A Manifestly Causal Approach to Quantum Field Theory

TL;DR

This paper introduces a causal formalism for quantum field theory calculations that emphasizes probability-level summations, offering new insights into IR divergence cancellations, particle scattering, and the Unruh effect.

Contribution

It develops a manifestly causal, probability-based formalism for QFT, including new diagrammatic rules and applications to scattering and the Unruh effect.

Findings

Causal formalism reproduces known scattering cross sections.

Transition rates of Unruh detectors are consistent across perspectives.

Numerical results show transient effects in detector responses.

Abstract

We develop a probability-level, manifestly causal formalism for calculations in QFT. The approach involves an implicit summation over final states, which makes causality manifest since retarded propagators emerge naturally. This inclusive summation over final states may also offer insights into the cancellation of IR divergences in physical observables within gauge theories, in accordance with the BN and KLN theorems. To study this, we first conduct particle scattering calculations using conventional methods, determining the quark-antiquark production cross section at first-order in gluon corrections, with careful tracking and cancellation of both IR and UV divergences. We then apply the causal formalism to analogous processes in scalar field theory, introducing novel diagrams that represent algebraic terms at the probability level, akin to Feynman diagrams at the amplitude level. We present a list of rules that generate all probability-level diagrams for particle scattering processes in which one is fully inclusive over final states that contain no initial-state particles. We also investigate the Unruh effect through the lens of the causal formalism. We calculate the transition rate of an accelerating UdW detector coupled to a massive scalar field, from both the perspective of an inertial observer and an accelerating observer. We confirm that the two perspectives give the same transition rate, despite the Rindler observer describing the Minkowski vacuum state as a thermal bath of particles. Numerical results for the transition rate are presented, highlighting the transient effects caused by forcing the field to initially be in the Minkowski vacuum state. Finally, we review the literature regarding the response of an UdW detector on various trajectories in the spacetime of a (3+1)-D Schwarzschild black hole, with a view to extending the analysis in the future using our causal formalism.