12 - Quantization of Gauge Theories

Summary

Extending the technique of functional integral quantization to gauge field theories through the Faddeev—Popov principle; quantization of QED and preparation for quantization of the Standard Model of the electroweak and strong interactions.

Introduction

The previous chapter explained functional integral quantization of scalar, spinor, and massive vector fields and demonstrated that the Feynman rule for a derivative interactions proposed in Section 10.8 was correct. In this chapter, we extend functional integral quantization to gauge field theories. There are two elements in this extension. The first element is the Faddeev—Popov principle of integrating over orbits of the gauge group. This technique proposes a functional integral formula for the generating functional which is based on the Lagrangian form of the action. As a consequence, the Feynman rules are covariant and gauge invariance of the quantum theory is obvious. The second element is the use of Lagrangians which are first order in derivatives to prove equivalence between Faddeev—Popov quantization and canonical quantization in axial gauge. After this proof, we cheerfully accept the Faddeev—Popov principle for quantization of gauge theories.

Section 11.10 established the Lagrangian form of the generating functional for the free vector field. In Sections 12.1 and 12.2, we explain the use of first-order Lagrangians in the context of spinor and scalar QED with a massive photon.