Dirac string
Main article: 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 + iAµdxµ 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 2p / 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 2p / 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.
[edit] Grand unified theories
In a U(1) with quantized charge, the gauge group is a circle of radius 2p / 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:
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.
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.
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.
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.
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 2pN / e. This is the Dirac quantization condition, and it is a topological condition which demands that the long distance U(1) gauge field configurations be consistent.
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.
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.
[edit] 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, of which 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.
[edit] 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 pd-2(G).
[edit] 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.
[edit] Monopole searches
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