My research interests are in the area of mathematical physics, in particular, applications of Clifford algebras and spinors to differential geometry. This has great utility in general relativity where there is a natural relationship between spinors and vectors, and hence to higher order tensors. This relationship coupled with the comparative simplicity of spinor space in four dimensions means that many problems in relativity are simpler when described in terms of spinors rather than tensors. My PhD thesis is mainly concerned with spinors corresponding to shear-free vector fields. Such vector fields generate a conformal isometry on their conjugate space. Usually, the treatment of shear-free vectors depends on whether the vector field is null or timelike. My thesis contains a unified treatment of spinors corresponding to either type of shear-free vector, expressed as a single spinor equation. Spinors corresponding to conformal Killing vectors satisfy a special case of this equation.
From a pair of shear-free spinors one may construct a tensor known as a conformal Killing-Yano (CKY) 2-form. This is a tensor satisfying a generalisation of Killing's equation to totally antisymmetric tensors. A CKY 2-form may be used to construct symmetry operators for certain field equations. In my thesis I show how symmetry operators for the vacuum Maxwell equations, the massive Dirac equation and the neutrino equation may be constructed from a CKY 2-form. I then show how the symmetry operator for the massive Dirac and neutrino equations may be extended to arbitrary dimensions and signatures.
It is possible to generalise Killing's equation to forms of degree higher than 2. In future work I intend to investigate the role of such generalised CKY tensors in relativity as well as in higher dimensions. Preliminary work indicates that this may allow the construction of more general symmetry operators for fields of arbitrary spin in higher dimensions. Furthermore, the existence of such tensors imposes non-trivial integrabality conditions which may allow the massless field equation to be solved via separation of variables.