Unitarity and the LSZ Formalism

Abstract

In this chapter, we discuss how the LSZ formalism naturally incorporates unitarity, a central physical requirement for the S-matrix, as also emphasized by Heisenberg when he introduced his S-matrix for the first time. The importance of this for the analytic S-matrix and Dispersion Relations(DR) is also commented upon. The chapter begins by introducing the well-known Mandelstam Variables and their properties. It then identifies the physical and unphysical regions for various channels. It then shows the equivalence between the T-matrix as follows from the LSZ formalism and a form introduced by Lehmann in terms of the so-called $R^\prime$-product. The unitarity condition is stated explicitly in terms of the T-matrix. The proof of unitarity in the LSZ formalism is then worked out in detail. This includes the demonstration that the LSZ formalism yields unitarity in all the channels.