Equations of Motion as Constraints: Superselection Rules, Ward Identities

TL;DR

This paper explores how equations of motion act as constraints in gauge theories, revealing superselection sectors and the role of the BMS group in infrared effects, charge conservation, and low energy theorems.

Contribution

It demonstrates that equations of motion can be viewed as constraints generating gauge transformation groups, extending to infrared sectors and revealing superselection structures.

Findings

Equations of motion serve as constraints generating gauge groups.

Infrared effects are captured by extended operators not vanishing at infinity.

Superselection sectors are distinguished by the quotient of gauge groups.

Abstract

The meaning of local observables is poorly understood in gauge theories, not to speak of quantum gravity. As a step towards a better understanding we study asymptotic (infrared) transformation in local quantum physics. Our observables are smeared by test functions, at first vanishing at infinity. In this context we show that the equations of motion can be seen as constraints, which generate a group, the group of space and time dependent gauge transformations.This is one of the main points of the paper. Infrared nontrivial effects are captured allowing test functions which do not vanish at infinity. These extended operators generate a larger group. The quotient of the two groups generate superselection sectors, which differentiate different infrared sectors. The BMS group changes the superselection sector, a result long known for its Lorentz subgroup. It is hence spontaneously broken. Ward identities implied by the gauge invariance of the S-matrix generalize the standard results and lead to charge conservation and low energy theorems. Their validity does not require Lorentz invariance.