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.