4 Spontaneous symmetry breaking

In this chapter, we are going to look at spontaneous symmetry breaking. The

general setting is as follows — our Lagrangian

L

enjoys some symmetry. However,

the potential has multiple minima. Usually, in perturbation theory, we imagine

we are sitting near an equilibrium point of the system (the “vacuum”), and then

look at what happens to small perturbations near the equilibrium.

When there are multiple minima, we have to arbitrarily pick a minimum

to be our vacuum, and then do perturbation around it. In many cases, this

choice of minimum is not invariant under our symmetries. Thus, even though

the theory itself is symmetric, the symmetry is lost once we pick a vacuum. It