Seminario Dottorato 2023/24 - Elena Danesi (05.06.2024)

The Dirac equation is a first order partial differential equation. It was first derived by Paul Dirac in 1928 in order to describe the free motion of a spin 1/2 particle on R^3, in according with the principles of quantum mechanics and special relativity. In the following years its definition has been generalized in order to be adapted to curved backgrounds. From the mathematical side, it can be listed within the class of dispersive equations, together with the Schrödinger, wave and Klein-Gordon equations. In the years, because of the study of nonlinear systems, a lot of effort has been devoted to developing tools to quantify the dispersion of a system. Among these tools we find a priori estimates on the solutions, such as Strichartz or local smoothing estimates. 

In the first part of the talk I will focus on the Schrödinger and wave equations in order to present these kind of estimates as well as some classical tools to prove them. In the second part, I will first introduce the Dirac equation on R x R^3 and describe its connection with the above mentioned equations. I will then present the equation in curved spacetimes. To conclude, I will survey some recent results concerning the validity of Strichartz estimates for the “curved” Dirac equation in specific settings, in particular compact or asymptotically flat manifolds.