11 - Functional Integral Quantization

Summary

Providing a classical functional calculus for investigating interacting quantum fields from the level of the generating functional, proving the equivalence of this new quantization to perturbative canonical quantization, developing an efficient procedure for obtaining Feynman rules directly from a Lagrangian density, and preparing for functional quantization of gauge theories.

Introduction

So far, we have used the perturbative canonical quantization of Chapter 4 as the method for transforming classical field theories into quantum field theories. By the end of Chapter 9, this program had encountered three problems. First, the nature of the adiabatic turning on and off function makes it theoretically unsound to apply that method to a theory with unstable particles or bound states or gauge invariance. Second, derivative interactions in massive scalar QED gave rise to non-covariant interactions and Feynman rules. Third, we cannot readily see that the gauge invariance of QED is preserved by this method of quantization. These troubles are greatly aggravated when we attempt to apply perturbative canonical quantization to non-abelian gauge theories. Clearly, a new and more powerful approach to quantization is needed.

Chapter 10 provided the first step, the theory of the generating functional. Taking the generating functional as the source of the perturbation expansion, we were able to eliminate the adiabatic turning on and off function, propose convenient renormalization conditions, make sense of divergent Feynman integrals, and extend perturbation theory to include fields with unstable particles.