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Asymptotic symmetries of electromagnetism at spatial infinity

Marc Henneaux, Cédric Troessaert

TL;DR

The paper develops a Hamiltonian formulation of Maxwell theory at spatial infinity with precise boundary conditions that admit an infinite-dimensional angle-dependent u(1) symmetry and nonzero charges. A key innovation is the introduction of a boundary surface degree of freedom Ψ and a modified symplectic form, which together render Lorentz boosts canonical and yield a rich symmetry algebra that semi-directly extends the Poincaré group. The framework accommodates magnetic monopoles through twisted parity conditions on angular data and yields a consistent set of Poincaré generators with explicit boundary terms. It further connects spatial infinity symmetries to those at null infinity and outlines potential generalizations to gravity, Yang–Mills, and supergravity. The results illuminate the infrared structure of electromagnetism and the role of boundary data in realizing nontrivial asymptotic symmetries.

Abstract

We analyse the asymptotic symmetries of Maxwell theory at spatial infinity through the Hamiltonian formalism. Precise, consistent boundary conditions are explicitly given and shown to be invariant under asymptotic angle-dependent $u(1)$-gauge transformations. These symmetries generically have non-vanishing charges. The algebra of the canonical generators of this infinite-dimensional symmetry with the Poincaré charges is computed. The treatment requires the addition of surface degrees of freedom at infinity and a modification of the standard symplectic form by surface terms. We extend the general formulation of well-defined generators and Hamiltonian vector fields to encompass such boundary modifications of the symplectic structure. Our study covers magnetic monopoles.

Asymptotic symmetries of electromagnetism at spatial infinity

TL;DR

The paper develops a Hamiltonian formulation of Maxwell theory at spatial infinity with precise boundary conditions that admit an infinite-dimensional angle-dependent u(1) symmetry and nonzero charges. A key innovation is the introduction of a boundary surface degree of freedom Ψ and a modified symplectic form, which together render Lorentz boosts canonical and yield a rich symmetry algebra that semi-directly extends the Poincaré group. The framework accommodates magnetic monopoles through twisted parity conditions on angular data and yields a consistent set of Poincaré generators with explicit boundary terms. It further connects spatial infinity symmetries to those at null infinity and outlines potential generalizations to gravity, Yang–Mills, and supergravity. The results illuminate the infrared structure of electromagnetism and the role of boundary data in realizing nontrivial asymptotic symmetries.

Abstract

We analyse the asymptotic symmetries of Maxwell theory at spatial infinity through the Hamiltonian formalism. Precise, consistent boundary conditions are explicitly given and shown to be invariant under asymptotic angle-dependent -gauge transformations. These symmetries generically have non-vanishing charges. The algebra of the canonical generators of this infinite-dimensional symmetry with the Poincaré charges is computed. The treatment requires the addition of surface degrees of freedom at infinity and a modification of the standard symplectic form by surface terms. We extend the general formulation of well-defined generators and Hamiltonian vector fields to encompass such boundary modifications of the symplectic structure. Our study covers magnetic monopoles.

Paper Structure

This paper contains 22 sections, 2 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Some background
  3. Action and gauge symmetries
  4. Poincaré transformations
  5. Liénard-Wiechert solution
  6. Magnetic monopoles
  7. How to make the symplectic form finite
  8. Condition on the radial components
  9. Conditions on the angular components
  10. Asymptotic analysis: I. No magnetic charge
  11. Generalization of the boundary conditions
  12. Lorentz boosts: need to add a boundary term to the symplectic structure
  13. Complete Action
  14. New global symmetry - Alternative form of the action
  15. Poincaré charges
  16. ...and 7 more sections

Key Result

Theorem A.1

The Poisson bracket of two allowed functionals $F_1$ and $F_2$ is an allowed functional with

Theorems & Definitions (2)

  • Theorem A.1
  • Theorem A.2