In physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

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

Paul Dirac

Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

, stretching between two Dirac magnetic monopole

Magnetic monopole

A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...

s with opposite magnetic charges, or from one magnetic monopole out to infinity. The gauge potential cannot be defined on the Dirac string, but it is defined everywhere else. The Dirac string acts as the solenoid

Solenoid

A solenoid is a coil wound into a tightly packed helix. In physics, the term solenoid refers to a long, thin loop of wire, often wrapped around a metallic core, which produces a magnetic field when an electric current is passed through it. Solenoids are important because they can create...

 in the Aharonov-Bohm effect

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 field , despite being confined to a region in which both the magnetic field B and electric field E are...

, 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 . The magnetic flux running along the interior of the string maintains the validity of Maxwell's equations

Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

. If Maxwell equations are modified to allow magnetic charges at the fundamental level then the magnetic monopoles are Dirac monopoles no longer and do not require attached Dirac strings.

Details

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

Cohomology

In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...

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

Field strength

In physics, the field strength of a field is the magnitude of its vector value.In theoretical physics, field strength is another name for the curvature form...

s , which are manifestly exact forms, modulo all possible gauge transformations, given that the field strength F must be a closed form

Closed and exact differential forms

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 that is the exterior derivative of another differential form β...

: . Here, A is the vector potential

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 negative gradient is a given vector field....

 and d represents the gauge-covariant derivative

Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

, and F the field strength or curvature form

Curvature form

In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.-Definition:...

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 everywhere except at the location of the monopole.