Schroeck, Franklin E.

In this work, we discuss the conceptual framework of quantum mechanics in the Hilbert space formalism from a group representation point of view. After a brief review of the main results of the theory of groups and their representations, we describe mathematical models of the subject, and show the applications of this theory for getting numerical answers to problems in elementary particle physics.

A Test Space is a mathematical object which models the process of scientific inquiry. We examine the motivation of defining the Test Spaces and discuss connections between the Test Spaces and Metamathematics. A variant of Test Spaces called orthogonal partitions is introduced and we draw comparisons between the two. The combinatorial problems of counting finite Test Spaces and orthogonal partitions is highlighted. Some issues in manipulating infinite Test Spaces are discussed as well.

The Riesz Representation is of great importance in analysis. But in certain situations such as in infinite dimensional Hilbert spaces it fails to apply. Here we present an extensive proof of a relevant generalization using sigma topology as discovered by Edwin Beggs.

A general method for the geometric quantization of connected and simply connected symplectic manifolds and the lifting of symplectic Lie group actions is developed. In particular, a geometric construction of multipliers for a Lie group based on the action of the group on a potential of the symplectic form on the manifold is given. These methods are then employed to quantize the 'massive' symplectic homogeneous spaces of the Galilei group and the group action, thereby emphazising the affine structure of the group and deriving a novel form of phase space representations. In the case of nonzero spin we quantize the action of the covering group of the Galilei group. We derive the spin bundles needed from frame bundles over spheres equipped with their natural Levi Civita connection. Furthermore we give a new geometric description of the 'massless' symplectic homogeneous spaces (the coadjoint orbits) of the Galilei group including a description of the group actions and the symplectic forms. We then describe their geometric quantization as well as the lifting of the group action.