Magnetic monopole

In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa).^[1]^[2] A magnetic monopole would have a net north or south "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring theories, which predict their existence.^[3]^[4]^{[full citation needed]} The known elementary particles that have electric charge are electric monopoles.

Magnetism in bar magnets and electromagnets is not caused by magnetic monopoles, and indeed, there is no known experimental or observational evidence that magnetic monopoles exist.

Some condensed matter systems contain effective (non-isolated) magnetic monopole quasi-particles,^[5] or contain phenomena that are mathematically analogous to magnetic monopoles.^[6]

Historical background

Early science and classical physics

Many early scientists attributed the magnetism of lodestones to two different "magnetic fluids" ("effluvia"), a north-pole fluid at one end and a south-pole fluid at the other, which attracted and repelled each other in analogy to positive and negative electric charge.^[7]^[8] However, an improved understanding of electromagnetism in the nineteenth century showed that the magnetism of lodestones was properly explained not by magnetic monopole fluids, but rather by a combination of electric currents, the electron magnetic moment, and the magnetic moments of other particles. Gauss's law for magnetism, one of Maxwell's equations, is the mathematical statement that magnetic monopoles do not exist. Nevertheless, Pierre Curie pointed out in 1894^[9] that magnetic monopoles could conceivably exist, despite not having been seen so far.

Quantum mechanics

The quantum theory of magnetic charge started with a paper by the physicist Paul Dirac in 1931.^[10] In this paper, Dirac showed that if any magnetic monopoles exist in the universe, then all electric charge in the universe must be quantized (Dirac quantization condition).^[11] The electric charge is, in fact, quantized, which is consistent with (but does not prove) the existence of monopoles.^[11]

Since Dirac's paper, several systematic monopole searches have been performed. Experiments in 1975^[12] and 1982^[13] produced candidate events that were initially interpreted as monopoles, but are now regarded as inconclusive.^[14] Therefore, it remains an open question whether monopoles exist. Further advances in theoretical particle physics, particularly developments in grand unified theories and quantum gravity, have led to more compelling arguments (detailed below) that monopoles do exist. Joseph Polchinski, a string theorist, described the existence of monopoles as "one of the safest bets that one can make about physics not yet seen".^[15] 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 create in particle accelerators (see § Searches for magnetic monopoles below), and also too rare in the Universe to enter a particle detector with much probability.^[15]

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. Since 2009, numerous news reports from the popular media^[16]^[17] have incorrectly described these systems as the long-awaited discovery of the magnetic monopoles, but the two phenomena are only superficially related to one another.^[18]^[19] These condensed-matter systems remain an area of active research. (See § "Monopoles" in condensed-matter systems below.)

Poles and magnetism in ordinary matter

All matter 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 do not derive from magnetic monopoles.

Instead, magnetism in ordinary matter is due to 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

Maxwell's equations of electromagnetism relate the electric and magnetic fields to each other and to the distribution of electric charge and current. The standard equations provide for electric charge, but they posit zero magnetic charge and current. Except for this constraint, the equations are symmetric under the interchange of the electric and magnetic fields. Maxwell's equations are symmetric when the charge and electric current density are zero everywhere, as in vacuum.

Maxwell's equations can also be written in a fully symmetric form if one allows for "magnetic charge" analogous to electric charge.^[20] With the inclusion of a variable for the density of magnetic charge, say $ρ m$ , there is also a "magnetic current density" variable in the equations, $j m$ .

If magnetic charge does not exist – or if it exists but is absent 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 $\nabla \cdot B = 0$ (where $\nabla\cdot$ is the divergence operator and $B$ is the magnetic flux density).

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.

Top: E field due to an electric dipole moment d.
Bottom left: B field due to a magnetic dipole m formed by two hypothetical 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 charge (black/white) and magnetic charge (red/blue).^[21]^[22]

In Gaussian cgs units

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

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

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 the International System of Quantities used with the SI, there are two conventions for defining magnetic charge $q m$ , each with different units: weber (Wb) and ampere-meter (A⋅m). The conversion between them is $q m$ = $μ 0 q m$ , since the units are 1 Wb = 1 H⋅A = (1 H⋅m⁻¹)(1 A⋅m), where H is the henry – the SI unit of inductance.

Maxwell's equations then take the following forms (using the same notation above):^{[notes 1]}

Maxwell's equations and Lorentz force equation with magnetic monopoles: SI units
Name	Without magnetic monopoles	With magnetic monopoles
Name	Without magnetic monopoles	Weber convention	Ampere-meter convention
Gauss's law	$\nabla \cdot \mathbf {E} ={\frac {\rho _{\mathrm {e} }}{\varepsilon _{0}}}$
Ampère's law (with Maxwell's extension)	$\nabla \times \mathbf {B} -{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}=\mu _{0}\mathbf {j} _{\mathrm {e} }$
Gauss's law for magnetism	$\nabla \cdot \mathbf {B} =0$	$\nabla \cdot \mathbf {B} =\rho _{\mathrm {m} }$	$\nabla \cdot \mathbf {B} =\mu _{0}\rho _{\mathrm {m} }$
Faraday's law of induction	$-\nabla \times \mathbf {E} -{\frac {\partial \mathbf {B} }{\partial t}}=0$

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[notes 1]