In theoretical physics, the Dirac monopole is another name for the magnetic monopole; the article about the magnetic monopoles contains a lot of useful information. Theoretical physics attempts to understand the world by making a model of reality, used for rationalizing, explaining, and predicting physical phenomena through a physical theory. There are three types of theories in physics: mainstream theories, proposed theories and fringe theories. ... In physics, magnetic monopole is a term describing a hypothetical particle that could be quickly clarified to a person familiar with magnets but not electromagnetic theory as a magnet with only one pole. In more accurate terms, it would have net magnetic charge. Interest in the concept stems from particle...

In 1931, Paul Dirac realized that there is a symmetry between the electric fields and the magnetic fields in Maxwell's equations. However, only the electric charges appear as charged monopoles; the magnetic sources only appear as magnetic dipoles: the Southern pole of a magnet cannot be separated from the Northern pole. 1931 is a common year starting on Thursday. ... Paul Adrien Maurice Dirac Paul Adrien Maurice Dirac, (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... In physics, an electric field or E-field is an effect produced by an electric charge that exerts a force on charged objects in its vicinity. ... Current flowing through a wire produces a magnetic field (M) around the wire. ... Maxwells equations are the set of four equations, attributed to James Clerk Maxwell (written by Oliver Heaviside), that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... This article is about the electromagnetic phenomenon. ...

Dirac investigated the question whether the magnetic sources can appear as monopoles. He considered a point-like magnetic charge whose magnetic field behaves as μ / r² and is directed in the radial direction. Because the divergence of B is equal to zero almost everywhere, except for the locus of the magnetic monopole at r = 0, one can locally define the vector potential such that the curl of the vector potential A equals the magnetic field B. In vector calculus, a vector potential is a vector field whose curl is a given vector field. ... In vector calculus, curl is a vector operator that shows a vector fields rate of rotation about a point. ...

However, the vector potential cannot be defined globally precisely because the divergence of the magnetic field is proportional to the delta function at the origin. We must define one set of functions for the vector potential on the Northern hemisphere, and another set of functions for the Southern hemispheres. These two vector potentials are matched at the equator, and they differ by a gauge transformation. The wave function of an electrically charged particle (a probe) that orbits the equator generally changes by a phase, much like in the Aharonov-Bohm effect. This phase is proportional to the electric charge q_e of the probe, as well as to the magnetic charge q_m of the source. Dirac was originally considering an electron whose wave function is described by the Dirac equation. The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared... The Aharonov-Bohm effect is a quantum mechanical phenomenon by which a charged particle is affected by electromagnetic fields in regions from which the particle is excluded, proposed by Aharonov and Bohm in 1959. ... Properties The electron is a subatomic particle. ... The Dirac equation is a relativistic quantum mechanical wave equation invented by Paul Dirac in 1928. ...

Because the electron returns to the same point after the full trip around the equator, the phase exp(iφ) of its wave function must be unchanged, which implies that the phase φ added to the wave function must be a multiple of 2π:

This is known as the Dirac quantization condition. In certain units, the condition is exactly given by the relation above. The hypothetical existence of a magnetic monopole would imply that the electric charge must be quantized in certain units; also, the existence of the electric charges implies that the magnetic charges of the hypothetical magnetic monopoles, if they exist, must be quantized in units inverse to the elementary electric charge.

If we maximally extend the definition of the vector potential for the Southern hemisphere, it will be defined everywhere except for a semi-infinite line stretched from the origin in the direction towards the Northern pole. This semi-infinite line is called the Dirac string and its effect on the wave function is analogous to the effect of the solenoid in the Aharonov-Bohm effect. The quantization condition comes from the requirement that the phases around the Dirac string are trivial, which means that the Dirac string must be unphysical. The Dirac string is merely an artifact of the coordinate chart used and should not be taken seriously. In engineering, a solenoid is a mechanical device that converts energy into linear motion. ...

The Dirac monopole is a singular solution of Maxwell's equation (because it requires removing the worldline from spacetime); in more complicated theories, it is superseded by a smooth solution such as the 't Hooft-Polyakov monopole. In theoretical physics, the t Hooft-Polyakov monopole is a topological soliton similar to the Dirac monopole but without any singularities. ...

Mathematical details

Classically, gauge theory is described by a connection over a principal G-bundle over spacetime. Ordinary spacetime has the topology of R⁴, which is topologically trivial. So, the space of all possible connections over the principal G-bundle is connected. But let's see what happens when we remove a timelike worldline from spacetime. The resulting spacetime is homotopically equivalent to the topological sphere S². So, it suffices to classify the connected components of the space of all connections over a principal G-bundle over S². To do this, consider covering S² by two charts, each homeomorphic to the open 2-ball such that their intersection is homeomorphic to the strip S¹×I. 2-balls are homotopically trivial and the strip is homotopically equivalent to the circle S¹. So, a topological classification of the possible connections is reduced to classifying the transition functions, which is given by the first homotopy group of G. In other words, a gauge theory can only admit Dirac monopoles provided G isn't simply connected. for instance, U(1), which has quantized charges isn't simply connected and can have Dirac monopoles while R, its universal covering group, is simply connected, doesn't have quantized charges and does not admit Dirac monopoles even in principal. Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... A world line of an object or person is the sequence of events labeled with time and place, that marks the history of the object or person. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... A sphere is, roughly speaking, a ball-shaped object. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... This word should not be confused with homomorphism. ... In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... The universal covering space of a topological group is also a topological group. ...

It's easy to see how this argument generalizes to d+1 dimensions with . We look at the homotopy group π_d-2(G).