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Equations of Motion as Constraints: Superselection Rules, Ward Identities

M. Asorey, A. P. Balachandran, F. Lizzi, G. Marmo

TL;DR

The paper addresses the challenge of defining local observables in gauge theories by formulating Maxwell equations as covariant, spacetime-dependent Gauss-law constraints and examining infrared (IR) transformations via smeared observables. It distinguishes two gauge-like groups, G0∞ and G0, whose quotient labels superselection sectors; infrared dressing with non-vanishing-at-infinity test functions leads to a Sky-group structure Maps(S^2,U(1)) that labels IR sectors. The BMS group is shown to act on this superselection algebra and is spontaneously broken (to rotations), while Ward identities derived from gauge invariance yield charge conservation and soft-photon low-energy theorems independent of Lorentz invariance. These results clarify the IR structure of QED and provide insights into asymptotic symmetries with potential implications for quantum gravity.

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.

Equations of Motion as Constraints: Superselection Rules, Ward Identities

TL;DR

The paper addresses the challenge of defining local observables in gauge theories by formulating Maxwell equations as covariant, spacetime-dependent Gauss-law constraints and examining infrared (IR) transformations via smeared observables. It distinguishes two gauge-like groups, G0∞ and G0, whose quotient labels superselection sectors; infrared dressing with non-vanishing-at-infinity test functions leads to a Sky-group structure Maps(S^2,U(1)) that labels IR sectors. The BMS group is shown to act on this superselection algebra and is spontaneously broken (to rotations), while Ward identities derived from gauge invariance yield charge conservation and soft-photon low-energy theorems independent of Lorentz invariance. These results clarify the IR structure of QED and provide insights into asymptotic symmetries with potential implications for quantum gravity.

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.

Paper Structure

This paper contains 10 sections, 91 equations.

Table of Contents

  1. Introduction
  2. Equations of motion as the Covariant Gauss law
  3. Smearing
  4. Vacua and Coherent States
  5. The Superselection Algebra
  6. Gauge Invariance and Ward identities
  7. The superselection operator $Q(\zeta)$ and charge conservation
  8. Charge conservation and low energy theorems
  9. The BMS Group
  10. Conclusions