N-symplectic analysis of field theory

by 1972- McLean, Michael A.

MCLEAN, MICHAEL ALLEN. N-Symplectic Analysis of Field Theory. (Under the direction of L. K. Norris.) Two techniques for relating n-symplectic geometry to the jet bundle formulations of classical field theory are presented. The tangent bundle of the frame bundle of a manifold M is shown to be a principal fiber bundle over the jet bundle of the tangent bundle of M. We are able to generalize this result to symmetric and antisymmetric tensor bundles of rank p. Using this GL(m) gauge freedom, we interpret the standard free field Lagrangian as a symmetric type (0, 2) tensor on LM. The adapted frame bundle of an arbitrary fiber bundle ? is shown to be a principal bundle over the jet bundle of ?. Using this GL(m) × GL(k) gauge freedom we generate a modified m + k-symplectic geometry from a lifted Lagrangian. The modified soldering form is shown to induce the Cartan-Hamilton-Poincaré m-form on J 1?. We derive generalized Hamilton-Jacobi and Hamilton equations on L?E, and show that the Hamilton-Jacobi and canonical equations of Carathéodory-Rund and de Donder-Weyl are obtained as special cases. These results demonstrate that by introducing additional gauge freedom into the standard jet bundle formalism one can obtain a great of of additional geometric and algebraic structure. Such additional structure may be key in achieving a greater understanding of field theory or in attempts at quantization. N-SYMPLECTIC ANALYSIS OF FIELD THEORY by Michael A. McLean a thesis submitted to the graduate faculty of north carolina state university in partial fulfillment of the requirements for the degree of doctor of philosophy department of mathematics raleigh, north carolina March 23, 2001 approved by: