magnetic monopole

In physics, a magnetic monopole is a hypothetical particle^[1] that is a magnet with only one pole (see Maxwell's equations for more on magnetic poles). In more technical terms, it would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unification theory and superstring theories, which predict their existence.^[2][3]

The magnetic monopole was first hypothesized by Pierre Curie in 1894^[4], but the quantum theory of magnetic charge started with a 1931 paper by Paul Dirac.^[5] In this paper, Dirac showed that the existence of magnetic monopoles was consistent with Maxwell's equations only if electric charges are quantized, which is observed. Since then, several systematic monopole searches have been performed. Experiments in 1975^[6] and 1982^[7] produced candidate events that were initially interpreted as monopoles, but are now regarded as inconclusive^[8].

Monopole detection is an open problem in experimental physics. Within theoretical physics, some modern approaches assume their existence. Joseph Polchinski, a prominent string-theorist, described the existence of monopoles as "one of the safest bets that one can make about physics not yet seen".^[9] These theories are not necessarily inconsistent with the experimental evidence: in some models magnetic monopoles are unlikely to be observed, because they are too massive to be created in particle accelerators, and too rare in the universe to enter a particle detector.^[9]

Some condensed matter systems propose a superficially similar structure, 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. In late 2009 a large number of popular publications incorrectly reported this phenomenon as the long-awaited discovery of magnetic monopoles. However, the two phenomena are not related.^[10] However, they are considered interesting in their own right, and are an area of active research. (See below.)

Background

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 such 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 provides for an electric charge, but posits no magnetic charge. Except for this, the equations are symmetric under interchange of electric and magnetic fields.^[11] 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.^[12] With the inclusion of a variable for the density of these magnetic charges, say , there will also be a "magnetic current density" variable in the equations, .

If magnetic charges do not exist, or if they exist but are not present in a region, then the new variables are zero, and the extended equations reduce to the conventional equations of electromagnetism such as . Classically, the question is "Why does the magnetic charge always seem to be zero?"

In cgs units

The extended Maxwell's equations are as follows, in cgs units:^[13]

Maxwell's equations in cgs

Name	Without magnetic monopoles	With magnetic monopoles
Gauss's law:
Gauss's law for magnetism:
Faraday's law of induction:
Ampère's law (with Maxwell's extension):
Note: For the equations in nondimensionalized form, remove the factors of c.

The Lorentz force becomes^[13][14]

In these equations is the magnetic charge density, 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.

In SI units

In SI units, there are two conflicting conventions in use for magnetic charge. In one, magnetic charge has units of webers, while in the other, magnetic charge has units of ampere-meters. Maxwell's equations then take the following forms:^[15]

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

Name	Without magnetic monopoles	Weber convention	Ampere·meter convention
Gauss's Law
Gauss's Law for magnetism
Faraday's Law of induction
Ampère's Law
Lorentz force

In these equations is the magnetic charge density, 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.

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, that is to say, we can maintain the current 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, and 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 current 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, which led to new ideas. He considered a point-like magnetic charge whose magnetic field behaves as q_m / r² 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(iφ) of its wave function must be unchanged, which implies that the phase φ added to the wave function must be a multiple of 2π:

(cgs units)

(SI units, weber convention)^[16]

(SI units, ampere·meter convention)

where

• is vacuum permittivity
• is reduced Planck's constant
• is speed of light
• 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 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.

Topological interpretation

Dirac string

A gauge theory like electromagnetism is defined by a gauge field, which associates a group element to each path in space time. For infinitesimal paths, the group element is close to the identity, while for longer paths the group element is the successive product of the infinitesimal group elements along the way.

In electrodynamics, the group is U(1), unit complex numbers under multiplication. For infinitesimal paths, the group element is 1 + iA_μdx^μ which implies that for finite paths parametrized by s, the group element is:

The map from paths to group elements is called the Wilson loop or the holonomy, and for a U(1) gauge group it is the phase factor which the wavefunction of a charged particle acquires as it traverses the path. For a loop:

So that the phase a charged particle gets when going in a loop is the magnetic flux through the loop. When a small solenoid has a magnetic flux, there are interference fringes for charged particles which go around the solenoid, or around different sides of the solenoid, which reveal its presence.

But if all particle charges are integer multiples of e, solenoids with a flux of 2π / e have no interference fringes, because the phase factor for any charged particle is . Such a solenoid, if thin enough, is quantum mechanically invisible. If such a solenoid were to carry a flux of 2π / e, when the flux leaked out from one of its ends it would be indistinguishable from a monopole.

Dirac's monopole solution in fact describes an infinitesimal line solenoid ending at a point, and the location of the solenoid is the singular part of the solution, the Dirac string. Dirac strings link monopoles and antimonopoles of opposite magnetic charge, although in Dirac's version, the string just goes off to infinity. The string is unobservable, so you can put it anywhere, and by using two coordinate patches, the field in each patch can be made nonsingular by sliding the string to where it cannot be seen.

Grand unified theories

In a U(1) with quantized charge, the gauge group is a circle of radius 2π / e. Such a U(1) is called compact. Any U(1) which comes from a Grand Unified Theory is compact, because only compact higher gauge groups make sense. The size of the gauge group is a measure of the inverse coupling constant, so that in the limit of a large volume gauge group, the interaction of any fixed representation goes to zero.

The U(1) case is special because all its irreducible representations are the same size--- the charge is bigger by an integer amount but the field is still just a complex number--- so that in U(1) gauge field theory it is possible to take the decompactified limit with no contradiction. The quantum of charge becomes small, but each charged particle has a huge number of charge quanta so its charge stays finite. In a non-compact U(1) gauge theory, the charges of particles are generically not integer multiples of a single unit. Since charge quantization is an experimental certainty, it is clear that the U(1) of electromagnetism is compact.

GUTs lead to compact U(1)s, so they explain charge quantization in a way that seems to be logically independent from magnetic monopoles. But the explanation is essentially the same, because in any GUT which breaks down to a U(1) at long distances, there are magnetic monopoles.

The argument is topological:

1. The holonomy of a gauge field maps loops to elements of the gauge group. Infinitesimal loops are mapped to group elements infinitesimally close to the identity.
2. If you imagine a big sphere in space, you can deform an infinitesimal loop which starts and ends at the north pole as follows: stretch out the loop over the western hemisphere until it becomes a great circle (which still starts and ends at the north pole) then let it shrink back to a little loop while going over the eastern hemisphere. This is called lassoing the sphere.
3. Lassoing is a sequence of loops, so the holonomy maps it to a sequence of group elements, a continuous path in the gauge group. Since the loop at the beginning of the lassoing is the same as the loop at the end, the path in the group is closed.
4. If the group path associated to the lassoing procedure winds around the U(1), the sphere contains magnetic charge. During the lassoing, the holonomy changes by the amount of magnetic flux through the sphere.
5. Since the holonomy at the beginning and at the end is the identity, the total magnetic flux is quantized. The magnetic charge is proportional to the number of windings N, the magnetic flux through the sphere is equal to 2πN / e. This is the Dirac quantization condition, and it is a topological condition which demands that the long distance U(1) gauge field configurations are consistent.
6. When the U(1) comes from breaking a compact Lie group, the path which winds around the U(1) enough times is topologically trivial in the big group. In a non-U(1) compact lie group, the covering space is a Lie group with the same Lie algebra but where all closed loops are contractible. Lie groups are homogenous, so that any cycle in the group can be moved around so that it starts at the identity, then its lift to the covering group ends at P, which is a lift of the identity. Going around the loop twice gets you to P², three times to P³, all lifts of the identity. But there are only finitely many lifts of the identity, because the lifts can't accumulate. This number of times one has to traverse the loop to make it contractible is small, for example if the GUT group is SO(3), the covering group is SU(2), and going around any loop twice is enough.
7. This means that there is a continuous gauge-field configuration in the GUT group allows the U(1) monopole configuration to unwind itself at short distances, at the cost of not staying in the U(1). In order to do this with as little energy as possible, you should only leave the U(1) in the neighborhood of one point, which is called the core of the monopole. Outside the core, the monopole has only magnetic field energy.

So the Dirac monopole is a topological defect in a compact U(1) gauge theory. When there is no GUT, the defect is a singularity--- the core shrinks to a point. But when there is some sort of short distance regulator on space time, the monopoles have a finite mass. Monopoles occur in lattice U(1), and there the core size is the lattice size. In general, they are expected to occur whenever there is a short-distance regulator.

String theory

In our universe, quantum gravity provides the regulator. When gravity is included, the monopole singularity can be a black hole, and for large magnetic charge and mass, the black hole mass is equal to the black hole charge, so that the mass of the magnetic black hole is not infinite. If the black hole can decay completely by Hawking radiation, the lightest charged particles can't be too heavy. The lightest monopole should have a mass less than or comparable to its charge in natural units.

So in a consistent holographic theory, and string theory is the only known example, there are always finite mass monopoles. For ordinary electromagnetism, the mass bound is not very useful because it is about the Planck mass.

Mathematical formulation

In mathematics, a gauge field is defined as a connection over a principal G-bundle over spacetime. G is the gauge group, and it acts on each fiber of the bundle separately.

A connection on a G bundle tells you how to glue F's together at nearby points of M. It starts with a continuous symmetry group G which acts on F, and then it associates a group element with each infinitesimal path. Group multiplication along any path tells you how to move from one point on the bundle to another, by acting the G element of a path on the fiber F.

In mathematics, the definition of bundle is designed to emphasize topology, so the notion of connection is added on as an afterthought. In physics, the connection is the fundamental physical object. Once you have a connection, there are nontrivial bundles which occur as connections of a trivial bundle. For example, the twisted torus is a connection on a U(1) bundle of a circle on a circle.

If space time has no topology, if it is R⁴ the space of all possible connections of the 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.

A principal G-bundle over S² is defined by 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. The transition function maps the strip to G, and the different ways of mapping a strip into G is given by the first homotopy group of G.

So in the G-bundle formulation, a gauge theory admits Dirac monopoles provided G is not simply connected, whenever there are paths that go around the group that cannot be deformed to nothing. 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. The mathematical definition is equivalent to the physics definition provided that, following Dirac, gauge fields are allowed which are only defined patch-wise and the gauge field on different patches are glued after a gauge transformation.

This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It generalizes to d + 1 dimensions with in several ways. One way is to extend everything into the extra dimensions, so that U(1) monopoles become sheets of dimension d-3. Another way is to examine the type of topological singularity at a point with 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 to 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. Other arguments based on the critical density of the universe indicate that monopoles should be fairly common; the apparent problem of the observed scarcity of monopoles is resolved by cosmic inflation in the early universe, which greatly reduces the expected abundance of magnetic monopoles. For these reasons, monopoles became a major interest in the 1970s and 80s, along with the other "approachable" prediction 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.

Monopole searches

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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 device", 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.^[7] The lack of such events places a limit on the number of monopoles of about 1 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 of Price.^[6] Price later retracted his claim, and a possible alternative explanation was offered by Alvarez.^[17] In his paper it was demonstrated that the path of the cosmic ray event that was claimed to be due to a magnetic monopole could be reproduced by a path followed by a platinum nucleus fragmenting to osmium and then to tantalum.

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.

"Monopoles" in condensed-matter systems

While a magnetic monopole particle has never been conclusively observed, there are a number of phenomena in condensed-matter physics where a material, due to the collective behavior of its electrons and ions, can show emergent phenomena that resemble magnetic monopoles in some respect^{[18][19][20][21]}. These should not be confused with actual monopole particles; in particular, the divergence of the microscopic magnetic B-field is zero everywhere in these systems, unlike in the presence of a true magnetic monopole particle. The behavior of these quasiparticles would only become indistinguishable from true magnetic monopoles — and they would truly deserve the name — if the so-called magnetic fluxtubes connecting these would-be monopoles became unobservable which also means that these flux tubes would have to be infinitely thin, obey the Dirac quantization rule, and deserve to be called Dirac strings.

In a paper published in Science in September 2009, researchers Jonathan Morris and Alan Tennant from the Helmholtz-Zentrum Berlin für Materialien und Energie (HZB) along with Santiago Grigera from Instituto de Física de Líquidos y Sistemas Biológicos (IFLYSIB, CONICET) and other colleagues from Dresden University of Technology, University of St. Andrews and Oxford University described the observation of quasiparticles resembling monopoles. A single crystal of dysprosium titanate in a highly frustrated pyrochlore lattice (F d -3 m) was cooled to 0.6 to 2 K. Using neutron scattering, the magnetic moments were shown to align in the spin ice into interwoven tube-like 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 is also described.^[22][23]

See also

• Dirac string
• Dyon
• Felix Ehrenhaft
• Gauss's law for magnetism
• Halbach array
• Instanton
• Meron
• Soliton
• 't Hooft-Polyakov monopole
• Wu-Yang monopole

Notes

References

• Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University Press. ISBN 0-19-514665-4. 
• Jackson, John David (1999). Classical Electrodynamics (3rd ed.). New York: Wiley. ISBN 0-471-30932-X. 
• Milton, Kimball A. (June 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. http://arxiv.org/abs/hep-ex/0602040. 
• 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.