Where does this description of matter and radiation need to go next? The answer is that it needs to include interactions between different physical fields and between different excitations in the same field.

In order to pursue this course, one needs to extend the simple linear response analysis of classical field theory to a non-linear response analysis. In the presence of interactions, linear response is only a first-order approximation to the response of the field to a source. Interactions turn the fields themselves into sources (sources for the fields themselves and others to scatter from). Non-linear response theory requires quantum field theory, because the products of fields, which arise in interactions, bring up the issue of the ordering of fields, which only the second quantization can resolve. It means the possibility of creation and annihilation of particle pairs and negative probabilities, which the anti-particle concept and the vacuum concept repair the consistency.

An area which has not been touched upon in this book is that of Grassman variables [136], which describe fermionic interactions. These arose fleetingly in connection with TCP invariance (see section 10.5). In interacting theories, one needs to account for their anti-commuting properties.

A full discussion of quantum field theory, and all of its computational algorithms, is far beyond the scope of this book. The purpose of this chapter is only to indicate briefly how the quantum theory of fields leads to more of the same. As usual, the most pleasing way to derive corrections to the classical theory within a dynamically complete framework is, of course, through an action principle.