–Bohm effect induced by a gauge field acting in the space of control parameters.^{ [34] }

In the Standard Model of particle physics, the **Higgs mechanism** is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other being fermions) would be considered massless, but measurements show that the W^{+}, W^{−}, and Z^{0} bosons actually have relatively large masses of around 80 GeV/c^{2}. The Higgs field resolves this conundrum. The simplest description of the mechanism adds a quantum field (the Higgs field) that permeates all space to the Standard Model. Below some extremely high temperature, the field causes spontaneous symmetry breaking during interactions. The breaking of symmetry triggers the Higgs mechanism, causing the bosons it interacts with to have mass. In the Standard Model, the phrase "Higgs mechanism" refers specifically to the generation of masses for the W^{±}, and Z weak gauge bosons through electroweak symmetry breaking. The Large Hadron Collider at CERN announced results consistent with the Higgs particle on 14 March 2013, making it extremely likely that the field, or one like it, exists, and explaining how the Higgs mechanism takes place in nature.

**Magnetic vector potential**, **A**, is the vector quantity in classical electromagnetism defined so that its curl is equal to the magnetic field: . Together with the electric potential *φ*, the magnetic vector potential can be used to specify the electric field **E** as well. Therefore, many equations of electromagnetism can be written either in terms of the fields **E** and **B**, or equivalently in terms of the potentials *φ* and **A**. In more advanced theories such as quantum mechanics, most equations use potentials rather than fields.

The **Little–Parks effect** was discovered in 1962 by William A. Little and Roland D. Parks in experiments with empty and thin-walled superconducting cylinders subjected to a parallel magnetic field. It was one of the first experiments that indicate the importance of Cooper-pairing principle in BCS theory.

In quantum mechanics, a **parity transformation** is the flip in the sign of *one* spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates :

In electromagnetism, the **Lorenz gauge condition** or **Lorenz gauge**, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring The name is frequently confused with Hendrik Lorentz, who has given his name to many concepts in this field. The condition is Lorentz invariant. The condition does not completely determine the gauge: one can still make a gauge transformation where is a harmonic scalar function. The Lorenz condition is used to eliminate the redundant spin-0 component in the (1/2, 1/2) representation theory of the Lorentz group. It is equally used for massive spin-1 fields where the concept of gauge transformations does not apply at all.

In the physics of gauge theories, **gauge fixing** denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.

The **electric-field integral equation** is a relationship that allows the calculation of an electric field (**E**) generated by an electric current distribution (**J**).

In physics, **charge conservation** is the principle that the total electric charge in an isolated system never changes. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always *conserved*. Charge conservation, considered as a physical conservation law, implies that the change in the amount of electric charge in any volume of space is exactly equal to the amount of charge flowing into the volume minus the amount of charge flowing out of the volume. In essence, charge conservation is an accounting relationship between the amount of charge in a region and the flow of charge into and out of that region, given by a continuity equation between charge density and current density .

In physics, a **quantum vortex** represents a quantized flux circulation of some physical quantity. In most cases, quantum vortices are a type of topological defect exhibited in superfluids and superconductors. The existence of quantum vortices was first predicted by Lars Onsager in 1949 in connection with superfluid helium. Onsager reasoned that quantisation of vorticity is a direct consequence of the existence of a superfluid order parameter as a spatially continuous wavefunction. Onsager also pointed out that quantum vortices describe the circulation of superfluid and conjectured that their excitations are responsible for superfluid phase transitions. These ideas of Onsager were further developed by Richard Feynman in 1955 and in 1957 were applied to describe the magnetic phase diagram of type-II superconductors by Alexei Alexeyevich Abrikosov. In 1935 Fritz London published a very closely related work on magnetic flux quantization in superconductors. London's fluxoid can also be viewed as a quantum vortex.

The **London equations,** developed by brothers Fritz and Heinz London in 1935, are constitutive relations for a superconductor relating its superconducting current to electromagnetic fields in and around it. Whereas Ohm's law is the simplest constitutive relation for an ordinary conductor, the London equations are the simplest meaningful description of superconducting phenomena, and form the genesis of almost any modern introductory text on the subject. A major triumph of the equations is their ability to explain the Meissner effect, wherein a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold.

The **quantum potential** or **quantum potentiality** is a central concept of the de Broglie–Bohm formulation of quantum mechanics, introduced by David Bohm in 1952.

In physics, **relativistic quantum mechanics** (**RQM**) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light *c*, and can accommodate massless particles. The theory has application in high energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics. *Non-relativistic quantum mechanics* refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. *Relativistic quantum mechanics* (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them also work with special relativity.

A **composite fermion** is the topological bound state of an electron and an even number of quantized vortices, sometimes visually pictured as the bound state of an electron and, attached, an even number of magnetic flux quanta. Composite fermions were originally envisioned in the context of the fractional quantum Hall effect, but subsequently took on a life of their own, exhibiting many other consequences and phenomena.

In physics, a **gauge theory** is a type of field theory in which the Lagrangian does not change under local transformations from certain Lie groups.

In physics, **Berry connection** and **Berry curvature** are related concepts which can be viewed, respectively, as a local gauge potential and gauge field associated with the Berry phase or geometric phase. These concepts were introduced by Michael Berry in a paper published in 1984 emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics.

The **Aharonov–Casher effect** is a quantum mechanical phenomenon predicted in 1984 by Yakir Aharonov and Aharon Casher, in which a traveling magnetic dipole is affected by an electric field. It is dual to the Aharonov–Bohm effect, in which the quantum phase of a charged particle depends upon which side of a magnetic flux tube it comes through. In the Aharonov–Casher effect, the particle has a magnetic moment and the tubes are charged instead. It was observed in a gravitational neutron interferometer in 1989 and later by fluxon interference of magnetic vortices in Josephson junctions. It has also been seen with electrons and atoms.

The **angular momentum of light** is a vector quantity that expresses the amount of dynamical rotation present in the electromagnetic field of the light. While traveling approximately in a straight line, a beam of light can also be rotating around its own axis. This rotation, while not visible to the naked eye, can be revealed by the interaction of the light beam with matter.

In quantum mechanics, the **Byers-Yang theorem** states that all physical properties of a doubly connected system enclosing a magnetic flux through the opening are periodic in the flux with period . The theorem was first stated and proven by Nina Byers and Chen-Ning Yang (1961), and further developed by Felix Bloch (1970).

In theoretical physics, the **dual photon** is a hypothetical elementary particle that is a dual of the photon under electric–magnetic duality which is predicted by some theoretical models, including M-theory.

The **Maxwell-Lodge effect** is a phenomenon of elctromagnetic induction in which an electric charge, near a solenoid in which current changes slowly, feels an electromotive force (e.m.f.) even if the magnetic field is practically static inside and null outside. It can be considered a classical analogue of the quantum mechanical Aharonov–Bohm effect, where instead the field is exactly static inside and null outside.

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*and*...is the vector potential a "real" field? ... a real field is a mathematical device for avoiding the idea of action at a distance. .... for a long time it was believed that**A**was not a "real" field. .... there are phenomena involving quantum mechanics which show that in fact**A**is a "real" field in the sense that we have defined it.....**E**and**B**are slowly disappearing from the modern expression of physical laws; they are being replaced by**A**[the vector potential] and [the scalar potential] - ↑ London, F (1948). "On the Problem of the Molecular Theory of Superconductivity".
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- D. J. Thouless (1998). "§2.2 Gauge invariance and the Aharonov–Bohm effect".
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