Seminar on the Dirac equation

Next week Tadeusz Kulczycki and I start a seminar on the Dirac equation. The full title is Mathematical models in relativistic quantum physics, but actually we plan to focus on just one particular model. We will roughly follow (parts of) The Dirac equation book by B. Thaller, aiming at understanding the hydrogen atom model. From time to time I will post my notes on this blog. Hopefully this will become a mathematician-friendly introduction to the Dirac equation.

The Dirac equation is a first-order differential equation for a \mathbf{C}^4-valued wavefunction \psi(x, t), which describes the evolution of the state of an electron. It is famous for a nice (but not perfect!) description of the hydrogen atom, and the prediction of existence of positrons, or, more generally, antimatter.

With no interaction with external fields, the Dirac equation is believed to be fully correct. However, for many-particle systems, like the hydrogen atom, it is in a sense an intermediate step between classical quantum mechanics and quantum field theory: it is Lorentz invariant (hence relativistic) and describes a spin-½ particle in an electromagnetic field, but it fails to catch the influence of (for example) the spin of the proton. Therefore, the Dirac equation explains most of the fine structure of the hydrogen atom spectrum, but it says nothing about its hyperfine structure. (Honestly, these Wikipedia articles did not explain me much. I found this blog post much more informative for such a greenhorn.)

The Dirac equation is related to the Klein-Gordon equation and the Klein-Gordon square-root operator \sqrt{-\Delta+m^2}. In my recent work I study operators of this kind, in the context of subordinate Brownian motion. I hope that my results may have something to do with the motion of an electron in the presence of an infinite potential wall, but I know too little about quantum physics to state this formally. My basic motivation to study the Dirac equation is to find a physical application of my mathematical work. But even if this fails, I will enjoy the seminar — in fact, I always dreamed to be a physicist.

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