ABSTRACT

The theory of electromagnetism exhibits a gauge invariance, or more elaborately, a local gauge invariance. This can be maintained even if interactions are introduced. In fact, quantum electrodynamics, which is the first quantum field theory to have been fully developed, involves fermions coupled to the electromagnetic field in a gauge invariant way. The quantization procedure is complicated because of the gauge invariance and needs gauge fixing, as we have seen. The gauge field is massless and has only two transverse degrees of freedom as in the free theory. However, gauge interactions can drastically alter the theory. If a scalar field is coupled to the gauge field in a gauge invariant way, normally the situation will be similar, but special forms of the potential may cause the global gauge invariance to be broken. This happens when the scalar field acquires a nonvanishing vacuum expectation value in contrast to the normal situation where the vacuum expectation value of a field vanishes. A nonvanishing vacuum expectation value corresponds to one of an infinite set which is invariant under global gauge transformations. The choice of a single element of the set breaks the invariance. This kind of symmetry breaking is called spontaneous symmetry breaking. This is similar to what happens in ferromagnetism. The elementary magnets are randomly oriented and result in zero total magnetic moment, but on cooling, the elementary magnets become aligned in some direction, producing a total magnetic moment. The selection of a direction breaks the rotational symmetry.