Real-Time Path Integrals, Caustics and Interference in Cosmology

Abstract

Interference is one of the most universal phenomena in nature, as exemplified by the real-time Feynman path integral. Despite the ubiquity of interference patterns, their evaluation has often proven challenging. This is especially apparent when considering lensing in astrophysics in the wave optics regime and when studying quantum cosmology using the path integral for gravity. The oscillatory integrals involved are frequently conditionally convergent, converge slowly, and artefacts such as dependence on unphysical cut-offs can be difficult to avoid. Traditionally, these oscillatory integrals are approximated with saddle point methods. However, determining which saddle points to include can be a tricky exercise. Using Picard-Lefschetz theory — a general, exact method for handling multidimensional oscillatory integrals — I will present an unambiguous definition of the real-time path integral and an efficient numerical method for its evaluation. I will show that we can unambiguously identify the relevant instantons using resurgence theory. By going beyond the WKB approximation, we obtain asymptotic series whose interrelations enable us to detect which instantons contribute to the path integral. The resulting propagator consists of an interference pattern governed by the caustics of the underlying classical system. These methods pave the way to the study of real-time dynamical systems in our quantum Universe