Chapter 45

U(1) Gauge Symmetry and

Electromagnetic Fields

We have now constructed both relativistic and non-relativistic quantum field
theories for free scalar particles. In the non-relativistic case we had to use
complex-valued fields, and found that the theory came with an action of aU(1)
group, the group of phase transformations on the fields. In the relativistic case
real-valued fields could be used, but if we took complex-valued ones (or used
pairs of real-valued fields), again there was an action of aU(1) group of phase
transformations. This is the simplest example of a so-called “internal symmetry”
and it is reflected in the existence of an operatorQ̂called the “charge”.
In this chapter we’ll see how to go beyond the theory of free quantized
charged particles, by introducing background electromagnetic fields that the
charged particles will interact with. It turns out that this can be done using
theU(1) group action, but now acting independently at each point in space-
time, giving a large, infinite dimensional group called the “gauge group”. This
requires introducing a new sort of space-time dependent field, called a “vector
potential” by physicists, a “connection” by mathematicians. Use of this field
allows the construction of a Hamiltonian dynamics invariant under the gauge
group. This fixes the way charged particles interact with electromagnetic fields,
which are described by the vector potential.
Most of our discussion will be for the case of theU(1) group, but we will also
indicate how this generalizes to the case of non-Abelian groups such asSU(2).

45.1U(1) gauge symmetry

In sections 38.2.1 and 44.1 we saw that the existence of aU(1) group action by
overall phase transformations on the complex field values led to the existence of
an operatorQ̂, which commuted with the Hamiltonian and acted with integral
eigenvalues on the space of states. Instead of acting on fields by multiplication by
a constant phaseeiφ, one can imagine multiplying by a phase that varies with the