thumb|It is impossible to make magnetic monopoles from a [[bar magnet. If a bar magnet is cut in half, it is not the case that one half has the north pole and the other half has the south pole. Instead, each piece has its own north and south poles. A magnetic monopole cannot be created from normal matter such as atoms and electrons, but would instead be a new particle.]]
In particle physics, a magnetic monopole is a hypothetical particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magnetic charge". Modern interest in the concept stems from particle theories, notably grand unified and superstring theories, which predict their existence.
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. A magnetic monopole is not necessarily an elementary particle, and models for magnetic monopole production can include (but are not limited to) spin-0 monopoles or spin-1 massive vector mesons. The term "magnetic monopole" only refers to the nature of the particle, rather than a designation for a single particle.
Some condensed matter systems contain effective (non-isolated) magnetic monopole quasi-particles,
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. 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 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. 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). The electric charge is, in fact, quantized, which is consistent with (but does not prove) the existence of monopoles. and 1982 produced candidate events that were initially interpreted as monopoles, but are now regarded as inconclusive. Therefore, whether monopoles exist remains an open question. 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". 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 below), and also too rare in the Universe to enter a particle detector with much probability. These condensed-matter systems remain an area of active research. (See ' 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. With the inclusion of a variable for the density of magnetic charge, say , there is also a "magnetic current density" variable in the equations, .
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 (where is the divergence operator and is the magnetic flux density).
In SI units
In the International System of Quantities used with the SI, there are two conventions for defining magnetic charge , each with different units: weber (Wb) and ampere-meter (A⋅m). The conversion between them is , since the units are , where H is the henry – the SI unit of inductance.
Maxwell's equations then take the following forms (using the same notation above):
{| class="wikitable" style="text-align: center;"
|+ Maxwell's equations and Lorentz force equation with magnetic monopoles: SI units
|-
! rowspan=2 scope="col" width="200px" | Name
! rowspan=2 | Without magnetic <br/>monopoles
! colspan=2 | With magnetic monopoles
|-
! Weber convention
! Ampere-meter convention
|-
! Gauss's law
| colspan="3" | <math>\nabla \cdot \mathbf{E} = \frac{\rho_{\mathrm e{\varepsilon_0}</math>
|-
! Ampère's law (with Maxwell's extension)
| colspan="3" | <math>\nabla \times \mathbf{B} - \frac{1}{c^2} \frac{\partial \mathbf{E {\partial t} = \mu_0 \mathbf{j}_{\mathrm e}</math>
|-
! Gauss's law for magnetism
| <math>\nabla \cdot \mathbf{B} = 0</math>
| <math>\nabla \cdot \mathbf{B} = \rho_{\mathrm m}</math>
| <math>\nabla \cdot \mathbf{B} = \mu_0\rho_{\mathrm m}</math>
|-
! Faraday's law of induction
| <math>-\nabla \times \mathbf{E} - \frac{\partial \mathbf{B {\partial t} = 0</math>
| <math>-\nabla \times \mathbf{E} - \frac{\partial \mathbf{B {\partial t} = \mathbf{j}_{\mathrm m}</math>
| <math>-\nabla \times \mathbf{E} - \frac{\partial \mathbf{B {\partial t} = \mu_0\mathbf{j}_{\mathrm m}</math>
|-
! Lorentz force equation
| <math>\mathbf{F} = q_{\mathrm e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)</math>
| <math>\begin{align}
\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{align}</math>
| <math>\begin{align}
\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{align}</math>
|-
|}
Potential formulation
Maxwell's equations can also be expressed in terms of potentials as follows:
{| class="wikitable"
|-
! Name
! Gaussian units
! SI units (Wb)
! SI units (A⋅m)
|-
! Maxwell's equations <br /> (assuming Lorenz gauge)
| <math>\begin{align}
\Box \varphi_{\mathrm e} =& -4\pi \rho_{\mathrm e} \\
\Box \mathbf{A}_{\mathrm e} =& -\frac{4\pi}{c} \mathbf{j}_{\mathrm e} \\
\Box \varphi_{\mathrm m} =& -4\pi \rho_{\mathrm m} \\
\Box \mathbf{A}_{\mathrm m} =& -\frac{4\pi}{c} \mathbf{j}_{\mathrm m} \\
\end{align}</math>
| <math>\begin{align}
\Box \varphi_{\mathrm e} =& -\frac{\rho_{\mathrm e{\varepsilon_0} \\
\Box \mathbf{A}_{\mathrm e} =& -\mu_0 \mathbf{j}_{\mathrm e} \\
\Box \varphi_{\mathrm m} =& -\frac{\rho_{\mathrm m{\mu_0} \\
\Box \mathbf{A}_{\mathrm m} =& -\varepsilon_0 \mathbf{j}_{\mathrm m} \\
\end{align}</math>
| <math>\begin{align}
\Box \varphi_{\mathrm e} =& -\frac{\rho_{\mathrm e{\varepsilon_0} \\
\Box \mathbf{A}_{\mathrm e} =& -\mu_0 \mathbf{j}_{\mathrm e} \\
\Box \varphi_{\mathrm m} =& -\rho_{\mathrm m} \\
\Box \mathbf{A}_{\mathrm m} =& -\frac{\mathbf{j}_{\mathrm m{c^2} \\
\end{align}</math>
|-
! Lorenz gauge condition
| <math>\begin{align}
&\frac{1}{c}\frac{\partial}{\partial t}\varphi_{\mathrm e} + \nabla \cdot \mathbf{A}_{\mathrm e} = 0 \\
&\frac{1}{c}\frac{\partial}{\partial t}\varphi_{\mathrm m} + \nabla \cdot \mathbf{A}_{\mathrm m} = 0 \\
\end{align}</math>
| colspan="2" | <math>\begin{align}
&\frac{1}{c^2}\frac{\partial}{\partial t}\varphi_{\mathrm e} + \nabla \cdot \mathbf{A}_{\mathrm e} = 0 \\
&\frac{1}{c^2}\frac{\partial}{\partial t}\varphi_{\mathrm m} + \nabla \cdot \mathbf{A}_{\mathrm m} = 0 \\
\end{align}</math>
|-
! Relation to fields
| <math>\begin{align}
\mathbf{E} =& -\nabla \varphi_{\mathrm e} - \frac{1}{c}\frac{\partial \mathbf{A}_{\mathrm e{\partial t} - \nabla \times \mathbf{A}_{\mathrm m} \\
\mathbf{B} =& -\nabla \varphi_{\mathrm m} - \frac{1}{c}\frac{\partial \mathbf{A}_{\mathrm m{\partial t} + \nabla \times \mathbf{A}_{\mathrm e} \\
\end{align}</math>
| colspan="2" | <math>\begin{align}
\mathbf{E} =& -\nabla \varphi_{\mathrm e} - \frac{\partial \mathbf{A}_{\mathrm e{\partial t} - \frac{1}{\varepsilon_0} \nabla \times \mathbf{A}_{\mathrm m} \\
\mathbf{B} =& -\mu_0 \nabla \varphi_{\mathrm m} - \mu_0\frac{\partial \mathbf{A}_{\mathrm m{\partial t} + \nabla \times \mathbf{A}_{\mathrm e} \\
\end{align}</math>
|}
where
: <math>\Box = \nabla^2 - \frac{1}{c^2}\frac{\partial^2} (\mathbf{F}\cdot\mathbf{v},\; -\mathbf{F})</math>
|}
where:
- The signature of the Minkowski metric is .
- The electromagnetic tensor and its Hodge dual are antisymmetric tensors:
- : <math>F^{\alpha\beta} = -F^{\beta\alpha},\quad {\tilde F}^{\alpha\beta} = -{\tilde F}^{\beta\alpha}</math>
The generalized equations are:
{| class="wikitable"
|-
! Maxwell equations
! Gaussian units
! SI units (Wb)
! SI units (A⋅m)
|-
! Ampère–Gauss law
| <math>\partial_\alpha F^{\alpha\beta} = \frac{4\pi}{c}J^\beta_{\mathrm e}</math>
| colspan="2" | <math>\partial_\alpha F^{\alpha\beta} = \mu_0 J^\beta_{\mathrm e}</math>
|-
! Faraday–Gauss law
| <math>\partial_\alpha {\tilde F^{\alpha\beta = \frac{4\pi}{c} J^\beta_{\mathrm m}</math>
| <math>\partial_\alpha {\tilde F^{\alpha\beta = \frac{1}{c} J^\beta_{\mathrm m}</math>
| <math>\partial_\alpha {\tilde F^{\alpha\beta = \frac{\mu_0}{c} J^\beta_{\mathrm m}</math>
|-
! Lorentz force law
| <math>f_\alpha = \left[ q_{\mathrm e} F_{\alpha\beta} + q_{\mathrm m} {\tilde F_{\alpha\beta \right] \frac{v^\beta}{c} </math>
| <math>f_\alpha = \left[ q_{\mathrm e} F_{\alpha\beta} + \frac{q_{\mathrm m{\mu_0 c} {\tilde F_{\alpha\beta \right] v^\beta </math>
| <math>f_\alpha = \left[ q_{\mathrm e} F_{\alpha\beta} + \frac{q_{\mathrm m{c} {\tilde F_{\alpha\beta \right] v^\beta </math>
|}
Alternatively,
{| class="wikitable"
|-
! Name
! Gaussian units
! SI units (Wb)
! SI units (A⋅m)
|-
! rowspan="2" | Maxwell's equations
| <math>\partial^\alpha \partial_\alpha A^\beta_{\mathrm e} - \partial^\beta \partial_\alpha A^\alpha_{\mathrm e} = \frac{4\pi}{c}J^\beta_{\mathrm e}</math>
| colspan="2" | <math>\partial^\alpha \partial_\alpha A^\beta_{\mathrm e} - \partial^\beta \partial_\alpha A^\alpha_{\mathrm e} = \mu_0 J^\beta_{\mathrm e}</math>
|-
| <math>\partial^\alpha \partial_\alpha A^\beta_{\mathrm m} - \partial^\beta \partial_\alpha A^\alpha_{\mathrm m} = \frac{4\pi}{c}J^\beta_{\mathrm m}</math>
| <math>\partial^\alpha \partial_\alpha A^\beta_{\mathrm m} - \partial^\beta \partial_\alpha A^\alpha_{\mathrm m} = \varepsilon_0 J^\beta_{\mathrm m}</math>
| <math> \cdots = \frac{1}{c^2} J^\beta_{\mathrm m}</math>
|-
! Lorenz gauge condition
| colspan="3" | <math>\partial_\alpha A^\alpha_{\mathrm e} = 0,\quad \partial_\alpha A^\alpha_{\mathrm m} = 0 </math>
|-
! Relation to fields<br />(Cabibbo–Ferrari-Shanmugadhasan relation)
| <math>F^{\alpha\beta} = \partial^\alpha A_{\mathrm e}^\beta - \partial^\beta A_{\mathrm e}^\alpha - \varepsilon^{\alpha\beta\mu\nu} \partial_\mu A_