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Null Hamiltonian Yang-Mills theory: Soft symmetries and memory as superselection

Aldo Riello, Michele Schiavina

Abstract

Soft symmetries for Yang-Mills theory are shown to correspond to the residual Hamiltonian action of the gauge group on the Ashtekar-Streubel phase space, which is the result of a partial symplectic reduction. The associated momentum map is the electromagnetic memory in the abelian theory, or a nonlinear, gauge-equivariant, generalization thereof in the nonabelian case. This result follows from an application of Hamiltonian reduction by stages, enabled by the existence of a natural normal subgroup of the gauge group on a null codimension-1 submanifold with boundaries. The first stage is coisotropic reduction of the Gauss constraint, and it yields a symplectic extension of the Ashtekar-Streubel phase space (up to a covering). Hamiltonian reduction of the residual gauge action leads to the fully-reduced phase space of the theory. This is a Poisson manifold, whose symplectic leaves, called superselection sectors, are labelled by the (gauge classes of the generalized) electric flux across the boundary. In this framework, the Ashtekar-Streubel phase space arises as an intermediate reduction stage that enforces the superselection of the electric flux at only one of the two boundary components. These results provide a natural, purely Hamiltonian, explanation of the existence of soft symmetries as a byproduct of partial symplectic reduction, as well as a motivation for the expected decomposition of the quantum Hilbert space of states into irreducible representations labelled by the Casimirs of the Poisson structure on the reduced phase space.

Null Hamiltonian Yang-Mills theory: Soft symmetries and memory as superselection

Abstract

Soft symmetries for Yang-Mills theory are shown to correspond to the residual Hamiltonian action of the gauge group on the Ashtekar-Streubel phase space, which is the result of a partial symplectic reduction. The associated momentum map is the electromagnetic memory in the abelian theory, or a nonlinear, gauge-equivariant, generalization thereof in the nonabelian case. This result follows from an application of Hamiltonian reduction by stages, enabled by the existence of a natural normal subgroup of the gauge group on a null codimension-1 submanifold with boundaries. The first stage is coisotropic reduction of the Gauss constraint, and it yields a symplectic extension of the Ashtekar-Streubel phase space (up to a covering). Hamiltonian reduction of the residual gauge action leads to the fully-reduced phase space of the theory. This is a Poisson manifold, whose symplectic leaves, called superselection sectors, are labelled by the (gauge classes of the generalized) electric flux across the boundary. In this framework, the Ashtekar-Streubel phase space arises as an intermediate reduction stage that enforces the superselection of the electric flux at only one of the two boundary components. These results provide a natural, purely Hamiltonian, explanation of the existence of soft symmetries as a byproduct of partial symplectic reduction, as well as a motivation for the expected decomposition of the quantum Hilbert space of states into irreducible representations labelled by the Casimirs of the Poisson structure on the reduced phase space.
Paper Structure (43 sections, 39 theorems, 274 equations)

This paper contains 43 sections, 39 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. A note on boundary gauge transformations
  4. Specifics
  5. A note on quantisation
  6. Structure of the paper
  7. Theoretical framework
  8. Locally Hamiltonian gauge theory
  9. Constraint and flux forms
  10. Symplectic reduction: overview
  11. Constraint reduction
  12. Flux superselection
  13. Yang--Mills theory on null boundaries: geometric setup
  14. The null manifold
  15. The geometric phase space
  16. ...and 28 more sections

Key Result

Lemma 2.8

Let $\mathcal{W}$ be as above, and $\mathcal{X}\subset \mathcal{W}$ be a closed vector subspace, then

Theorems & Definitions (126)

  • Remark 2.1: Null $\Sigma$
  • Definition 2.2: Equivariant locally Hamiltonian gauge theory RielloSchiavina
  • Remark 2.3: Duals
  • Definition 2.4: Constraint and flux forms
  • Remark 2.5: Noether current, charges, and the constraints
  • Definition 2.6: Reduced phase space
  • Definition 2.7: Annihilators
  • Lemma 2.8
  • proof
  • Definition 2.9
  • ...and 116 more