<P> GUTs lead to compact U (1) gauge groups, so they explain charge quantization in a way that seems logically independent from magnetic monopoles . However, the explanation is essentially the same, because in any GUT that breaks down into a U (1) gauge group at long distances, there are magnetic monopoles . </P> <P> The argument is topological: </P> <Ol> <Li> 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 . </Li> <Li> If you imagine a big sphere in space, you can deform an infinitesimal loop that 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 . </Li> <Li> 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 . </Li> <Li> 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 . </Li> <Li> 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 that demands that the long distance U (1) gauge field configurations be consistent . </Li> <Li> When the U (1) gauge group comes from breaking a compact Lie group, the path that winds around the U (1) group 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 P, three times to P, 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 . </Li> <Li> 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). To do this with as little energy as possible, you should leave only the U (1) gauge group in the neighborhood of one point, which is called the core of the monopole . Outside the core, the monopole has only magnetic field energy . </Li> </Ol> <Li> 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 . </Li>

When was the first magnetic monopole observed and by whom