Canonical Quantisation of Bound and Unbound WQFT

Abstract

Using canonical quantisation, and eschewing the Schwinger-Keldysh path integral, we derive a version of the Worldline Quantum Field Theory (WQFT) formalism suitable for both scattering and bound configurations of the classical two-body problem. Focusing on a pair of charged particles interacting via a scalar field, we quantise Hamilton's equations both in flat space and around a non-zero background, perturbing in post-Lorentzian (PL) and self-force (SF) expansions respectively. Our quantisation procedure provides access to the Magnus series, and is perfectly suited for computing matrix elements of $\hat{N}(t,t_0)=- i \hbar \log\hat{U}(t,t_0)$, both with and without external scalar states, for finite time intervals (bound orbits) and infinite time intervals (scattering). Doing so, we provide a complete set of gauge-invariant matrix elements describing the 1SF scattering dynamics up to 3PL order, and corresponding matrix elements for bound orbits. We also demonstrate how $\hat{N}$-matrix elements encode physical observables, providing a unified operator-based framework for conservative and radiative dynamics of binary systems. The new WQFT formalism generalises naturally to both gravity and electromagnetism.