S. S. Somaroo.
Applications of the Geometric Algebra to Relativistic Quantum Theory
Ph.D. thesis, University of Cambridge (1996).

Abstract: This thesis deals primarily with three areas of relativistic quantum theory. These areas are relativistic quantum mechanical tunnelling and the tunnelling time problem, relativistic multiparticle dynamics and higher-spin quantum physics. The principal goal is to investigate new insights and approaches given by use of a novel mathematical tool called the Geometric Algebra (GA). We conclude that use of the GA has profound implications for the foundations of quantum theory and provides avenues for further research that are poorly accessed or inaccessible via any other means.

In an opening chapter we review aspects of GA and in particular, the GA of spacetime - the Spacetime Algebra (STA). A discussion of the Dirac-Hestenes equation, which forms much of the basis of this thesis, is given.

In a subsequent chapter we use the Dirac-Hestenes equation to study relativistic tunnelling through electrostatic step potential barriers. The approach takes full account of relativity and quantum spin. A powerful method using GA-valued operators is developed to aid solution. Results are used in numerical simulations of the tunnelling process. In addition to implications for the interpretations of quantum theory, an appraisal of the relativistic and spin effects with regard to the tunnelling time problem is made . As a prelude, a review of other approaches to the tunnelling time problem is given in a separate chapter.

Of three chapters dedicated to relativistic multiparticle dynamics, one focuses on classical relativistic dynamics. A brief review of previous attempts is given after which we present a relativistic extension of the work of Hestenes and Pappas to provide a GA-based relativistic Hamiltonian dynamics. Use of a universal evolutionary parameter is central to the extension. Another chapter then looks at a relativistic two-particle quantum wave equation proposed in Doran et. al. [1]. This is based on a multiparticle extension of the STA called the multiparticle spacetime algebra (MSTA). The symmetries of this equation are studied in detail and a novel approach to incorporating the Pauli exclusion principle via a symmetry requirement is presented. The equation is then used to numerically model the Pauli exchange `force'. The third chapter in this group compares the MSTA with a multiparticle causal approach due to Holland. The computational and conceptual benefits of the GA are identified and interpretational aspects of the MSTA consolidated.

One of the remaining pair of chapters establishes a new foundation for the representation theory of the Lorentz Group. We discuss thoroughly how our use of multilinear STA spinor valued functions corresponds to more conventional approaches. We emphasise the role played by complexification in conventional approaches. Finally, we apply this work to formulating Hestenes type equations to describe relativistic higher spin fermions. These equations are GA analogues of the Rarita-Schwinger higher-spin wave equations.

[1] Chris Doran, Anthony Lasenby, Stephen Gull, Shyamal Somaroo and Anthony Challinor. Spacetime Algebra and Electron Physics. In P. W. Hawkes, editor, Advances in Imaging and Electron Physics, Vol. 95, (Academic Press).

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