Grassmannian variables in physics

The five elements of this thesis are linked by the concept of Grassmannian variables. I begin with a brief introductory chapter discussing the general setting and then go on to deal with five topics each of which features anti-commuting co-ordinates in some guise. Chapter 2 uses the conventional, space-time supersymmetry, admissible for relativistic field theories. In the same way that the Dirac equation can be regarded as a square root of the Klein-Gordon equation, I have obtained a square root of the Dirac equation. This equation involves spinor-valued superfields and is given in terms of two component Grassmannian spinors. It has a larger component field content than the Dirac equation, just as the Dirac equation has a larger content than the KleinGordon equation. After setting it up and solving the constraints for both the massless and massive cases, I have gone on for the massless case to solve the equation itself. The next chapter is concerned with Grassmannian variables in the context of path integrals. Specifically, I have studied the derivation of the index of the twisted Dirac operator via a supersymmetric quantum mechanics and taken great care to establish how certain ambiguities in the path integral can arise and how they can be circumvented. As an aside I have also obtained the general expression for the index of fields of arbitrary spin from the Atiyah-Singer index theorem itself. Chapter 4 uses anti-commuting co-ordinates as an appendage to the four commuting space-time co-ordinates. The Kaluza-Klein idea of force unification via general relativity is applied to a (4+N)-dimensional superspace. It is possible to give a consistent ansatz for a higher-dimensional metric which reproduces the standard model of elementary particles. I have considered the extension to grand unified theory and examined the SU(5) and SO(lO) models, showing how the former is more natural and more economical than the latter within such a framework. The consistent quantization of a gauge theory requires the inclusion of \ghost\" fields having \"wrong\" spin and statistics. The resulting gauge-fixed quantized theory is endowed with a BRST symmetry which replaces the classical gauge invariance. This symmetry can be best understood when considered with a partner the anti-BRST symmetry. The two are both supersymmetries as they mix commuting and anti-commuting fields and can therefore be formulated on a superspace with two Grassmannian co-ordinates. In chapter 5 I suggest that it is useful to do this in an Sp(2)-symmetric manner - that is with the ghosts and anti-ghosts and the BRST and anti-BRST symmetries themselves treated symmetrically - but that extending the symmetry group to 0Sp(4/2) is more of a hindrance than a help. The final chapter of this thesis is concerned with a theory of massive nonabelian vector fields based on the Stueckelberg approach. Here renormalizability and unitarity are found to be conflicting requirements. Either one may be satisfied but not both. In particular the violation of unitarity comes about either because of the failure of the BRST operator to be nilpotent or diagrammatically from the incomplete cancellation of the negative-norm ghost contributions."