Magnetic monopole

In physics, a magnetic monopole is a hypothetical particle that may be loosely described as "a magnet with only one pole" (see electromagnetic theory for more on magnetic poles). In more accurate terms, it would have net magnetic charge. Interest in the concept stems from particle theories, notably Grand Unified Theories and superstring theories, that predict either the existence or the possibility of magnetic monopoles.

Despite systematic searches since 1931, as of 2007, magnetic monopoles have never been observed. It therefore remains possible that monopoles do not exist at all. The failure of given experiments to find magnetic monopoles also places constraints on their possible properties and hence on the physical theories that predict them. Some current models suggest that while magnetic monopoles could exist, they are so massive that they may never be observed in practice.

Background

Unsolved problem in physics:

Are there any particles that carry "magnetic charge", and if so, why are they so difficult to detect?

(more unsolved problems in physics)

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 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

Maxwell's equations of electromagnetism relate the electric and magnetic fields to the motions of electric charges. The standard form of the equations provide for an electric charge, but posit no magnetic charge. Except for this, the equations are symmetric under interchange of electric and magnetic field. 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.[1] With the inclusion of a variable for these magnetic charges, say $\rho _{m}\,$, there will also be "magnetic current" variable in the equations, ${\vec {J}}_{m}\,$. The extended Maxwell's equations, simplified by nondimensionalization, are as follows:

Name	Without Magnetic Monopoles	With Magnetic Monopoles
Gauss's law:	${\vec {\nabla }}\cdot {\vec {E}}=4\pi \rho _{e}$	${\vec {\nabla }}\cdot {\vec {E}}=4\pi \rho _{e}$
Gauss' law for magnetism:	${\vec {\nabla }}\cdot {\vec {B}}=0$	${\vec {\nabla }}\cdot {\vec {B}}=4\pi \rho _{m}$
Faraday's law of induction:	$-{\vec {\nabla }}\times {\vec {E}}={\frac {\partial {\vec {B}}}{\partial t}}$	$-{\vec {\nabla }}\times {\vec {E}}={\frac {\partial {\vec {B}}}{\partial t}}+4\pi {\vec {J}}_{m}$
Ampère's law (with Maxwell's extension):	${\vec {\nabla }}\times {\vec {B}}={\frac {\partial {\vec {E}}}{\partial t}}+4\pi {\vec {J}}_{e}$	${\vec {\nabla }}\times {\vec {B}}={\frac {\partial {\vec {E}}}{\partial t}}+4\pi {\vec {J}}_{e}$
Note: the Bivector notation embodies the sign swap, and these four equations can be written as only one equation.

If magnetic charges do not exist, or if they exist but where they are not present in a region, then the new variables are zero, and the symmetric equations reduce to the conventional equations of electromagnetism such as ${\vec {\nabla }}\cdot {\vec {B}}=0$. Such extended Maxwell's equations would symmetric in three dimensional space, so that electric and magnetic field would be interchangeable. However this would be true only for 3D, because magnetic field is an pseudovector and can be generally described as an rank 2 antisymmetric tensor in general number of spatial dimensions, while electric field is true vector and it is vector in any number of dimensions.

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 QM, but in 1931 Dirac showed that a discrete charge naturally "falls out" of QM.

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, 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 point-like 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 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 $\exp(i\phi )$ of its wave function must be unchanged, which implies that the phase $\phi$ added to the wave function must be a multiple of $2\pi$:

$q_{e}q_{m}\in Z$ , where Z is the set of integers

This is known as the Dirac quantization condition. In certain units, the condition is exactly given by the relation above. 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 inverse 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 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.

Mathematical approach to Dirac monopole

Classically, gauge theory is described by a connection over a principal G-bundle over spacetime. Ordinary spacetime has the topology of R⁴, which is topologically trivial. So, the space of all possible connections over the principal G-bundle is connected. But consider what happens when we remove a timelike worldline from spacetime. The resulting spacetime is homotopically equivalent to the topological sphere S². So, it suffices to classify the connected components of the space of all connections over a principal G-bundle over S². To do this, consider covering S² by two charts, each homeomorphic to the open 2-ball such that their intersection is homeomorphic to the strip S¹×I. 2-balls are homotopically trivial and the strip is homotopically equivalent to the circle S¹. So a topological classification of the possible connections is reduced to classifying the transition functions, which is given by the first homotopy group of G. In other words, a gauge theory can only admit Dirac monopoles provided G is not simply connected. For instance, 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 even in principle.

It is easy to see how this argument generalizes to d + 1 dimensions with $d\geq 2$. We look at the homotopy group π_d−2(G).

Grand Unified Theories

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 for them to decay to.

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. This leads to a direct prediction of the amount of monopoles in the universe today, which is about 10¹¹ times the critical density of our universe. The universe appears to be close to critical density, so monopoles should be fairly common. For this reason, monopoles became a major interest in the 1970s and 80s, along with the other "approachable" prediction of GUTs, proton decay. The apparent problem with monopoles is resolved by cosmic inflation that greatly reduces the expected abundance of magnetic monopoles.

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.

Attempts to find monopoles

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 so-called "superconducting quantum interference detector", 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. The lack of such events places a limit on the number of monopoles of about 1 monopole per 10²⁹ nucleons.

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/c² 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 10¹⁷ GeV/c².

Non-inflationary Big Bang cosmology suggests that monopoles should be plentiful, and the failure to find magnetic monopoles is one of the main problems that led to the creation of cosmic inflation theory. In inflation, the visible universe was much smaller in the period before inflation, and despite the very short time before inflation, it would have been small enough for the whole visible universe to have been within the horizon, and thus not requiring many monopoles, perhaps only one. At the moment, versions of inflation seem to be the most likely cosmological theories.

Interestingly, the other major prediction of GUTs, proton decay, has also not been observed. The absence of these two key pieces of evidence has generally led to a decline in work on "classic" GUTs, and the introduction of more "radical" proposals, such as superstrings.

See also

• Dirac string
• Dyon
• Felix Ehrenhaft
• Halbach array
• Instanton
• Meron
• Soliton
• 't Hooft-Polyakov monopole
• Wu-Yang monopole

References

• Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University Press. ISBN 0-19-514665-4.
• Milton, Kimball A. (2006). "Theoretical and experimental status of magnetic monopoles". Reports on Progress in Physics. 69 (6): 1637–1711. doi:10.1088/0034-4885/69/6/R02. {{cite journal}}: Unknown parameter |month= ignored (help)
• Shnir, Yakov M. (2005). Magnetic Monopoles. Springer-Verlag. ISBN 3-540-25277-0.

External links

• 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.