4 - From Dyson's Formula to Feynman Rules

Summary

On the basis of canonical quantization of free fields, by means of the interaction picture which expresses the evolution of interacting fields in terms of the free field, the scattering matrix is first expressed in Dyson's formula as a perturbation series in a free-field Hamiltonian, then converted to Wick's operator expansion, and finally reduced to matrix elements and the Feynman perturbation series.

Introduction

Now that, through symmetries and conserved quantities, we have some constraints on the possibilities for the evolution of a state, it is time to develop the machinery for computing the structure of an evolving state. At the end of Chapter 2, we observed that canonical quantization of an interacting field theory could not be carried out on account of the normal-ordering problem. This chapter describes Dyson's technique for viewing an interacting field in terms of a free field, thereby quantizing an interacting theory on the basis of the canonical quantization of a free theory. This technique is intrinsically perturbative, being in effect equivalent to the asymptotic series defined by Feynman diagrams and rules, and could be called perturbative canonical quantization.

Section 4.1 presents the fundamental principle of perturbative canonical quantization, namely the interaction picture, in which the states evolve in a modified Schrödinger fashion, and the operators evolve like Heisenberg operators in a free-field theory.