A note on electric-magnetic duality and soft charges
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Springer
Received: April 21, 2020 Accepted: May 25, 2020 Published: June 12, 2020
Marc Henneauxa,b and C´ edric Troessaertc a
Universit´e Libre de Bruxelles and International Solvay Institutes, Physique Math´ematique des Interactions Fondamentales, Campus Plaine — CP 231, Bruxelles B-1050, Belgium b Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, France c Haute-Ecole Robert Schuman, Rue Fontaine aux Mˆ ures, 13b, B-6800, Belgium
E-mail: [email protected], [email protected] Abstract: We derive the asymptotic symmetries of the manifestly duality invariant formulation of electromagnetism in Minkoswki space. We show that the action is invariant under two algebras of angle-dependent u(1) transformations, one electric and the other magnetic. As in the standard electric formulation, Lorentz invariance requires the addition of additional boundary degrees of freedom at infinity, found here to be of both electric and magnetic types. A notable feature of this duality symmetric formulation, which we comment upon, is that the on-shell values of the zero modes of the gauge generators are equal to only half of the electric and magnetic fluxes (the other half is brought in by Diracstring type contributions). Another notable feature is the absence of central extension in the angle-dependent u(1)2 -algebra. Keywords: Gauge Symmetry, Global Symmetries ArXiv ePrint: 2004.05668
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)081
JHEP06(2020)081
A note on electric-magnetic duality and soft charges
Contents 1
2 Starting point 2.1 Action and presymplectic form 2.2 Hamiltonian vector fields 2.3 Boundary conditions 2.4 Improper gauge symmetries 2.5 Equations of motion
3 3 4 4 5 7
3 Poincar´ e invariance 3.1 Boundary degrees of freedom 3.2 More improper gauge transformations 3.3 Poincar´e transformations 3.4 Poincar´e transformations of the improper gauge generators 3.5 SO(2) duality generator
8 8 9 10 12 12
4 Sources
12
5 Conclusions
13
1
Introduction
The asymptotic structure of electromagnetism in Minkowski space has been a subject of great interest in the last years, with the discovery that soft photon theorems could be viewed as Ward identities of the corresponding asymptotic symmetries [1], triggering a lot of insightful activity [2–6] reviewed in [7]. (Earlier work on the asymptotic symmetries of electromagnetism at null infinity involves [8, 9].) While this work was originally focused on null infinity, the structure of the asymptotic symmetry algebra, which is given by arbitrary functions on the 2-sphere (“angle-dependent u(1) transformations”) was also explored at spatial infinity [10–12] and equivalence between the two formulations demonstrated. In particular the antipodal matching conditions of the null infinity approaches, relating fields at the past of I + to fields at the future of I − could be justified on a dynamical basis [12]. The proof of equivalence involves an interesting change of basis in the algebra based on a parity decomposition. T
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