27 - Interacting discrete time scalar fields

Summary

In this chapter we explore the discrete time (DT) quantum field theory (QFT) of a system under observation described via a self-interacting neutral scalar field, as an illustration of the sort of techniques and results we encounter in such a theory.

We have commented several times in other chapters on the importance of recognizing the architecture of an experiment or theory describing that experiment. The same is true here. Even in continuous time (CT) QFT, it was found necessary to modify the architecture used in scattering calculations. It was found to be no longer sufficient to keep to a single Hilbert-space architecture as in Figure 5.1(a), so the architecture had to be extended to incorporate the observers in some way. The usual abbreviation for the enhancement of this architecture is LSZ, after Lehmann, Symanzik and Zimmermann, who added an ‘in’ free-particle Hilbert space to model the processes of state preparation and an ‘out’ free-particle Hilbert space to model the process of state detection (Lehmann et al., 1955).

An important factor in the LSZ formalism is the recognition that the ultimate objectives in QFT are scattering amplitudes, rather than solutions to operator equations of motion. The former are objects from which physical quantities can be calculated, whilst the latter are ill-defined at best and would have to be processed further anyway by the taking of matrix elements in order to be of any use.