Title:

Geometric algebra, group theory and theoretical physics

This dissertation applies the language of geometric algebra to group theory and theoretical physics. Geometric algebra, which is introduced in Chapter 2, provides a natural extension of the concept of multiplication from real numbers to geometric objects such as line segments and planes. It is based on Clifford algebra and augmented by auxiliary definitions which give it a geometric interpretation. Since geometric algebra provides a natural encoding of the concepts of directed quantities, it has the potential to unify many of the disparate systems of notation that are used in mathematics. In Chapter 3, the properties of multilinear functions are investigated and the theory is developed to make them useful for formulating the representation of groups. It will be found that multilinear functions are more flexible than their tensor or matrix counterparts in traditional linear algebra. Multilinear functions can be classified according to the symmetry class of their arguments and their behaviour under the monogenic or harmonic decomposition. It is found that the previous definitions of monogenic and harmonic functions need some modification if they are to be defined consistently. Polynomial projection is also discussed, a technique that is useful in constructing nonlinear functions from linear functions, an operation outside the scope of conventional linear algebra. In Chapter 4, multilinear functions are used to construct the irreducible representations of the three regular classes of classical groups; rotation groups, the special unitary and special linear group, and the symplectic group. In each case it is found that a decomposition must be applied to the multilinear functions in order to find the irreducible representations of the groups. For the representations of some of the groups this entails finding the harmonic or monogenic parts of the functions. The groups can be realised as subgroups of the spin group of some dimension and signature. However, geometric algebra provides such a rich algebraic structure that the representations of the groups can be realised in more than one way. In Chapter 7 a brief review is given of computer software for performing symbolic calculations with geometric algebra. A new software package which performs semisymbolic manipulation of multivectors in spaces of any dimension and signature is presented.
