Abstract

This chapter corresponds to lectures given at the Villa de Leyva Summer School in Colombia (July 2007). The main purpose in this short treatment of BRS invariance of gauge theories is to illuminate corners of the theory left in the shade by standard treatments. The plan is as follows. First, a review is given of Utiyama's general gauge theory. Immediately we find a counterexample to it in the shape of the massive spin-1 Stückelberg gauge field. This is not fancy, as the massive case is the most natural one to introduce BRS invariance in the context of free quantum fields. Mathematically speaking, the first part of the chapter uses Utiyama's notation, and thus has the flavour and nonintrinsic notation of standard physics textbooks. Next we deal with boson fields on Fock space and BRS invariance in connection with the existence of Krein operators; the attending rigour points are then addressed.

Utiyama's method in classical gauge theory

A historical note

Ryoyu Utiyama developed non-abelian gauge theory early in 1954 in Japan, almost at the same time that Yang and Mills [1] did at the Institute for Advanced Study (IAS) in Princeton, which Utiyama was to visit later in the year. Unfortunately, Utiyama chose not to publish immediately, and upon his arrival at IAS in September of that year, he was greatly discouraged to find he had apparently just been ‘scooped’.