A magnetic monopole is a hypothetical particle in physics that is a magnet with only one pole (see Maxwell's equations for more on magnetic poles).^{[1]} In more technical terms, it would have a net "magnetic charge." Modern interest in the concept stems from particle theories, notably the grand unification theory and superstring theories, which predict their existence.^{[2]}^{[3]}
The magnetic monopole was first hypothesized by Pierre Curie in 1894,^{[4]} but the quantum theory of magnetic charge started with a 1931 paper by Paul Dirac.^{[5]} In this paper, Dirac showed that the existence of magnetic monopoles was consistent with Maxwell's equations only if electric charges are quantized, which is observed. Since then, several systematic monopole searches have been performed. Experiments in 1975^{[6]} and 1982^{[7]} produced candidate events that were initially interpreted as monopoles, but are now regarded as inconclusive.^{[8]}
Monopole detection is an open problem in experimental physics. Within theoretical physics, some modern approaches assume their existence. Joseph Polchinski, a prominent stringtheorist, described the existence of monopoles as "one of the safest bets that one can make about physics not yet seen."^{[9]} These theories are not necessarily inconsistent with the experimental evidence: in some models magnetic monopoles are unlikely to be observed, because they are too massive to be created in particle accelerators, and too rare in the universe to enter a particle detector.^{[9]}
Some condensed matter systems propose a superficially similar structure, known as a flux tube. The ends of a flux tube form a magnetic dipole, but since they move independently, they can be treated for many purposes as independent magnetic monopole quasiparticles. In late 2009 a large number of popular publications incorrectly reported this phenomenon as the longawaited discovery of magnetic monopoles, but the two phenomena are not related.^{[10]} However, they are considered interesting in their own right, and are an area of active research. (See "Monopoles" in condensedmatter systems below.)
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Magnets exert forces on one another, similar to electric charges. Like poles will repel each other, and unlike poles will attract. When a magnet (an object conventionally described as having magnetic north and south poles) is cut in half across the axis joining those "poles", the resulting pieces are two normal (albeit smaller) magnets. Each has its own north pole and south pole.
Even atoms have tiny magnetic fields. In the Bohr model of an atom, electrons orbit the nucleus. The constant change in their motion gives rise to a magnetic field. Permanent magnets have measurable magnetic fields because the atoms and molecules in them are arranged in such a way that their individual magnetic fields align, combining to form large aggregate fields. In this model, the lack of a single pole makes intuitive sense; cutting a bar magnet in half does nothing to the arrangement of the molecules within. The end result is two magnetic bars whose atoms have the same orientation as before, and therefore generate a magnetic field with the same orientation as the original larger magnet.
Maxwell's equations of electromagnetism relate the electric and magnetic fields to the motions of electric charges. The standard form of the equations provides for an electric charge, but posits no magnetic charge. Except for this, the equations are symmetric under interchange of electric and magnetic fields.^{[11]} In fact, symmetric equations can be written when all charges are zero, and this is how the wave equation is derived.
Fully symmetric equations can also be written if one allows for the possibility of "magnetic charges" analogous to electric charges.^{[12]} With the inclusion of a variable for the density of these magnetic charges, say , there will also be a "magnetic current density" variable in the equations, .
If magnetic charges do not exist, or if they exist but are not present in a region, then the new variables are zero, and the extended equations reduce to the conventional equations of electromagnetism such as . Classically, the question is "Why does the magnetic charge always seem to be zero?"
The extended Maxwell's equations are as follows, in cgs units:^{[13]}
Name  Without magnetic monopoles  With magnetic monopoles  

Gauss's law:  
Gauss's law for magnetism:  
Faraday's law of induction:  
Ampère's law (with Maxwell's extension): 

Note: For the equations in nondimensionalized form, remove the factors of c. 
The Lorentz force becomes^{[13]}^{[14]}
In these equations is the magnetic charge density, is the magnetic current density, and q_{m} is the magnetic charge of a test particle, all defined analogously to the related quantities of electric charge and current.
In SI units, there are two conflicting conventions in use for magnetic charge. In one, magnetic charge has units of webers, while in the other, magnetic charge has units of amperemeters. Maxwell's equations then take the following forms:^{[15]}
Name  Without magnetic monopoles  Weber convention  Ampere·meter convention 

Gauss's Law  
Gauss's Law for magnetism  
Faraday's Law of induction  
Ampère's Law  
Lorentz force 
In these equations is the magnetic charge density, is the magnetic current density, and q_{m} is the magnetic charge of a test particle, all defined analogously to the related quantities of electric charge and current.
One of the defining advances in quantum theory was Paul Dirac's work on developing a relativistic quantum electromagnetism. Before his formulation, the presence of electric charge was simply "inserted" into QM, but in 1931 Dirac showed that a discrete charge naturally "falls out" of QM; that is to say, we can maintain the current form of Maxwell's equations and still have magnetic charges.
Consider a system consisting of a single stationary electric monopole (an electron, say) and a single stationary magnetic monopole. Classically, the electromagnetic field surrounding them has a momentum density given by the Poynting vector, and it also has a total angular momentum, which is proportional to the product q_{e}q_{m}, and independent of the distance between them.
Quantum mechanics dictates, however, that angular momentum is quantized in units of ħ, and therefore the product q_{e}q_{m} must also be quantized. This means that if even a single magnetic monopole existed in the universe, and the current form of Maxwell's equations is valid, all electric charges would then be quantized.
What are the units in which magnetic charge would be quantized? Although it would be possible simply to integrate over all space to find the total angular momentum in the above example, Dirac took a different approach, which led to new ideas. He considered a pointlike magnetic charge whose magnetic field behaves as q_{m} / r^{2} 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.
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 AharonovBohm 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.
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:
where
This is known as the Dirac quantization condition. 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 inversely proportional to the elementary electric charge.
At the time it was not clear if such a thing existed, or even had to. After all, another theory could come along that would explain charge quantization without need for the monopole. The concept remained something of a curiosity. However, in the time since the publication of this seminal work, no other widely accepted explanation of charge quantization has appeared. (The concept of local gauge invariance—see gauge theory below—provides a natural explanation of charge quantization, without invoking the need for magnetic monopoles; but only if the U(1) gauge group is compact, in which case we will have magnetic monopoles anyway.)
If we maximally extend the definition of the vector potential for the Southern hemisphere, it will be defined everywhere except for a semiinfinite line stretched from the origin in the direction towards the Northern pole. This semiinfinite line is called the Dirac string and its effect on the wave function is analogous to the effect of the solenoid in the AharonovBohm 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.
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 HooftPolyakov monopole.
A gauge theory like electromagnetism is defined by a gauge field, which associates a group element to each path in space time. For infinitesimal paths, the group element is close to the identity, while for longer paths the group element is the successive product of the infinitesimal group elements along the way.
In electrodynamics, the group is U(1), unit complex numbers under multiplication. For infinitesimal paths, the group element is 1 + iA_{μ}dx^{μ} which implies that for finite paths parametrized by s, the group element is:
The map from paths to group elements is called the Wilson loop or the holonomy, and for a U(1) gauge group it is the phase factor which the wavefunction of a charged particle acquires as it traverses the path. For a loop:
So that the phase a charged particle gets when going in a loop is the magnetic flux through the loop. When a small solenoid has a magnetic flux, there are interference fringes for charged particles which go around the solenoid, or around different sides of the solenoid, which reveal its presence.
But if all particle charges are integer multiples of e, solenoids with a flux of 2π / e have no interference fringes, because the phase factor for any charged particle is . Such a solenoid, if thin enough, is quantum mechanically invisible. If such a solenoid were to carry a flux of 2π / e, when the flux leaked out from one of its ends it would be indistinguishable from a monopole.
Dirac's monopole solution in fact describes an infinitesimal line solenoid ending at a point, and the location of the solenoid is the singular part of the solution, the Dirac string. Dirac strings link monopoles and antimonopoles of opposite magnetic charge, although in Dirac's version, the string just goes off to infinity. The string is unobservable, so you can put it anywhere, and by using two coordinate patches, the field in each patch can be made nonsingular by sliding the string to where it cannot be seen.
In a U(1) with quantized charge, the gauge group is a circle of radius 2π / e. Such a U(1) is called compact. Any U(1) which comes from a Grand Unified Theory is compact, because only compact higher gauge groups make sense. The size of the gauge group is a measure of the inverse coupling constant, so that in the limit of a large volume gauge group, the interaction of any fixed representation goes to zero.
The U(1) case is special because all its irreducible representations are the same size—the charge is bigger by an integer amount but the field is still just a complex number—so that in U(1) gauge field theory it is possible to take the decompactified limit with no contradiction. The quantum of charge becomes small, but each charged particle has a huge number of charge quanta so its charge stays finite. In a noncompact U(1) gauge theory, the charges of particles are generically not integer multiples of a single unit. Since charge quantization is an experimental certainty, it is clear that the U(1) of electromagnetism is compact.
GUTs lead to compact U(1)s, so they explain charge quantization in a way that seems to be logically independent from magnetic monopoles. But the explanation is essentially the same, because in any GUT which breaks down to a U(1) at long distances, there are magnetic monopoles.
The argument is topological:
So the Dirac monopole is a topological defect in a compact U(1) gauge theory. When there is no GUT, the defect is a singularity—the core shrinks to a point. But when there is some sort of short distance regulator on space time, the monopoles have a finite mass. Monopoles occur in lattice U(1), and there the core size is the lattice size. In general, they are expected to occur whenever there is a shortdistance regulator.
In our universe, quantum gravity provides the regulator. When gravity is included, the monopole singularity can be a black hole, and for large magnetic charge and mass, the black hole mass is equal to the black hole charge, so that the mass of the magnetic black hole is not infinite. If the black hole can decay completely by Hawking radiation, the lightest charged particles can't be too heavy. The lightest monopole should have a mass less than or comparable to its charge in natural units.
So in a consistent holographic theory, and string theory is the only known example, there are always finite mass monopoles. For ordinary electromagnetism, the mass bound is not very useful because it is about the Planck mass.
In mathematics, a gauge field is defined as a connection over a principal Gbundle over spacetime. G is the gauge group, and it acts on each fiber of the bundle separately.
A connection on a G bundle tells you how to glue F's together at nearby points of M. It starts with a continuous symmetry group G which acts on F, and then it associates a group element with each infinitesimal path. Group multiplication along any path tells you how to move from one point on the bundle to another, by acting the G element of a path on the fiber F.
In mathematics, the definition of bundle is designed to emphasize topology, so the notion of connection is added on as an afterthought. In physics, the connection is the fundamental physical object. Once you have a connection, there are nontrivial bundles which occur as connections of a trivial bundle. For example, the twisted torus is a connection on a U(1) bundle of a circle on a circle.
If space time has no topology, if it is R^{4} the space of all possible connections of the Gbundle is connected. But consider what happens when we remove a timelike worldline from spacetime. The resulting spacetime is homotopically equivalent to the topological sphere S^{2}.
A principal Gbundle over S^{2} is defined by covering S^{2} by two charts, each homeomorphic to the open 2ball such that their intersection is homeomorphic to the strip S^{1}×I. 2balls are homotopically trivial and the strip is homotopically equivalent to the circle S^{1}. So a topological classification of the possible connections is reduced to classifying the transition functions. The transition function maps the strip to G, and the different ways of mapping a strip into G is given by the first homotopy group of G.
So in the Gbundle formulation, a gauge theory admits Dirac monopoles provided G is not simply connected, whenever there are paths that go around the group that cannot be deformed to nothing. U(1), which has quantized charges, is not 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. The mathematical definition is equivalent to the physics definition provided that, following Dirac, gauge fields are allowed which are only defined patchwise and the gauge field on different patches are glued after a gauge transformation.
This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It generalizes to d + 1 dimensions with in several ways. One way is to extend everything into the extra dimensions, so that U(1) monopoles become sheets of dimension d3. Another way is to examine the type of topological singularity at a point with the homotopy group π_{d−2}(G).
In more recent years, a new class of theories has also suggested the presence of a magnetic monopole.
In the early 1970s, the successes of quantum field theory and gauge theory in the development of electroweak and the strong nuclear force led many theorists to move on to attempt to combine them in a single theory known as a grand unified theory, or GUT. Several GUTs were proposed, most of which had the curious feature of suggesting the presence of a real magnetic monopole particle. More accurately, GUTs predicted a range of particles known as dyons, of which the most basic state is a monopole. The charge on magnetic monopoles predicted by GUTs is either 1 or 2gD, depending on the theory.
The majority of particles appearing in any quantum field theory are unstable, and decay into other particles in a variety of reactions that have to conserve various values. Stable particles are stable because there are no lighter particles to decay into that still conserve these values. For instance, the electron has a lepton number of 1 and an electric charge of 1, and there are no lighter particles that conserve these values. On the other hand, the muon, essentially a heavy electron, can decay into the electron and is therefore not stable.
The dyons in these same theories are also stable, but for an entirely different reason. The dyons are expected to exist as a side effect of the "freezing out" of the conditions of the early universe, or symmetry breaking. In this model the dyons arise due to the vacuum configuration in a particular area of the universe, according to the original Dirac theory. They remain stable not because of a conservation condition, but because there is no simpler topological state to which they can decay.
The length scale over which this special vacuum configuration exists is called the correlation length of the system. A correlation length cannot be larger than causality would allow, therefore the correlation length for making magnetic monopoles must be at least as big as the horizon size determined by the metric of the expanding universe. According to that logic, there should be at least one magnetic monopole per horizon volume as it was when the symmetry breaking took place. Other arguments based on the critical density of the universe indicate that monopoles should be fairly common; the apparent problem of the observed scarcity of monopoles is resolved by cosmic inflation in the early universe, which greatly reduces the expected abundance of magnetic monopoles. For these reasons, monopoles became a major interest in the 1970s and 80s, along with the other "approachable" prediction of GUTs such as proton decay.
Many of the other particles predicted by these GUTs were beyond the abilities of current experiments to detect. For instance, a wide class of particles known as the X and Y bosons are predicted to mediate the coupling of the electroweak and strong forces, but these particles are extremely heavy and well beyond the capabilities of any reasonable particle accelerator to create.
A number of attempts have been made to detect magnetic monopoles. One of the simplest is to use a loop of superconducting wire that can look for even tiny magnetic sources, a socalled "superconducting quantum interference device", or SQUID. Given the predicted density, loops small enough to fit on a lab bench would expect to see about one monopole event per year. Although there have been tantalizing events recorded, in particular the event recorded by Blas Cabrera on the night of February 14, 1982 (thus, sometimes referred to as the "Valentine's Day Monopole"), there has never been reproducible evidence for the existence of magnetic monopoles.^{[7]} The lack of such events places a limit on the number of monopoles of about 1 monopole per 10^{29} nucleons.
Another experiment in 1975 resulted in the announcement of the detection of a moving magnetic monopole in cosmic rays by the team of Price.^{[6]} Price later retracted his claim, and a possible alternative explanation was offered by Alvarez.^{[17]} In his paper it was demonstrated that the path of the cosmic ray event that was claimed to be due to a magnetic monopole could be reproduced by a path followed by a platinum nucleus fragmenting to osmium and then to tantalum.
Other experiments rely on the strong coupling of monopoles with photons, as is the case for any electrically charged particle as well. In experiments involving photon exchange in particle accelerators, monopoles should be produced in reasonable numbers, and detected due to their effect on the scattering of the photons. The probability of a particle being created in such experiments is related to their mass —heavier particles are less likely to be created—so by examining such experiments limits on the mass can be calculated. The most recent such experiments suggest that monopoles with masses below 600 GeV/c^{2} do not exist, while upper limits on their mass due to the existence of the universe (which would have collapsed by now if they were too heavy) are about 10^{17} GeV/c^{2}.
While a magnetic monopole particle has never been conclusively observed, there are a number of phenomena in condensedmatter physics where a material, due to the collective behavior of its electrons and ions, can show emergent phenomena that resemble magnetic monopoles in some respect.^{[18]}^{[19]}^{[20]}^{[21]}^{[22]} These should not be confused with actual monopole particles; in particular, the divergence of the microscopic magnetic Bfield is zero everywhere in these systems, unlike in the presence of a true magnetic monopole particle. The behavior of these quasiparticles would only become indistinguishable from true magnetic monopoles — and they would truly deserve the name — if the socalled magnetic fluxtubes connecting these wouldbe monopoles became unobservable which also means that these flux tubes would have to be infinitely thin, obey the Dirac quantization rule, and deserve to be called Dirac strings.
In a paper published in Science in September 2009, researchers Jonathan Morris and Alan Tennant from the HelmholtzZentrum Berlin für Materialien und Energie (HZB) along with Santiago Grigera from Instituto de Física de Líquidos y Sistemas Biológicos (IFLYSIB, CONICET) and other colleagues from Dresden University of Technology, University of St. Andrews and Oxford University described the observation of quasiparticles resembling monopoles. A single crystal of dysprosium titanate in a highly frustrated pyrochlore lattice (F d 3 m) was cooled to 0.6 to 2 K. Using neutron scattering, the magnetic moments were shown to align in the spin ice into interwoven tubelike bundles resembling Dirac strings. At the defect formed by the end of each tube, the magnetic field looks like that of a monopole. Using an applied magnetic field to break the symmetry of the system, the researchers were able to control the density and orientation of these strings. A contribution to the heat capacity of the system from an effective gas of these quasiparticles is also described.^{[23]}^{[24]}
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole (see Maxwell's equations for more on magnetic poles).^{[1]} 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 unification and superstring theories, which predict their existence.^{[2]}^{[3]}
The magnetic monopole was first hypothesized by Pierre Curie in 1894,^{[4]} but the quantum theory of magnetic charge started with a paper by the physicist Paul A.M. Dirac in 1931.^{[5]} In this paper, Dirac showed that the existence of magnetic monopoles was consistent with Maxwell's equations only if electric charges are quantized, which is always observed. Since then, several systematic monopole searches have been performed. Experiments in 1975^{[6]} and 1982^{[7]} produced candidate events that were initially interpreted as monopoles, but are now regarded as inconclusive.^{[8]}
The detection of magnetic monopoles is an open problem in experimental physics. Within theoretical physics, some modern approaches predict the existence of magnetic monopoles. Joseph Polchinski, a prominent stringtheorist, described the existence of monopoles as "one of the safest bets that one can make about physics not yet seen".^{[9]} These theories are not necessarily inconsistent with the experimental evidence. In some theoretical models, magnetic monopoles are unlikely to be observed, because they are too massive to be created in particle accelerators, and also too rare in the Universe to enter a particle detector with much probability.^{[9]}
Some condensed matter systems propose a structure superficially similar to a magnetic monopole, known as a flux tube. The ends of a flux tube form a magnetic dipole, but since they move independently, they can be treated for many purposes as independent magnetic monopole quasiparticles. In late 2009, numerous news reports from the popular media incorrectly described this phenomenon as the longawaited discovery of the magnetic monopoles, but the two phenomena are only superficially related to one another.^{[10]} Nevertheless, these condensedmatter quasiparticles are considered to be interesting in their own right, and they are an area of active research. (See "Monopoles" in condensedmatter systems below.)
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Magnets exert forces on one another, similar to the force associated with electric charges. Like poles will repel each other, and unlike poles will attract. When any magnet (an object conventionally described as having magnetic north and south poles) is cut in half across the axis joining those "poles", the resulting pieces are two normal (albeit smaller) magnets. Each has its own north pole and south pole.
Even atoms and subatomic particles have tiny magnetic fields. In the Bohr model of an atom, electrons orbit the nucleus. Their constant motion gives rise to a magnetic field. Permanent magnets have measurable magnetic fields because the atoms and molecules in them are arranged in such a way that their individual magnetic fields align, combining to form large aggregate fields. In this model, the lack of a single pole makes intuitive sense: cutting a bar magnet in half does nothing to the arrangement of the molecules within. The end result is two bar magnetics whose atoms have the same orientation as before, and therefore generate a magnetic field with the same orientation as the original larger magnet.
Maxwell's equations of electromagnetism relate the electric and magnetic fields to each other and to the motions of electric charges. The standard equations provide for electric charges, but they posit no magnetic charges. Except for this difference, the equations are symmetric under the interchange of the electric and magnetic fields.^{[11]} In fact, symmetric Maxwell's equations can be written when all charges (and hence electric currents) are zero, and this is how the electromagnetic wave equation is derived.
Fully symmetric Maxwell's equations can also be written if one allows for the possibility of "magnetic charges" analogous to electric charges.^{[12]} With the inclusion of a variable for the density of these magnetic charges, say $\backslash \; \backslash rho\_m$, there will also be a "magnetic current density" variable in the equations, $\backslash \; \backslash mathbf\{j\}\_m$.
If magnetic charges do not exist  or if they do exist but are not present in a region of free space  then the new terms in Maxwell's equations are all zero, and the extended equations reduce to the conventional equations of electromagnetism such as $\backslash nabla\backslash cdot\backslash mathbf\{B\}\; =\; 0$ (where $\backslash nabla\; \backslash cdot$ is divergence and B is the magnetic B field).
For a long time, the open question has been "Why does the magnetic charge always seem to be zero?"
The extended Maxwell's equations are as follows, in cgs units:^{[13]}
Name  Without magnetic monopoles  With magnetic monopoles  

Gauss's law:  $\backslash nabla\; \backslash cdot\; \backslash mathbf\{E\}\; =\; 4\; \backslash pi\; \backslash rho\_e$  $\backslash nabla\; \backslash cdot\; \backslash mathbf\{E\}\; =\; 4\; \backslash pi\; \backslash rho\_e$  
Gauss's law for magnetism:  $\backslash nabla\; \backslash cdot\; \backslash mathbf\{B\}\; =\; 0$  $\backslash nabla\; \backslash cdot\; \backslash mathbf\{B\}\; =\; 4\; \backslash pi\; \backslash rho\_m$  
Faraday's law of induction:  $\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; \backslash frac\{1\}\{c\}\backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}$  $\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; \backslash frac\{1\}\{c\}\backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}\; +\; \backslash frac\{4\; \backslash pi\}\{c\}\backslash mathbf\{j\}\_m$  
Ampère's law (with Maxwell's extension):  $\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash frac\{1\}\{c\}\backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}\; +\; \backslash frac\{4\; \backslash pi\}\{c\}\; \backslash mathbf\{j\}\_e$  $\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash frac\{1\}\{c\}\backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}\; +\; \backslash frac\{4\; \backslash pi\}\{c\}\; \backslash mathbf\{j\}\_e$  
$\backslash rho\_m$ and $j\_m$ are defined above. For all other definitions and details, see Maxwell's equations article.  
Note: For the equations in nondimensionalized form, remove the factors of c. 
The equallyimportant Lorentz force equation becomes^{[13]}^{[14]}
In these equations $\backslash \; \backslash rho\_m$ is the magnetic charge density, $\backslash \; \backslash mathbf\{j\}\_m$ is the magnetic current density, and $q\_m$ is the magnetic charge of a test particle, all defined analogously to the related quantities of electric charge and current; v is the particle's velocity and c is the speed of light.
In SI units, there are two conflicting conventions in use for magnetic charge. In one, magnetic charge has units of webers, while in the other, magnetic charge has units of amperemeters. Maxwell's equations then take the following forms:^{[15]}
Name  Without magnetic monopoles  Weber convention  Ampere·meter convention 

Gauss's Law  $\backslash nabla\; \backslash cdot\; \backslash mathbf\{E\}\; =\; \backslash rho\_e/\backslash epsilon\_0$  $\backslash nabla\; \backslash cdot\; \backslash mathbf\{E\}\; =\; \backslash rho\_e/\backslash epsilon\_0$  $\backslash nabla\; \backslash cdot\; \backslash mathbf\{E\}\; =\; \backslash rho\_e/\backslash epsilon\_0$ 
Gauss's Law for magnetism  $\backslash nabla\; \backslash cdot\; \backslash mathbf\{B\}\; =\; 0$  $\backslash nabla\; \backslash cdot\; \backslash mathbf\{B\}\; =\; \backslash rho\_m$  $\backslash nabla\; \backslash cdot\; \backslash mathbf\{B\}\; =\; \backslash mu\_0\backslash rho\_m$ 
Faraday's Law of induction  $\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}$  $\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}\; +\; \backslash mathbf\{j\}\_m$  $\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}\; +\; \backslash mu\_0\backslash mathbf\{j\}\_m$ 
Ampère's Law  $\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash mu\_0\; \backslash epsilon\_0\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}\; +\; \backslash mu\_0\; \backslash mathbf\{j\}\_e$  $\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash mu\_0\; \backslash epsilon\_0\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}\; +\; \backslash mu\_0\; \backslash mathbf\{j\}\_e$  $\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash mu\_0\; \backslash epsilon\_0\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}\; +\; \backslash mu\_0\; \backslash mathbf\{j\}\_e$ 
Lorentz force equation  $\backslash mathbf\{F\}=q\_e\backslash left(\backslash mathbf\{E\}+\backslash mathbf\{v\}\backslash times\backslash mathbf\{B\}\backslash right)$  $\backslash mathbf\{F\}=q\_e\backslash left(\backslash mathbf\{E\}+\backslash mathbf\{v\}\backslash times\backslash mathbf\{B\}\backslash right)\; +$ $+\; \backslash frac\{q\_m\}\{\backslash mu\_0\}\backslash left(\backslash mathbf\{B\}\backslash mathbf\{v\}\backslash times(\backslash mathbf\{E\}/c^2)\backslash right)$  $\backslash mathbf\{F\}=q\_e\backslash left(\backslash mathbf\{E\}+\backslash mathbf\{v\}\backslash times\backslash mathbf\{B\}\backslash right)\; +$ $+\; q\_m\backslash left(\backslash mathbf\{B\}\backslash mathbf\{v\}\backslash times(\backslash mathbf\{E\}/c^2)\backslash right)$ 
In these equations $\backslash \; \backslash rho\_m$ is the magnetic charge density, $\backslash \; \backslash mathbf\{j\}\_m$ is the magnetic current density, and $q\_m$ is the magnetic charge of a test particle, all defined analogously to the related quantities of electric charge and current.
One of the defining advances in quantum theory was Paul Dirac's work on developing a relativistic quantum electromagnetism. Before his formulation, the presence of electric charge was simply "inserted" into the equations of quantum mechanics (QM), but in 1931 Dirac showed that a discrete charge naturally "falls out" of QM. That is to say, we can maintain the form of Maxwell's equations and still have magnetic charges.
Consider a system consisting of a single stationary electric monopole (an electron, say) and a single stationary magnetic monopole. Classically, the electromagnetic field surrounding them has a momentum density given by the Poynting vector, and it also has a total angular momentum, which is proportional to the product $q\_e\; q\_m$, and independent of the distance between them.
Quantum mechanics dictates, however, that angular momentum is quantized in units of ħ, so therefore. the product $q\_e\; q\_m$ must also be quantized. This means that if even a single magnetic monopole existed in the universe, and the form of Maxwell's equations is valid, all electric charges would then be quantized.
What are the units in which magnetic charge would be quantized? Although it would be possible simply to integrate over all space to find the total angular momentum in the above example, Dirac took a different approach. This led him to new ideas. He considered a pointlike magnetic charge whose magnetic field behaves as $q\_m\; /r^2$ 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 $B$.
However, the vector potential cannot be defined globally precisely because the divergence of the magnetic field is proportional to the Dirac 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 electricallycharged particle (a "probe charge") 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.
Because the electron returns to the same point after the full trip around the equator, the phase $\backslash exp(i\backslash phi)$ of its wave function must be unchanged, which implies that the phase $\backslash phi$ added to the wave function must be a multiple of $2\backslash pi$:
where
This is known as the Dirac quantization condition. 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 inversely proportional to the elementary electric charge.
At the time it was not clear if such a thing existed, or even had to. After all, another theory could come along that would explain charge quantization without need for the monopole. The concept remained something of a curiosity. However, in the time since the publication of this seminal work, no other widely accepted explanation of charge quantization has appeared. (The concept of local gauge invariance—see gauge theory below—provides a natural explanation of charge quantization, without invoking the need for magnetic monopoles; but only if the U(1) gauge group is compact, in which case we will have magnetic monopoles anyway.)
If we maximally extend the definition of the vector potential for the southern hemisphere, it will be defined everywhere except for a semiinfinite line stretched from the origin in the direction towards the northern pole. This semiinfinite 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.
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.
A gauge theory like electromagnetism is defined by a gauge field, which associates a group element to each path in space time. For infinitesimal paths, the group element is close to the identity, while for longer paths the group element is the successive product of the infinitesimal group elements along the way.
In electrodynamics, the group is U(1), unit complex numbers under multiplication. For infinitesimal paths, the group element is $1+iA\_\backslash mu\; dx^\backslash mu$ which implies that for finite paths parametrized by s, the group element is:
The map from paths to group elements is called the Wilson loop or the holonomy, and for a U(1) gauge group it is the phase factor which the wavefunction of a charged particle acquires as it traverses the path. For a loop:
So that the phase a charged particle gets when going in a loop is the magnetic flux through the loop. When a small solenoid has a magnetic flux, there are interference fringes for charged particles which go around the solenoid, or around different sides of the solenoid, which reveal its presence.
But if all particle charges are integer multiples of e, solenoids with a flux of $2\backslash pi/e$ have no interference fringes, because the phase factor for any charged particle is $\backslash scriptstyle\; e^\{2\backslash pi\; i\}=1$. Such a solenoid, if thin enough, is quantummechanically invisible. If such a solenoid were to carry a flux of $2\backslash pi/e$, when the flux leaked out from one of its ends it would be indistinguishable from a monopole.
Dirac's monopole solution in fact describes an infinitesimal line solenoid ending at a point, and the location of the solenoid is the singular part of the solution, the Dirac string. Dirac strings link monopoles and antimonopoles of opposite magnetic charge, although in Dirac's version, the string just goes off to infinity. The string is unobservable, so you can put it anywhere, and by using two coordinate patches, the field in each patch can be made nonsingular by sliding the string to where it cannot be seen.
In a U(1) gauge group with quantized charge, the group is a circle of radius $2\backslash pi/e$. Such a U(1) gauge group is called compact. Any U(1) which comes from a Grand Unified Theory is compact  because only compact higher gauge groups make sense. The size of the gauge group is a measure of the inverse coupling constant, so that in the limit of a largevolume gauge group, the interaction of any fixed representation goes to zero.
The case of the U(1) gauge group is a special case because all its irreducible representations are of the same size — the charge is bigger by an integer amount, but the field is still just a complex number — so that in U(1) gauge field theory it is possible to take the decompactified limit with no contradiction. The quantum of charge becomes small, but each charged particle has a huge number of charge quanta so its charge stays finite. In a noncompact U(1) gauge group theory, the charges of particles are generically not integer multiples of a single unit. Since charge quantization is an experimental certainty, it is clear that the U(1) gauge group of electromagnetism is compact.
GUTs lead to compact U(1) gauge groups, so they explain charge quantization in a way that seems to be logically independent from magnetic monopoles. However, the explanation is essentially the same, because in any GUT which breaks down into a U(1) gauge group at long distances, there are magnetic monopoles.
The argument is topological:
Hence, the Dirac monopole is a topological defect in a compact U(1) gauge theory. When there is no GUT, the defect is a singularity — the core shrinks to a point. But when there is some sort of shortdistance regulator on space time, the monopoles have a finite mass. Monopoles occur in lattice U(1), and there the core size is the lattice size. In general, they are expected to occur whenever there is a shortdistance regulator.
In our universe, quantum gravity provides the regulator. When gravity is included, the monopole singularity can be a black hole, and for large magnetic charge and mass, the black hole mass is equal to the black hole charge, so that the mass of the magnetic black hole is not infinite. If the black hole can decay completely by Hawking radiation, the lightest charged particles cannot be too heavy. The lightest monopole should have a mass less than or comparable to its charge in natural units.
So in a consistent holographic theory, of which string theory is the only known example, there are always finitemass monopoles. For ordinary electromagnetism, the mass bound is not very useful because it is about same size as the Planck mass.
In mathematics, a gauge field is defined as a connection over a principal Gbundle over spacetime. G is the gauge group, and it acts on each fiber of the bundle separately.
A connection on a G bundle tells you how to glue F's together at nearby points of M. It starts with a continuous symmetry group G which acts on F, and then it associates a group element with each infinitesimal path. Group multiplication along any path tells you how to move from one point on the bundle to another, by acting the G element of a path on the fiber F.
In mathematics, the definition of bundle is designed to emphasize topology, so the notion of connection is added on as an afterthought. In physics, the connection is the fundamental physical object. One of the fundamental observations in the theory of characteristic classes in algebraic topology is that many homotopical structures of nontrivial principal bundles may be expressed as an integral of some polynomial over any connection over it. Note that any connection over a trivial bundle can never give us a nontrivial principal bundle.
If space time has no topology, if it is R^{4} the space of all possible connections of the Gbundle is connected. But consider what happens when we remove a timelike worldline from spacetime. The resulting spacetime is homotopically equivalent to the topological sphere S^{2}.
A principal Gbundle over S^{2} is defined by covering S^{2} by two charts, each homeomorphic to the open 2ball such that their intersection is homeomorphic to the strip S^{1}×I. 2balls are homotopically trivial and the strip is homotopically equivalent to the circle S^{1}. So a topological classification of the possible connections is reduced to classifying the transition functions. The transition function maps the strip to G, and the different ways of mapping a strip into G is given by the first homotopy group of G.
So in the Gbundle formulation, a gauge theory admits Dirac monopoles provided G is not simply connected, whenever there are paths that go around the group that cannot be deformed to nothing. U(1), which has quantized charges, is not 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. The mathematical definition is equivalent to the physics definition provided that, following Dirac, gauge fields are allowed which are only defined patchwise and the gauge field on different patches are glued after a gauge transformation.
The total magnetic flux is none other than the first Chern number of the principal bundle, and only depends upon the choice of the principal bundle, and not the specific connection over it. In other words, it's a topological invariant.
This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It generalizes to d + 1 dimensions with $d\backslash geq\; 2$ in several ways. One way is to extend everything into the extra dimensions, so that U(1) monopoles become sheets of dimension d3. Another way is to examine the type of topological singularity at a point with the homotopy group π_{d−2}(G).
In more recent years, a new class of theories has also suggested the existence of magnetic monopoles.
During the early 1970s, the successes of quantum field theory and gauge theory in the development of electroweak theory and the mathematics of the strong nuclear force led many theorists to move on to attempt to combine them in a single theory known as a Grand Unified Theory (GUT). Several GUTs were proposed, most of which had the curious feature of implying the presence of a real magnetic monopole particle. More accurately, GUTs predicted a range of particles known as dyons, of which the most basic state was a monopole. The charge on magnetic monopoles predicted by GUTs is either 1 or 2 gD, depending on the theory.
The majority of particles appearing in any quantum field theory are unstable, and they decay into other particles in a variety of reactions that must satisfy various conservation laws. Stable particles are stable because there are no lighter particles into which they can decay and still satisfy the conservation laws. For instance, the electron has a lepton number of one and an electric charge of one, and there are no lighter particles that conserve these values. On the other hand, the muon, essentially a heavy electron, can decay into the electron plus two quanta of energy, and hence it is not stable.
The dyons in these GUTs are also stable, but for an entirely different reason. The dyons are expected to exist as a side effect of the "freezing out" of the conditions of the early universe, or a symmetry breaking. In this scenerio, the dyons arise due to the configuration of the vacuum in a particular area of the universe, according to the original Dirac theory. They remain stable not because of a conservation condition, but because there is no simpler topological state into which they can decay.
The length scale over which this special vacuum configuration exists is called the correlation length of the system. A correlation length cannot be larger than causality would allow, therefore the correlation length for making magnetic monopoles must be at least as big as the horizon size determined by the metric of the expanding universe. According to that logic, there should be at least one magnetic monopole per horizon volume as it was when the symmetry breaking took place. Other arguments based on the critical density of the universe indicate that monopoles should be fairly common; the apparent problem of the observed scarcity of monopoles is resolved by cosmic inflation in the early universe, which greatly reduces the expected abundance of magnetic monopoles. For these reasons, monopoles became a major interest in the 1970s and 80s, along with the other "approachable" predictions of GUTs such as proton decay.
Many of the other particles predicted by these GUTs were beyond the abilities of current experiments to detect. For instance, a wide class of particles known as the X and Y bosons are predicted to mediate the coupling of the electroweak and strong forces, but these particles are extremely heavy and well beyond the capabilities of any reasonable particle accelerator to create.
A number of attempts have been made to detect magnetic monopoles. One of the simpler ones is to use a loop of superconducting wire to look for even tiny magnetic sources, a socalled "superconducting quantum interference device", or SQUID. Given the predicted density, loops small enough to fit on a lab bench would expect to see about one monopole event per year. Although there have been tantalizing events recorded, in particular the event recorded by Blas Cabrera on the night of February 14, 1982 (thus, sometimes referred to as the "Valentine's Day Monopole"^{[17]}), there has never been reproducible evidence for the existence of magnetic monopoles.^{[7]} The lack of such events places a limit on the number of monopoles of about one monopole per 10^{29} nucleons.
Another experiment in 1975 resulted in the announcement of the detection of a moving magnetic monopole in cosmic rays by the team lead by P. Buford Price.^{[6]} Price later retracted his claim, and a possible alternative explanation was offered by Alvarez.^{[18]} In his paper it was demonstrated that the path of the cosmic ray event that was claimed to have been be due to a magnetic monopole could be reproduced by the path followed by a platinum nucleus decaying first to osmium, and then to tantalum.
Other experiments rely on the strong coupling of monopoles with photons, as is the case for any electricallycharged particle as well. In experiments involving photon exchange in particle accelerators, monopoles should be produced in reasonable numbers, and detected due to their effect on the scattering of the photons. The probability of a particle being created in such experiments is related to their mass — with heavier particles being less likely to be created — so by examining the results of such experiments, limits on the mass of a magnetic monopole can be calculated. The most recent such experiments suggest that monopoles with masses below 600 GeV/c^{2} do not exist, while upper limits on their mass due to the very existence of the universe  which would have collapsed by now if they were too heavy  are about 10^{17} GeV/c^{2}.
While a magnetic monopole particle has never been conclusively observed, there are a number of phenomena in condensedmatter physics where a material, due to the collective behavior of its electrons and ions, can show emergent phenomena that resemble magnetic monopoles in some respect.^{[19]}^{[20]}^{[21]}^{[22]}^{[23]} These should not be confused with actual monopole particles. In particular, the divergence of the microscopic magnetic Bfield is zero everywhere in these systems, which it would not be in the presence of a true magnetic monopole particle. The behavior of these quasiparticles would only become indistinguishable from true magnetic monopoles — and they would truly deserve the name — if the socalled magnetic fluxtubes connecting these wouldbe monopoles became unobservable which also means that these flux tubes would have to be infinitely thin, obey the Dirac quantization rule, and thus deserve to be called Dirac strings.
In a paper published in the journal Science in September 2009, researchers Jonathan Morris and Alan Tennant from the HelmholtzZentrum Berlin für Materialien und Energie (HZB) along with Santiago Grigera from Instituto de Física de Líquidos y Sistemas Biológicos (IFLYSIB, CONICET) and other colleagues from Dresden University of Technology, University of St. Andrews and Oxford University described the observation of quasiparticles resembling magnetic monopoles. A single crystal of dysprosium titanate in a highly frustrated pyrochlore lattice (F d 3 m) was cooled to a temperature between 0.6 kelvin and 2.0 kelvin. Using observations of neutron scattering, the magnetic moments were shown to align in the spin ice into interwoven tubelike bundles resembling Dirac strings. At the defect formed by the end of each tube, the magnetic field looks like that of a monopole. Using an applied magnetic field to break the symmetry of the system, the researchers were able to control the density and orientation of these strings. A contribution to the heat capacity of the system from an effective gas of these quasiparticles was also described.^{[24]}^{[25]}
