Analysis & Applied Math

 This is a featured talk

The fine asymptotic properties of dispersive gauge theories, the archetypal examples of which are the Maxwell—Klein—Gordon and the Yang—Mills—Higgs equations, depend crucially on the choice of gauge as well as the scale and regularity of the initial data. In physics, these asymptotic properties are expected to play a role as some of the conditions for a good definition of a unitary S-matrix and an analytic understanding of infrared divergences. Motivated by these considerations, in this talk I will discuss the analytic properties of the Cauchy and Goursat problems for the Maxwell—Klein—Gordon equations, the appearance of null and weak null structures in the equations depending on gauge, discuss the role played by the total charge as an obstruction to defining “scattering states”, and discuss John Baez’s attempt from 1989 to use the conformal method to define a scattering operator in the case of vanishing charge, which ran into issues due to a lack of analytic theorems in low regularity at the time. I will mention recent work with J.-P. Nicolas (University of Brest) which resolves the issue Baez encountered in the case of Maxwell—Klein—Gordon. Time permitting, I will mention ongoing work which suggests that the charge obstruction is a gauge effect, and exhibit a novel gauge for the Maxwell—Klein—Gordon equations adapted to the scattering problem from infinity.