Correlation Structure

In this chapter we will perform a careful and detailed analysis of the two-point correlation function of the Gaussian model. In this way we will be using the calculational techniques described in the previous chapters in order to probe into the fundamental structure of the theory, regarding its correlation functions. This is where we depart from the purely traditional approach to the subject, because we shall see that the results obtained are not part of that traditional approach, and lead to the necessity of a fundamental reinterpretation of the theory. In fact, some aspects of the behavior of the two-point function are quite surprising at first sight.

We will also examine in detail the issue of the mathematical nature of the field configurations that contribute in a dominant way to the expectation values of the theory. The conclusion that we will be led to, that those configurations are not only non-differentiable functions, but that they are in fact discontinuous functions on all the points of their domains, will put the definition of the quantum theory of fields in terms of functional integrals defined on the lattice in sharp contrast to the usual path-integral approach to quantum mechanics. This important fact will have important future consequences for the mathematical treatment of the theory.

We will also introduce and examine in detail the concept of block variables, which will be instrumental in solving the conceptual problems posed by the singular structure of the two-point function. In fact, these variables will turn out to be of central importance for the physical interpretation of the theory. We will see that these variables are closely related to the Fourier components of the field in momentum space, and hence that these Fourier components are quantities more closely related to the observational aspects of the theory, and better instruments than their counterparts in position space for looking into the physical content of the theory.