Higher symmetries of relativistic wave equations in curved spacetime

For a little over a century now, symmetry has played a foundational role in the development of theoretical physics. Loosely, a higher symmetry is a differential operator which takes a solution of the equations of motion for a system to a new solution of those equations of motion. While originally studied by experts in general relativity analysing the Kerr spacetime and subsequently by mathematicians in the context of separation of variables on manifolds, in recent years higher symmetries have garnered renewed interest in high energy physics due to the parallels between their algebra and higher spin algebras. In this thesis, I developed techniques - especially emphasizing spinor methods - for computing higher symmetries in curved spacetimes. As illustrative examples, the equations of motion I considered were the relativistic wave equations for spin-0 and spin-1/2 massless particles. I mainly studied the cases when the higher symmetry was a 1st or 2nd order differential operator. For both equations of motion I was able to uniquely determine physically admissible candidates for 1st and 2nd order higher symmetries in terms of conformal Killing vectors/tensors. However, only the 1st order candidates actually proved to be higher symmetries on arbitrary manifolds possessing a conformal Killing vector. Provided a conformal Killing tensor exists on the manifold, conformal flatness was a sufficient, but perhaps not necessary, condition for the 2nd order candidates to be higher symmetries. I finished by briefly exploring the potential for “conformal geometry” to improve the efficiency of the calculations presented. All calculations were performed on an arbitrary, four-dimensional, orientable, connected manifold of Lorentzian metric signature.

The complete thesis can be downloaded from here. I also presented my work in a 30 minute presentation, the slides and script of which are available here and here respectively.