Simplified derivation of the crossing relations using the substitution rule as a guide

Abstract

A simple derivation of the Trueman–Wick crossing relations for center of mass helicity amplitudes is presented. The method uses only Wigner rotation matrices and does not deal specifically with more general Lorentz transformations. Each particle is treated individually in turn in its own rest frame in order to discover how its helicity transforms under crossing and the substitution rule is used as a guide in determining the outcome of the analytic continuation. Crossing relations for different paths of analytic continuation in the s–t plane are discussed and it is shown that care must be exercised when dealing with weak interaction amplitudes. The method presented may be extended straightforwardly to multiparticle amplitudes.