The field theory action is constructed to be invariant under Lorentz transformations and spacetime translations. These don't mix different fields. Now we consider internal symmetries, corresponding to transformations which mix different fields but don't involve spacetime.

Internal symmetries come in two kinds, called global symmetry and gauge symmetry. A gauge symmetry must be exact, but a global symmetry may only be approximate.

What is the motivation for considering a symmetry? One might first choose an action, guided by observation and/or aesthetics, and then look to see what is its symmetry group. That was the case when Quantum Electrodynamics was first formulated. Nowadays, the more usual procedure is to first choose a symmetry group (guided again by observation and/or aesthetics) and then to write down the most general action consistent with the symmetry group, which turns out to be quite strongly constrained. Often, the action arrived at in this way turns out to have additional global symmetries which were not imposed from the start, called accidental symmetries.

In this chapter we look at some examples of internal symmetry, as it applies to scalar fields at the classical level. Although we won't generally write down formulas involving spin-1/2 fields, it will be important to remember their existence.

Symmetry groups

To say that the action is invariant under some transformation is to say that it is the same before and after the transformation. It follows that the action is invariant also under the inverse of the transformation.