Real-Time Path Integrals, Caustics and Interference in Cosmology

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. The resulting propagator consists of an interference pattern governed by the caustics of the underlying classical system. After evaluating the path integral, I present methods to track the relevant real and complex saddle points (solutions to the classical boundary value problem, also known as instantons) while changing the boundary conditions of the path integral. I demonstrate that these instantons encounter singularity crossings and need to be generalised to an equivalent class of classical paths for the vast majority of physical theories. The path integral may receive contributions from instantons that were previously hiding from sight. These methods pave the way to the study of dynamical systems in our quantum universe.