Abstract
Large gauge symmetries in Minkowski spacetime are often studied in two distinct regimes: either at asymptotic (past or future) times or at spatial infinity. By working in harmonic gauge, we provide a unified description of large gauge symmetries (and their associated charges) that applies to both regimes. At spatial infinity the charges are conserved and interpolate between those defined at the asymptotic past and future. This explains the equality of asymptotic past and future charges, as recently proposed in connection with Weinberg’s soft photon theorem.
Highlights
Studied in relation to infrared divergences in QED [9]
By working in harmonic gauge, we provide a unified description of large gauge symmetries that applies to both regimes
Our treatment was purely classical; we did not need worry about possible subtleties in the definition of the quantum counterpart of (6.1) that is used in connection with the soft photon theorem
Summary
The system of study will be a Maxwell field Aa in four dimensional flat spacetime coupled to massive charge fields. Where ∇a is the spacetime covariant derivative, Fab = ∂aAb − ∂bAa the field strength, ja = ieφ(Daφ)∗ + c.c.,. The charge current and Da the gauge covariant derivative, Daφ = ∂aφ − ieAaφ. In the following subsections we describe expansions of the fields at null, space and time infinities. The scalar field being massive will only register at time-like infinity. The Maxwell field will have nontrivial components at all infinities. We here summarize the notation for coordinates, metric and derivative for each space:. Xdenotes the unit three-vector determine by a sphere point xA. Functions f on the sphere are denoted as f (x). We warn the reader that the coordinates ρ and τ have different meaning depending on whether we are discussing expansions at space or at time infinities
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