Magnetic monopole

All matter ever isolated to date, including every atom on the periodic table and every particle in the standard model, has zero magnetic monopole charge. Therefore, the ordinary phenomena of magnetism and magnets have nothing to do with magnetic monopoles.

Instead, magnetism in ordinary matter comes from two sources. First, electric currents create magnetic fields according to Ampère's law. Second, many elementary particles have an intrinsic magnetic moment, the most important of which is the electron magnetic dipole moment, which is related to its quantum-mechanical spin.

Mathematically, the magnetic field of an object is often described in terms of a multipole expansion. This is an expression of the field as the sum of component fields with specific mathematical forms. The first term in the expansion is called the monopole term, the second is called dipole, then quadrupole, then octupole, and so on. Any of these terms can be present in the multipole expansion of an electric field, for example. However, in the multipole expansion of a magnetic field, the "monopole" term is always exactly zero (for ordinary matter). A magnetic monopole, if it exists, would have the defining property of producing a magnetic field whose monopole term is non-zero.

A magnetic dipole is something whose magnetic field is predominantly or exactly described by the magnetic dipole term of the multipole expansion. The term dipole means two poles, corresponding to the fact that a dipole magnet typically contains a north pole on one side and a south pole on the other side. This is analogous to an electric dipole, which has positive charge on one side and negative charge on the other. However, an electric dipole and magnetic dipole are fundamentally quite different. In an electric dipole made of ordinary matter, the positive charge is made of protons and the negative charge is made of electrons, but a magnetic dipole does not have different types of matter creating the north pole and south pole. Instead, the two magnetic poles arise simultaneously from the aggregate effect of all the currents and intrinsic moments throughout the magnet. Because of this, the two poles of a magnetic dipole must always have equal and opposite strength, and the two poles cannot be separated from each other.

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.[notes 1] Maxwell's equations are symmetric when the charge and electric current density are zero everywhere, which is the case in vacuum.

Fully symmetric Maxwell's equations can also be written if one allows for the possibility of "magnetic charges" analogous to electric charges.[20] With the inclusion of a variable for the density of these magnetic charges, say ρ_m, there is also a "magnetic current density" variable in the equations, j_m.

If magnetic charges do not exist – or if they do exist but are not present in a region of 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 ∇⋅B = 0 (where ∇⋅ is divergence and B is the magnetic B field).

Left: Fields due to stationary electric and magnetic monopoles.
Right: In motion (velocity v), an electric charge induces a B field while a magnetic charge induces an E field. Conventional current is used.

Top: E field due to an electric dipole moment d.
Bottom left: B field due to a mathematical magnetic dipole m formed by two magnetic monopoles.
Bottom right: B field due to a natural magnetic dipole moment m found in ordinary matter (not from magnetic monopoles). (There should not be red and blue circles in the bottom right image.)

The E fields and B fields are due to electric charges (black/white) and magnetic poles (red/blue).[21][22]

In Gaussian cgs units

The extended Maxwell's equations are as follows, in Gaussian cgs units:[23]

Maxwell's equations and Lorentz force equation with magnetic monopoles: Gaussian cgs units

Name	Without magnetic monopoles	With magnetic monopoles
Gauss's law	${\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho _{\mathrm {e} }}$
Gauss's law for magnetism	${\displaystyle \nabla \cdot \mathbf {B} =0}$	${\displaystyle \nabla \cdot \mathbf {B} =4\pi \rho _{\mathrm {m} }}$
Faraday's law of induction	${\displaystyle -\nabla \times \mathbf {E} ={\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}}$	${\displaystyle -\nabla \times \mathbf {E} ={\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}+{\frac {4\pi }{c}}\mathbf {j} _{\mathrm {m} }}$
Ampère's law (with Maxwell's extension)	${\displaystyle \nabla \times \mathbf {B} ={\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}+{\frac {4\pi }{c}}\mathbf {j} _{\mathrm {e} }}$
Lorentz force law[23][24]	${\displaystyle \mathbf {F} =q_{\mathrm {e} }\left(\mathbf {E} +{\frac {\mathbf {v} }{c}}\times \mathbf {B} \right)}$	${\displaystyle \mathbf {F} =q_{\mathrm {e} }\left(\mathbf {E} +{\frac {\mathbf {v} }{c}}\times \mathbf {B} \right)+q_{\mathrm {m} }\left(\mathbf {B} -{\frac {\mathbf {v} }{c}}\times \mathbf {E} \right)}$

In these equations ρ_m is the magnetic charge density, 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. For all other definitions and details, see Maxwell's equations. For the equations in nondimensionalized form, remove the factors of c.

In SI units

In SI units, there are two conflicting definitions in use for magnetic charge q_m, with different units: weber (Wb) and ampere-meter (A⋅m). The conversion between them is q_m^[Wb] = μ₀q_m^[A⋅m], since the units are 1 Wb = 1 H⋅A = (1 H⋅m⁻¹)(1 A⋅m) by dimensional analysis (H is the henry – the SI unit of inductance).

Maxwell's equations then take the following forms (using the same notation above):[notes 2]

Maxwell's equations and Lorentz force equation with magnetic monopoles: SI units

Name	Without magnetic monopoles	With magnetic monopoles
		Weber convention	Ampere-meter convention
Gauss's law	${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{\mathrm {e} }}{\varepsilon _{0}}}}$
Gauss's law for magnetism	${\displaystyle \nabla \cdot \mathbf {B} =0}$	${\displaystyle \nabla \cdot \mathbf {B} =\rho _{\mathrm {m} }}$	${\displaystyle \nabla \cdot \mathbf {B} =\mu _{0}\rho _{\mathrm {m} }}$
Faraday's law of induction	${\displaystyle -\nabla \times \mathbf {E} ={\frac {\partial \mathbf {B} }{\partial t}}}$	${\displaystyle -\nabla \times \mathbf {E} ={\frac {\partial \mathbf {B} }{\partial t}}+\mathbf {j} _{\mathrm {m} }}$	${\displaystyle -\nabla \times \mathbf {E} ={\frac {\partial \mathbf {B} }{\partial t}}+\mu _{0}\mathbf {j} _{\mathrm {m} }}$
Ampère's law (with Maxwell's extension)	${\displaystyle \nabla \times \mathbf {B} ={\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}+\mu _{0}\mathbf {j} _{\mathrm {e} }}$
Lorentz force equation	${\displaystyle \mathbf {F} =q_{\mathrm {e} }\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)}$	${\displaystyle {\begin{aligned}\mathbf {F} ={}&q_{\mathrm {e} }\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)+\\&{\frac {q_{\mathrm {m} }}{\mu _{0}}}\left(\mathbf {B} -\mathbf {v} \times {\frac {\mathbf {E} }{c^{2}}}\right)\end{aligned}}}$	${\displaystyle {\begin{aligned}\mathbf {F} ={}&q_{\mathrm {e} }\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)+\\&q_{\mathrm {m} }\left(\mathbf {B} -\mathbf {v} \times {\frac {\mathbf {E} }{c^{2}}}\right)\end{aligned}}}$

Tensor formulation

Maxwell's equations in the language of tensors makes Lorentz covariance clear. The generalized equations are:[25][26]

Maxwell equations	Gaussian units	SI units (Wb)	SI units (A⋅m)
Faraday–Gauss law	${\displaystyle \partial _{\alpha }F^{\alpha \beta }={\frac {4\pi }{c}}J_{\mathrm {e} }^{\beta }}$	${\displaystyle \partial _{\alpha }F^{\alpha \beta }=\mu _{0}J_{\mathrm {e} }^{\beta }}$	${\displaystyle \partial _{\alpha }F^{\alpha \beta }=\mu _{0}J_{\mathrm {e} }^{\beta }}$
Ampère–Gauss law	${\displaystyle \partial _{\alpha }{{\tilde {F}}^{\alpha \beta }}={\frac {4\pi }{c}}J_{\mathrm {m} }^{\beta }}$	${\displaystyle \partial _{\alpha }{{\tilde {F}}^{\alpha \beta }}={\frac {1}{c}}J_{\mathrm {m} }^{\beta }}$	${\displaystyle \partial _{\alpha }{{\tilde {F}}^{\alpha \beta }}={\frac {\mu _{0}}{c}}J_{\mathrm {m} }^{\beta }}$
Lorentz force law	${\displaystyle {\frac {dp_{\alpha }}{d\tau }}=\left[q_{\mathrm {e} }F_{\alpha \beta }+q_{\mathrm {m} }{{\tilde {F}}_{\alpha \beta }}\right]{\frac {v^{\beta }}{c}}}$	${\displaystyle {\frac {dp_{\alpha }}{d\tau }}=\left[q_{\mathrm {e} }F_{\alpha \beta }+{\frac {q_{\mathrm {m} }}{\mu _{0}c}}{{\tilde {F}}_{\alpha \beta }}\right]v^{\beta }}$	${\displaystyle {\frac {dp_{\alpha }}{d\tau }}=\left[q_{\mathrm {e} }F_{\alpha \beta }+{\frac {q_{\mathrm {m} }}{c}}{{\tilde {F}}_{\alpha \beta }}\right]v^{\beta }}$

where

• F^αβ is the electromagnetic tensor, ~F^αβ = 1/2ε^αβγδF_γδ is the dual electromagnetic tensor,
• for a particle with electric charge q_e and magnetic charge q_m; v is the four-velocity and p the four-momentum,
• for an electric and magnetic charge distribution; J_e = (ρ_e, j_e) is the electric four-current and J_m = (ρ_m, j_m) the magnetic four-current.

For a particle having only electric charge, one can express its field using a four-potential, according to the standard covariant formulation of classical electromagnetism:

${\displaystyle F_{\alpha \beta }=\partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha }\,}$

However, this formula is inadequate for a particle that has both electric and magnetic charge, and we must add a term involving another potential P.[27][28]

${\displaystyle F_{\alpha \beta }=\partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha }\ +\partial ^{\mu }(\varepsilon _{\alpha \beta \mu \nu }P^{\nu }),}$

This formula for the fields is often called the Cabibbo–Ferrari relation, though Shanmugadhasan proposed it earlier.[28] The quantity ε^αβγδ is the Levi-Civita symbol, and the indices (as usual) behave according to the Einstein summation convention.

Duality transformation

The generalized Maxwell's equations possess a certain symmetry, called a duality transformation. One can choose any real angle ξ, and simultaneously change the fields and charges everywhere in the universe as follows (in Gaussian units):[29]

Charges and currents	Fields
${\displaystyle {\begin{pmatrix}\rho _{\mathrm {e} }\\\rho _{\mathrm {m} }\end{pmatrix}}={\begin{pmatrix}\cos \xi &-\sin \xi \\\sin \xi &\cos \xi \\\end{pmatrix}}{\begin{pmatrix}\rho _{\mathrm {e} }'\\\rho _{\mathrm {m} }'\end{pmatrix}}}$	${\displaystyle {\begin{pmatrix}\mathbf {E} \\\mathbf {H} \end{pmatrix}}={\begin{pmatrix}\cos \xi &-\sin \xi \\\sin \xi &\cos \xi \\\end{pmatrix}}{\begin{pmatrix}\mathbf {E'} \\\mathbf {H'} \end{pmatrix}}}$
${\displaystyle {\begin{pmatrix}\mathbf {J} _{\mathrm {e} }\\\mathbf {J} _{\mathrm {m} }\end{pmatrix}}={\begin{pmatrix}\cos \xi &-\sin \xi \\\sin \xi &\cos \xi \\\end{pmatrix}}{\begin{pmatrix}\mathbf {J} _{\mathrm {e} }'\\\mathbf {J} _{\mathrm {m} }'\end{pmatrix}}}$	${\displaystyle {\begin{pmatrix}\mathbf {D} \\\mathbf {B} \end{pmatrix}}={\begin{pmatrix}\cos \xi &-\sin \xi \\\sin \xi &\cos \xi \\\end{pmatrix}}{\begin{pmatrix}\mathbf {D'} \\\mathbf {B'} \end{pmatrix}}}$

where the primed quantities are the charges and fields before the transformation, and the unprimed quantities are after the transformation. The fields and charges after this transformation still obey the same Maxwell's equations. The matrix is a two-dimensional rotation matrix.

Because of the duality transformation, one cannot uniquely decide whether a particle has an electric charge, a magnetic charge, or both, just by observing its behavior and comparing that to Maxwell's equations. For example, it is merely a convention, not a requirement of Maxwell's equations, that electrons have electric charge but not magnetic charge; after a ξ = π/2 transformation, it would be the other way around. The key empirical fact is that all particles ever observed have the same ratio of magnetic charge to electric charge.[29] Duality transformations can change the ratio to any arbitrary numerical value, but cannot change the fact that all particles have the same ratio. Since this is the case, a duality transformation can be made that sets this ratio at zero, so that all particles have no magnetic charge. This choice underlies the "conventional" definitions of electricity and magnetism.[29]

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_eq_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_eq_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 point-like magnetic charge whose magnetic field behaves as q_m / r ² and is directed in the radial direction, located at the origin. 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" (the half-space z > 0 above the particle), and another set of functions for the "southern hemisphere". These two vector potentials are matched at the "equator" (the plane z = 0 through the particle), and they differ by a gauge transformation. The wave function of an electrically charged 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 φ of its wave function e^iφ must be unchanged, which implies that the phase φ added to the wave function must be a multiple of 2π:

Units	Condition
Gaussian-cgs units	${\displaystyle 2{\frac {q_{\mathrm {e} }q_{\mathrm {m} }}{\hbar c}}\in \mathbb {Z} }$
SI units (weber convention)[30]	${\displaystyle {\frac {q_{\mathrm {e} }q_{\mathrm {m} }}{2\pi \hbar }}\in \mathbb {Z} }$
SI units (ampere-meter convention)	${\displaystyle {\frac {q_{\mathrm {e} }q_{\mathrm {m} }}{2\pi \varepsilon _{0}\hbar c^{2}}}\in \mathbb {Z} }$

where ε₀ is the vacuum permittivity, ħ = h/2π is the reduced Planck's constant, c is the speed of light, and ℤ is the set of integers.

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—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 have magnetic monopoles anyway.)

If we maximally extend the definition of the vector potential for the southern hemisphere, it is 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.

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 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 implied 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 scenario, 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.

Cosmological models of the events following the Big Bang make predictions about what the horizon volume was, which lead to predictions about present-day monopole density. Early models predicted an enormous density of monopoles, in clear contradiction to the experimental evidence.[32][33] This was called the "monopole problem". Its widely accepted resolution was not a change in the particle-physics prediction of monopoles, but rather in the cosmological models used to infer their present-day density. Specifically, more recent theories of cosmic inflation drastically reduce the predicted number of magnetic monopoles, to a density small enough to make it unsurprising that humans have never seen one.[34] This resolution of the "monopole problem" was regarded as a success of cosmic inflation theory. (However, of course, it is only a noteworthy success if the particle-physics monopole prediction is correct.[35]) 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.

Experimental searches for magnetic monopoles can be placed in one of two categories: those that try to detect preexisting magnetic monopoles and those that try to create and detect new magnetic monopoles.

Passing a magnetic monopole through a coil of wire induces a net current in the coil. This is not the case for a magnetic dipole or higher order magnetic pole, for which the net induced current is zero, and hence the effect can be used as an unambiguous test for the presence of magnetic monopoles. In a wire with finite resistance, the induced current quickly dissipates its energy as heat, but in a superconducting loop the induced current is long-lived. By using a highly sensitive "superconducting quantum interference device" (SQUID) one can, in principle, detect even a single magnetic monopole.

According to standard inflationary cosmology, magnetic monopoles produced before inflation would have been diluted to an extremely low density today. Magnetic monopoles may also have been produced thermally after inflation, during the period of reheating. However, the current bounds on the reheating temperature span 18 orders of magnitude and as a consequence the density of magnetic monopoles today is not well constrained by theory.

There have been many searches for preexisting magnetic monopoles. Although there has been one tantalizing event recorded, by Blas Cabrera Navarro on the night of February 14, 1982 (thus, sometimes referred to as the "Valentine's Day Monopole"[36]), there has never been reproducible evidence for the existence of magnetic monopoles.[13] The lack of such events places an upper limit on the number of monopoles of about one monopole per 10²⁹ nucleons.

Another experiment in 1975 resulted in the announcement of the detection of a moving magnetic monopole in cosmic rays by the team led by P. Buford Price.[12] Price later retracted his claim, and a possible alternative explanation was offered by Alvarez.[37] In his paper it was demonstrated that the path of the cosmic ray event that was claimed due to a magnetic monopole could be reproduced by the path followed by a platinum nucleus decaying first to osmium, and then to tantalum.

High energy particle colliders have been used to try to create magnetic monopoles. Due to the conservation of magnetic charge, magnetic monopoles must be created in pairs, one north and one south. Due to conservation of energy, only magnetic monopoles with masses less than half of the center of mass energy of the colliding particles can be produced. Beyond this, very little is known theoretically about the creation of magnetic monopoles in high energy particle collisions. This is due to their large magnetic charge, which invalidates all the usual calculational techniques. As a consequence, collider based searches for magnetic monopoles cannot, as yet, provide lower bounds on the mass of magnetic monopoles. They can however provide upper bounds on the probability (or cross section) of pair production, as a function of energy.

The ATLAS experiment at the Large Hadron Collider currently has the most stringent cross section limits for magnetic monopoles of 1 and 2 Dirac charges, produced through Drell-Yan pair production. A team led by Wendy Taylor searches for these particles based on theories that define them as long lived (they don't quickly decay), as well as being highly ionizing (their interaction with matter is predominantly ionizing). In 2019 the search for magnetic monopoles in the ATLAS detector reported its first results from data collected from the LHC Run 2 collisions at center of mass energy of 13 TeV, which at 34.4 fb⁻¹ is the largest dataset analyzed to date.[38]

The MoEDAL experiment, installed at the Large Hadron Collider, is currently searching for magnetic monopoles and large supersymmetric particles using nuclear track detectors and aluminum bars around LHCb's VELO detector. The particles it is looking for damage the plastic sheets that comprise the nuclear track detectors along their path, with various identifying features. Further, the aluminum bars can trap sufficiently slowly moving magnetic monopoles. The bars can then be analyzed by passing them through a SQUID.

The Russian astrophysicist Igor Novikov claims the fields of macroscopic black holes are potential magnetic monopoles, representing the entrance to an Einstein–Rosen bridge.[39]

Since around 2003, various condensed-matter physics groups have used the term “magnetic monopole” to describe a different and largely unrelated phenomenon.[18][19]

A true magnetic monopole would be a new elementary particle, and would violate Gauss's law for magnetism ∇⋅B = 0. A monopole of this kind, which would help to explain the law of charge quantization as formulated by Paul Dirac in 1931,[40] has never been observed in experiments.[41][42]

The monopoles studied by condensed-matter groups have none of these properties. They are not a new elementary particle, but rather are an emergent phenomenon in systems of everyday particles (protons, neutrons, electrons, photons); in other words, they are quasi-particles. They are not sources for the B-field (i.e., they do not violate ∇⋅B = 0); instead, they are sources for other fields, for example the H-field,[5] the "B*-field" (related to superfluid vorticity),[6][43] or various other quantum fields.[44] They are not directly relevant to grand unified theories or other aspects of particle physics, and do not help explain charge quantization—except insofar as studies of analogous situations can help confirm that the mathematical analyses involved are sound.[45]

There are a number of examples in condensed-matter physics where collective behavior leads to emergent phenomena that resemble magnetic monopoles in certain respects,[17][46][47][48] including most prominently the spin ice materials.[5][49] While these should not be confused with hypothetical elementary monopoles existing in the vacuum, they nonetheless have similar properties and can be probed using similar techniques.

Some researchers use the term magnetricity to describe the manipulation of magnetic monopole quasiparticles in spin ice,[49][50] in analogy to the word “electricity”.

One example of the work on magnetic monopole quasiparticles is a paper published in the journal Science in September 2009, in which researchers described the observation of quasiparticles resembling magnetic monopoles. A single crystal of the spin ice material dysprosium titanate 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 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.[16][51] This research went on to win the 2012 Europhysics Prize for condensed matter physics.

In another example, a paper in the February 11, 2011 issue of Nature Physics describes creation and measurement of long-lived magnetic monopole quasiparticle currents in spin ice. By applying a magnetic-field pulse to crystal of dysprosium titanate at 0.36 K, the authors created a relaxing magnetic current that lasted for several minutes. They measured the current by means of the electromotive force it induced in a solenoid coupled to a sensitive amplifier, and quantitatively described it using a chemical kinetic model of point-like charges obeying the Onsager–Wien mechanism of carrier dissociation and recombination. They thus derived the microscopic parameters of monopole motion in spin ice and identified the distinct roles of free and bound magnetic charges.[50]

In superfluids, there is a field B*, related to superfluid vorticity, which is mathematically analogous to the magnetic B-field. Because of the similarity, the field B* is called a “synthetic magnetic field”. In January 2014, it was reported that monopole quasiparticles[52] for the B* field were created and studied in a spinor Bose–Einstein condensate.[6] This constitutes the first example of a quasi-magnetic monopole observed within a system governed by quantum field theory.[45]

• Magnetic Monopole Searches (lecture notes)
• Particle Data Group summary of magnetic monopole search
• 'Race for the Pole' Dr David Milstead Freeview 'Snapshot' video by the Vega Science Trust and the BBC/OU.
• Interview with Jonathan Morris about magnetic monopoles and magnetic monopole quasiparticles. Drillingsraum, April 16, 2010
• Nature, 2009
• Sciencedaily, 2009
• Kadowaki, H.; Doi, N.; Aoki, Y.; Tabata, Y.; Sato, T. J.; Lynn, J. W.; Matsuhira, K.; Hiroi, Z. (2009). "Observation of Magnetic Monopoles in Spin Ice". Journal of the Physical Society of Japan. 78 (10): 103706. arXiv:0908.3568. Bibcode:2009JPSJ...78j3706K. doi:10.1143/JPSJ.78.103706. S2CID 118373241.
• Video of lecture by Paul Dirac on magnetic monopoles, 1975 on YouTube

This article incorporates material from N. Hitchin (2001) [1994], "Magnetic Monopole", Encyclopedia of Mathematics, EMS Press, which is licensed under the Creative Commons Attribution/Share-Alike License and GNU Free Documentation License.