Non-Abelian Gauge Theories
Quantum electrodynamics is an example of a U(1) gauge theory. U(1) is the group of the unimodular complex numbers and determines the transformation of the charged fields
(x) → exp ( − iq (x))(x) ≡ g(x)(x). (18.1) It forms a group, which means that for any two elements g, h ∈ U(1), the product is also in U(1). Furthermore, any element has an inverse g−1, which satisfies gg−1 = g−1g = 1. The unit 1 satisfies g1 = 1g = g, for any g ∈ U(1). U(1) is called an Abelian group because its product is commutative. For every g, h ∈ U(1), gh = hg.