| he 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 ;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... |
