So far our approximations have been wholly analytic, essentially based on perturbation series. The calculation of (g − 2) for the electron has shown just how successful this approach can be.

However, in many ways (g − 2) is an exception. Most quantities that we would like to calculate (e.g. the pion mass in quantum chromodynamics, the field theory of the strong interactions) cannot be approached analytically. Although analytic methods do give us a lot of information (that is, after all, what this book is about) it is often of a qualitative nature. For this reason a variety of numerical methods have been developed for approximating the path integral directly. Bypassing problems of measure, they typically involve putting the theory on a Euclidean lattice, as in the case of the regularised ZΛ(j) of (6.74). The resulting integrals can then be calculated directly, although a sizeable lattice requires considerable computing resources. All the uncertainties of the method are displaced into the problem of recovering the continuum limit.

It is often difficult to link such tactics to the analytic ideas of continuum field theory. Before turning to more realistic theories we shall give a brief description of an alternative interpretation of formal Euclidean path integrals, proposed by Parisi and Wu (1981).