physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

, a Dirac string is a one-dimensional curve in space, conceived of by the physicist

Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...

, stretching between two hypothetical Dirac monopoles with opposite magnetic charges, or from one magnetic monopole out to infinity. The

gauge potential In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...

cannot be defined on the Dirac string, but it is defined everywhere else. The Dirac string acts as the solenoid in the

Aharonov–Bohm effect 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 confine ...

, and the requirement that the position of the Dirac string should not be observable implies the Dirac quantization rule: the product of a magnetic charge and an electric charge must always be an integer multiple of

2\pi\hbar

. Also, a change of position of a Dirac string corresponds to a gauge transformation. This shows that Dirac strings are not gauge invariant, which is consistent with the fact that they are not observable. The Dirac string is the only way to incorporate magnetic monopoles into

Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...

, since the magnetic flux running along the interior of the string maintains their validity. If Maxwell equations are modified to allow magnetic charges at the fundamental level then the magnetic monopoles are no longer Dirac monopoles, and do not require attached Dirac strings.

Details

The quantization forced by the Dirac string can be understood in terms of the

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

of the

fibre bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...

representing the gauge fields over the base manifold of space-time. The magnetic charges of a gauge field theory can be understood to be the group generators of the cohomology group

H^2(M)

for the fiber bundle ''M''. The cohomology arises from the idea of classifying all possible gauge field strengths

F=dA

, which are manifestly

exact form In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another diff ...

s, modulo all possible gauge transformations, given that the field strength ''F'' must be a closed form:

dF=0

. Here, ''A'' is the

vector potential In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field v, a ''vecto ...

and ''d'' represents the gauge- covariant derivative, and ''F'' the field strength or

curvature form In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie algeb ...

on the fiber bundle. Informally, one might say that the Dirac string carries away the "excess curvature" that would otherwise prevent ''F'' from being a closed form, as one has that

dF=0

everywhere except at the location of the monopole.

References

* {{DEFAULTSORT:Dirac String Gauge theories Magnetic monopoles