Speaker
Description
Interference is among the most universal phenomena in physics, as exemplified by the famous Feynman path integral describing quantum physics as a sum over histories. However, conditionally convergent oscillatory integrals are often delicate to define and even more difficult to evaluate. Moreover, the real-time Feynman path integral is still lacking a mathematically rigorous definition. In this talk, I will show steps towards defining the Feynman's path integral non-perturbatively, without a Wick rotation to imaginary time. I start by introducing a class of smooth regulators rendering the interference integral absolutely convergent. Subsequently, I continue the integration domain to its Lefschetz thimble associated with its relevant instantons. The resulting set of integrals can be formally defined in terms of the infinite-dimensional Wiener measure developed for the theory of Brownian motion. As I will show, the resulting path integral is completely governed by caustics as classified by catastrophe theory. The real-time propagator does not share this property with the Euclidean path integral. I will illustrate this approach by studying the real-time path integral of the one-dimensional Teller well and barrier and the quartic oscillator in quantum mechanics.
