Asymptotic structure of electromagnetism in higher spacetime dimensions

Abstract

We investigate the asymptotic structure of electromagnetism in Minkowski space in even and odd spacetime dimensions ≥ 4. We focus on d > 4 since the case d = 4 has been studied previously at length. We first consider spatial infinity where we provide explicit boundary conditions that admit the known physical solutions and make the formalism well defined (finite symplectic structure and charges). Contrary to the situation found in d = 4 dimensions, there is no need to impose parity conditions under the antipodal map on the leading order of the fields when d > 4. There is, however, the same need to modify the standard bulk symplectic form by a boundary term at infinity involving a surface degree of freedom. This step makes the Lorentz boosts act canonically. Because of the absence of parity conditions, the theory is found to be invariant under two independent algebras of angle-dependent u(1) transformations (d > 4). We then integrate the equations of motion in order to find the behaviour of the fields near null infinity. We exhibit the radiative and Coulomb branches, characterized by different decays and parities. The analysis yields generalized matching conditions between the past of I + and the future of I − . * On leave of absence from Collège de France, 11 place Marcelin Berthelot,

Generalized asymptotics for gauge fields

“…Specifically, with finite energy initial data, corresponding to an EM field that disperses to infinity, we might expect regular evolution. This seems at odds with claims [5][6][7][8]16] of singular behavior at I + .…”

“…Discussions of soft charges, and their conservation, have typically assumed the presence of such a symmetry more generally. There have also been attempts to prove the necessity of such symmetry, either as a consequence of regularity at infinity [5][6][7][8], or of the need for a finite symplectic form.…”

“…One expects this solution to be finite, and finiteness of the asymptotic behavior is confirmed by investigating bounds on this solution. Arguments for singular evolution of non-antipodal data [5][6][7][8]16] are reexamined, and found to not imply singularities in the usual electromagnetic fields. The question of the symplectic form is also briefly considered, and it is argued that the symplectic form has finite behavior.…”

Source and response soft charges for Maxwell theory on AdSd

“…We also discuss how the AdS d isometry charges act on our soft mode phase space. In section 7, we show how our results on AdS d and those studied in the literature for Maxwell theory on flat space [19][20][21][22][23][24][25][26][27] could be related to each other through an AdS (large radius) flat space limit. In particular, we show that only the source boundary gauge transformations and the associated soft charges survive the limit and the response charges become subdominant and do not appear in the limit.…”

“…This then led to the algebra of the corresponding soft charges (5.3). While we did our analysis in the action (Lagrangian) formulation, it can be instructive to repeat the same analysis in the Hamiltonian formulation, as was done in [23,24,36] for Maxwell theory on flat space. It is also desirable to explore the presence of magnetic sources and the role of electromagnetic duality in this context as was done in [37][38][39][40] on flat spacetime.…”