QED as a many-body theory of worldlines: I. General formalism and infrared structure

We discuss a reformulation of QED in which matter and gauge fields are integrated out explicitly, resulting in a many-body Lorentz covariant theory of 0+1 dimensional worldlines describing super-pairs of spinning charges interacting through Lorentz forces. This provides a powerful, string inspired definition of amplitudes to all loop orders. In particular, one obtains a more general formulation of Wilson loops and lines, with exponentiated dynamical fields and spin precession contributions, and worldline contour averages exactly defined through first quantized path integrals. We discuss in detail the attractive features of this formalism for high order perturbative computations. We show that worldline S-matrix elements, to all loop orders in perturbation theory, can be constructed to be manifestly free of soft singularities, with infrared (IR) divergences captured and removed by endpoint photon exchanges at infinity that are equivalent to the soft coherent dressings of the Dyson S-matrix proposed by Faddeev and Kulish. We discuss these IR structures and make connections with soft theorems, the Abelian exponentiation of IR divergences and cusp anomalous dimensions. Follow-up papers will discuss the efficient computation of cusp anomalous dimensions and universal features of soft theorems.