Construction of Amplitudes with Massless Particles and Gauge Invariance inS-Matrix Theory

Abstract

The fundamental statement of relativistic invariance for scattering amplitudes is that the amplitude remains invariant when the momentum and spin variables of each particle are transformed according to the corresponding irreducible, unitary representation of the inhomogeneous Lorentz group. To "construct an amplitude" is to find the most general function that has the required transformation properties. This construction, which had been previously effected for any number of massive particles of arbitrary spin, is extended here to include massless particles of arbitrary spin as well. In the case of photons, the resulting formalism is compared with the usual one that makes use of transverse polarization vectors and a gauge-invariance condition. The two formalisms are proven to be equivalent. It is concluded that the gauge condition is superfluous as an independent physical principle for the purpose of constructing amplitudes. Its use in the conventional formalism is simply a way of imposing the Lorentz-transformation properties appropriate to massless particles. In an Appendix, the known analogous construction for massive spin-one particles is shown to be equivalent to the usual formalism, and the requirement of Lorentz invariance is shown to be equivalent to the usual prescription for virtual photons as well.