Generalized Symplectic Geometry for Classical Fields and Spinors.

The purpose of this dissertation is twofold. First, we develop a generalized symplectic geometry to address classical field theories. Second, we develop a suitable prolongation of the linear frame bundle over spacetime so that we may perform geometric prequantization for spinning particles. Both use Norris's theory of n-symplectic geometry. For the first objective, consider a fiber bundle Y over a finite-dimensional manifold M. The bundle of linear frames LY reduces via symmetry breaking to a subbundle of vertically adapted frames L_{V}Y , and the generalized symplectic structure is pulled back from LY. Given a vector field on Y projectable to M, the solutions to the generalized symplectic equations are Hamiltonian vector fields on L_{V }Y. There exists an induced bundle tau^{*}{cal F}V over Y to which we can associate vector bundles describing linear field theories. The bundle L _{V}Y projects to tau ^{*}{cal F}V and equivariant sections of L_{V}Y over tau^{*} {cal F}V are in bijective correspondence with Ehresmann connections on Y. One may associate to L_{V}Y the multisymplectic vector bundle Z of affine cojets defined by Gotay, et al. There is a stratification of Z such that for each projectable vector field on Y the solutions to the generalized symplectic equations on L_{V}Y are mapped onto solutions of the multisymplectic equations on the target strata. Moreover, the generalized symplectic structure is invariant under the lifted action of Aut(Y). Thus, momentum mappings on L_{V}Y based on Aut(Y) induce momentum mappings on Z. Furthermore, we may better understand momentum observables on Z by representing them on L_{V}Y. For the second objective, we seek a program of geometric quantization for Norris's theory of n-symplectic geometry on the linear frame bundle LM over 4-dimensional metric spacetime (M,g). Specifically, we propose a geometrical model in which the Dirac equation emerges naturally from 4-symplectic geometry on the spin bundle SM over the orthonormal frame bundle OM. The vector fields corresponding to the metric g on OM are trivial, but through prolongation, a suitable bundle is found such that the structure equation admits nontrivial vector fields as solutions. Restriction of the Hamiltonian vector fields back to SM and representation as Hermitian operators on doubc^{4 -} spinors yields the Dirac equation.