Abstract
An infinite number of physically nontrivial symmetries are found for abelian gauge theories with massless charged particles. They are generated by large $U(1)$ gauge transformations that asymptotically approach an arbitrary function $\varepsilon(z,\bar{z})$ on the conformal sphere at future null infinity ($\mathscr I^+$) but are independent of the retarded time. The value of $\varepsilon$ at past null infinity ($\mathscr I^-$) is determined from that on $\mathscr I^+$ by the condition that it take the same value at either end of any light ray crossing Minkowski space. The $\varepsilon\neq$ constant symmetries are spontaneously broken in the usual vacuum. The associated Goldstone modes are zero-momentum photons and comprise a $U(1)$ boson living on the conformal sphere. The Ward identity associated with this asymptotic symmetry is shown to be the abelian soft photon theorem.
Highlights
Ε(z, z) on the conformal sphere at I but are constant along the null generators, even as they antipodally cross from I − to I + through spatial infinity
They are generated by large U(1) gauge transformations that asymptotically approach an arbitrary function ε(z, z) on the conformal sphere at future null infinity (I +) but are independent of the retarded time
The Ward identity associated with this asymptotic symmetry is shown to be the abelian soft photon theorem
Summary
The action on Γ+ is δεAz(u, z, z) = ∂zε(z, z). These comprise the asymptotic symmetries considered in this paper. The charge that generates this transformation can be determined by Noether’s procedure. Q+ww du ∂u (∂wAw + ∂wAw) + e2γwwju This is the total outgoing electric charge radiated into the fixed angle (w, w) on the asymptotic S2. The first term is a linear “soft” photon (by which we mean momentum is strictly zero, as opposed to just small) contribution to the fixed-angle charge. The second term is the accumulated matter charge flux at the angle (w, w). Q+ε generates the large gauge transformation on matter fields. Du d2wεγwwju , Φ(u, z, z) = −qε(z, z)Φ(u, z, z), where Φ is any massless charged matter field operator on I + with charge q
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