Dispersion Relations in RQFT

Abstract

This chapter is on Dispersion Relations (DR). It should be a source for many important aspects of DR. After highlighting DR in strong interactions, an exposition of J.S. Toll’s works on the logical foundations of DR, and his use of Causal Transforms is explained. He also introduces the notion of logical equivalences between analyticity, strict causality, certain integrability criteria, etc. The chapter then looks at certain general features of DR in QFT, focussing on Goldberger’s works and their relation to the LSZ formalism. After this, the DR for forward scattering of massless and massive particles is taken up. Goldberger’s and Symanzik’s treatments are separately discussed. Some aspects of Symanzik’s analysis like extension of DR to derivatives of scattering amplitudes are also pointed out. The chapter then turns to the more difficult DR for non-forward scattering. It first discusses a proposal by Salam wherein Goldberger’s forward scattering analysis is attempted to be generalized to the non-forward case. Many lacunae in this work are pointed out. A thorough discussion of fixed-t DR is taken up along the lines of Lehmann and Gustav Sommer. The Lehmann ellipses are made use of repeatedly. The chapter concludes with a discussion of the Mandelstam Double Spectral Representation.