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, generalisation thereof in the non-Abelian 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 generalised) 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.