View source for Bohm-Aharonov effect

A phenomenon in which the topological non-triviality of a gauge field is measurable physically [[#References|[a1]]]. Moreover, this topological non-triviality, which can be expressed as a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b1106901.png" />, say, is a global topological invariant and so is not expressible by a local formula; this latter point being in contrast to a simpler topological invariant such as the dimension of the underlying space, which is deducible locally.

To demonstrate this effect physically, one arranges that a non-simply-connected region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b1106902.png" /> of space has zero electromagnetic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b1106903.png" />. This electromagnetic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b1106904.png" /> is related to the gauge field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b1106905.png" /> by the usual relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b1106906.png" /></td> </tr></table>

where the usual [[Differential form|differential form]] language has been used for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b1106907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b1106908.png" /> so that, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b1106909.png" /> are local coordinates on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069010.png" />, one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069011.png" /></td> </tr></table>

Given this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069013.png" />, one can devise an experiment in which one measures a diffraction pattern associated with the parallel transport, or holonomy, of the gauge field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069014.png" /> round a non-contractible loop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069016.png" /> (cf. also [[Holonomy group|Holonomy group]]). The action of parallel transport on multi-linear objects, viewed either as vectors, spinors, tensors, etc., or as sections of the appropriate bundles, is via the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069017.png" />, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069018.png" /></td> </tr></table>

[[Differential topology|Differential topology]] provides immediately the means to see that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069019.png" /> is non-trivial. The argument goes as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069020.png" /> so that the vanishing of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069021.png" /> gives

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069022.png" /></td> </tr></table>

Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069023.png" /> determines a [[De Rham cohomology|de Rham cohomology]] class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069024.png" /> and one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069025.png" /></td> </tr></table>

It is clear from Stokes' theorem (cf. [[Stokes theorem|Stokes theorem]]) that the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069026.png" /> only depends on the homotopy class of the loop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069027.png" />. In addition, the loop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069028.png" /> determines a homology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069029.png" />, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069030.png" /></td> </tr></table>

This means that the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069031.png" /> (which can be taken equal to the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069032.png" />) is just the dual pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069033.png" /> between cohomology and homology, i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069034.png" /></td> </tr></table>

An experiment to test this for the electromagnetic field was done in [[#References|[a2]]]. The experimental setup is of the Young's slits type, where electrons replace photons and with the addition of a very thin solenoid. The electrons pass through the slits and on either side of the solenoid an interference pattern is then detected. The interference pattern is first measured with the solenoid off. This pattern is then found to change when the solenoid is switched on, even though the electrons always pass through a region where the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069035.png" /> is zero.

Mathematically speaking one realizes the solenoid by a cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069036.png" /> so that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069037.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069038.png" /></td> </tr></table>

The loop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069039.png" /> is the union of the electron paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069041.png" />.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b110690a.gif" />

Figure: b110690a

A schematic Bohm–Aharonov experiment

The Bohm–Aharonov effect shows that the gauge potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069042.png" /> is a more basic object than the electromagnetic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069043.png" />.

Geometrically speaking, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069044.png" /> is a connection and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069045.png" /> is a curvature. Hence, in purely geometrical language one can describe the situation by saying that a flat connection (i.e. one for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069046.png" />, cf. [[Parallel displacement(2)|Parallel displacement]]) need not be trivial if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069047.png" /> is not simply connected, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069048.png" />. This non-triviality of flat connections when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069049.png" /> extends to the non-Abelian or Yang–Mills case.

For a Yang–Mills <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069050.png" />-connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069051.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069052.png" /> non-Abelian), the curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069053.png" /> is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069054.png" /></td> </tr></table>

Further, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069055.png" /> is defined on a bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069056.png" /> over a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069057.png" /> and if it is assumed that, as in the Abelian case,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069058.png" /></td> </tr></table>

then the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069059.png" /> only depends on the homotopy class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069060.png" />, which is also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069061.png" /> (the context should prevent any confusion with homology classes). Adding to this the fact that the parallel transport operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069062.png" /> has the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069063.png" />, one can use <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069064.png" /> to define the following linear mapping:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069065.png" /></td> </tr></table>

One can check that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069066.png" /> so that, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069067.png" /> varies, the flat <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069068.png" />-connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069069.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069070.png" /> gives a representation of the [[Fundamental group|fundamental group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069071.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069072.png" />; it is easy to verify that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069073.png" /> under a gauge transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069074.png" />, then this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069075.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069076.png" /> via the adjoint action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069077.png" />. In other words, it acts by conjugation, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069078.png" />. But this means that the representations defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069080.png" /> are equivalent, thus the gauge equivalence class of flat connections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069081.png" /> is characterized by an element of the quotient

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069082.png" /></td> </tr></table>

Hence the moduli space of gauge-inequivalent flat connections is the space

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069083.png" /></td> </tr></table>

The holonomy group element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110690/b11069084.png" /> is also of central interest elsewhere. It occurs in the study of the adiabatic periodic change of parameters of a quantum system described in [[Quantum field theory|quantum field theory]], [[#References|[a3]]], [[#References|[a4]]], where it is called a Wilson line or Wilson loop, and in mathematics. One of the most important of these concerns the quantum field theory formulation of knot invariants [[#References|[a5]]], such as the Jones polynomial [[#References|[a6]]]. Flat connections also play a distinguished part in this theory.

See also [[Quantum field theory|Quantum field theory]]; [[Knot theory|Knot theory]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Y. Aharonov,   D. Bohm,   "Significance of electromagnetic potentials in quantum theory"  ''Phys. Rev.'' , '''115'''  (1959)  pp. 485–491</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.R. Brill,   F.G. Werner,   "Significance of electromagnetic potentials in the quantum theory in the interpretation of electron fringe interferometer observations"  ''Phys. Rev. Lett.'' , '''4'''  (1960)  pp. 344–347</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.V. Berry,   "Quantal phase factors accompanying adiabatic changes"  ''Proc. Roy. Soc. London A'' , '''392'''  (1984)  pp. 45–57</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. Simon,   "Holonomy, the quantum adiabatic theorem and Berry's phase"  ''Phys. Rev. Lett.'' , '''51'''  (1983)  pp. 2167–2170</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Witten,   "Topological quantum field theory"  ''Comm. Math. Phys.'' , '''117'''  (1988)  pp. 353–386</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  V.F.R. Jones,   "A polynomial invariant for knots via von Neumann algebras"  ''Bull. Amer. Math. Soc.'' , '''12'''  (1985)  pp. 103–111</TD></TR></table>