Spin Representations, Clifford Algebras and Spinors

Abstract: We begin by giving some theoretical background to the underlying concepts of spin representations and spinors. This is done from the perspective of Lie groups and Lie algebras. In particular, we discuss the functionality of Clifford algebras in the determination of the double-covering spin groups. An introduction to K-algebras and Clifford algebras is then given, focusing on the properties of pseudo-Euclidean spaces . Some low-dimensional examples are also included, culminating with a characterisation of some Clifford algebras as matrix algebras. Elementary representation theory is then introduced and quickly followed by the definition of the Clifford-Lipschitz and spin groups. The work of Lundholm and Svensson (2016), Vaz and da Rocha (2016), and Schwichtenberg (2018) is then united to construct a definition of the spin representations. An attempt at formulating a definition of spinors from a mathematical perspective is then given; formed by combining multiple approaches and definitions of the above-mentioned authors, as well as drawing inspiration from important cases in theoretical physics, in particular that of SO(3) and the Lorentz group SO(1,3).