Superstructures on graded phase space

In this thesis we study problems associated with the generalisation, to include Grassmann type variables, of the 'group theoretical' approach to quantisation of C.Isham [37]. Although a full generalisation of this quantisation scheme is not achieved, consideration of this problem leads us to make studies in four principle sectors: (A) Graded Poisson brackets and graded 'vector field like’ constructs. A graded version of the Hamiltonian vector field is defined and it is found that both left acting and right acting vector fields are necessary. Properties of these vector fields are investigated. (B) Local graded canonical transformations and graded function groups. Simple examples of these structures are studied. (C) The realisation of a general superalgebra by the use of graded 'functions' and the graded Poisson bracket. The graded generalisation of a standard classical result is presented. Also the - question of central, extensions to these algebras is studied and a partial generalisation of a classical result on this is given. (D) Investigations into a model of quantum mechanics on a2-sphere which incorporates fermions. This model is similarto that derived by Spiegelglas [56] and Barcelos-Neto et al.[6,7]from the 0(3) supersymmetrie sigma model first studied by Witten in [62,63], except that an additional primary constraint has been included. The graded Dirac brackets of this model are calculated.