10 - Asymptotic fields and the LSZ formalism

Summary

In this chapter, the Lehmann–Symanzik–Zimmermann (LSZ) formalism is presented. This was developed in 1955 and is named after its inventors, namely, the three German physicists H. Lehmann, K. Symanzik, and W. Zimmermann. Especially, the LSZ formalism includes the LSZ reduction formula that provides an explicit way of expressing physical S matrix elements (i.e. scattering amplitudes) in terms of T-ordered correlation functions of an interacting field within a quantum field theory to all orders in perturbation theory. Thus, the goal is to express the S matrix elements in terms of asymptotic free fields instead of the unknown interacting field. Therefore, given the Lagrangian of some quantum field theory, it leads to predictions of measurable quantities. Note that the LSZ reduction formula cannot treat massless particles, bound states, and so-called topological defects. For example, in quantum chromodynamics (QCD), the asymptotic states are bound, which means that they are not free states. However, the original formula can be generalized in order to include bound states, which are states described by non-local composite fields. In addition, in statistical physics, a general formulation of the fluctuation–dissipation theorem, which is a theorem that can be used to predict the non-equilibrium behaviour of a system, can be obtained using the LSZ formalism.