1 - Scalar Green functions and their perturbative solutions

Summary

In this book we shall be almost exclusively concerned with the interactions of relativistic particles that are the quanta of elementary fields. There is some ambiguity in the definition of ‘elementary’, but by it we mean local fields whose propagation and interactions can be described by a local Hamiltonian, or Lagrangian, density. Individual terms in these densities describe the basic transformations that the quanta can undergo. For example, if the classical Lagrangian density for a field A has a quartic gA⁴ interaction we assume that, quantum mechanically, one A-particle can turn directly into three (virtual) A-particles. The way in which these virtual particles further split or recombine determines the way in which A-particle interactions take place.

The aim of this first chapter is to indicate how canonical quantisation (i.e. the Hamiltonian formulation) can be reformulated as statements about how particles interact. The quantification of the qualitative statement that one A-particle can turn into three, or whatever, will occur through a set of relations termed the Dyson–Schwinger equations. In our approach these equations will play a critical role in formulating an alternative quantisation of field theory through path integrals. The path integral formulation, rather than the canonical approach, will be at the centre of all our calculational methods.