Magnetic monopole


In particle physics, a magnetic monopole is a hypothetical particle that is an isolated magnet with only one magnetic pole. 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 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 magnetic monopole quasi-particles, or contain phenomena that are mathematically analogous to magnetic monopoles.

Historical background

Early science and classical physics

Many early scientists attributed the magnetism of lodestones to two different "magnetic fluids", 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. The electric charge is, in fact, quantized, which is consistent with the existence of monopoles.
Since Dirac's paper, several systematic monopole searches have been performed. Experiments in 1975 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 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, and also too rare in the Universe to enter a particle detector with much probability.
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 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. These condensed-matter systems remain an area of active research.

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. 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 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. 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 .

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 and ampere-meter. 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 :

Potential formulation

Maxwell's equations can also be expressed in terms of potentials as follows:
where

Tensor formulation

Maxwell's equations in the language of tensors makes Lorentz covariance clear. We introduce electromagnetic tensors and preliminary four-vectors in this article as follows:
where:
The generalized equations are:
Alternatively,
where the is the Levi-Civita symbol.

Duality transformation

The generalized Maxwell's equations possess a certain symmetry, called a duality transformation. One can choose any real angle, and simultaneously change the fields and charges everywhere in the universe as follows :
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.
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 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. Duality transformations can change the ratio to any arbitrary numerical value, but cannot change 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.

Dirac's quantization

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, but in 1931 Dirac showed that a discrete charge is implied by 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 and a single stationary magnetic monopole, which would not exert any forces on each other. 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, and is independent of the distance between them.
Quantum mechanics dictates, however, that angular momentum is quantized as a multiple of, so therefore the product 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.
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 and is directed in the radial direction, located at the origin. Because the divergence of is equal to zero everywhere except for the locus of the magnetic monopole at, one can locally define the vector potential such that the curl of the vector potential equals the magnetic field.
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 hemisphere". These two vector potentials are matched at the "equator", and they differ by a gauge transformation. The wave function of an electrically charged particle that orbits the "equator" generally changes by a phase, much like in the Aharonov–Bohm effect. This phase is proportional to the electric charge of the probe, as well as to the magnetic charge 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 of its wave function must be unchanged, which implies that the phase added to the wave function must be a multiple of. This is known as the Dirac quantization condition. In various units, this condition can be expressed as:
where is the vacuum permittivity, is the reduced Planck constant, is the speed of light, and is the set of integers.
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.
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 ; in more sophisticated theories, it is superseded by a smooth solution such as the 't Hooft–Polyakov monopole.