S-matrix elements for gauge theories with and without implemented constraints

TL;DR

This paper derives a general relation between different scattering amplitudes in gauge theories, including non-perturbative states that implement constraints like Gauss's law, extending previous work to non-unitary transformations.

Contribution

It generalizes the relation between scattering amplitudes to non-unitary transformations and non-perturbative states in gauge theories, including those with constraints.

Findings

Derived a relation between two types of scattering amplitudes.

Extended previous unitary transformation results to non-unitary cases.

Discussed implications for Abelian and non-Abelian gauge theories.

Abstract

We derive an expression for the relation between two scattering transition amplitudes which reflect the same dynamics, but which differ in the description of their initial and final state vectors. In one version, the incident and scattered states are elements of a perturbative Fock space, and solve the eigenvalue problem for the `free' part of the Hamiltonian --- the part that remains after the interactions between particle excitations have been `switched off'. Alternatively, the incident and scattered states may be coherent states that are transforms of these Fock states. In earlier work, we reported on the scattering amplitudes for QED, in which a unitary transformation relates perturbative and non-perturbative sets of incident and scattered states. In this work, we generalize this earlier result to the case of transformations that are not necessarily unitary and that may not have unique inverses. We discuss the implication of this relationship for Abelian and non-Abelian gauge theories in which the `transformed', non-perturbative states implement constraints, such as Gauss's law.