UC Berkeley

Geometric approaches form the foundation of modern classical mechanics. The prototypical example of a geometric method in mechanics is symplectic geometry applied to the Hamiltonian formulation of a system of particles. Extending this approach to field theories leads to unattractive features, such as an infinite-dimensional phase space and loss of manifest covariance. These deficiencies are particularly glaring in general relativity, where manifest covariance is closely tied to the fundamental symmetries of the theory.

Recent progress on covariant Hamiltonian approaches for field theories has led to the development of multisymplectic geometry. Multisymplectic geometry generalizes the symplectic geometry of particle systems to covariant fields, producing a finite-dimensional phase space and retaining manifest covariance. The symplectic 2-form common to symplectic geometry generalizes to the multisymplectic 5-form. The Euler-Lagrange equations for the field can be written in geometric language using the 5-form in a way that is formally identical to the geometric form of Hamilton's equations in particle mechanics. The resulting approach is a powerful geometric tool for understanding classical field theories.

In this dissertation, we improve upon the current approach to performing a 3+1 decomposition (also known as space-time split) of multisymplectic geometry. We clarify the relationship between multisymplectic geometry, its 3+1 decomposition, and the traditional symplectic approach to field theory. The key observation is that there exist two intermediate phase spaces between the multisymplectic phase space and the traditional symplectic phase space. We show how a proper understanding of the geometry of these intermediate spaces clarifies aspects of the traditional symplectic formulation. Our improved 3+1 decomposition allows us to easily handle the case when the spatial manifold (in our space-time split) has a boundary. By careful consideration of what happens to the theory at the boundary, we can arrive at appropriate boundary conditions and boundary modifications to various 3+1 quantities. This is the first such decomposition of the multisymplectic phase space with boundaries in the literature.

Lastly, we develop a multisymplectic formalism for general relativity. Our approach here is new, and gives great insight into the geometric structure of the theory. In the course of developing multisymplectic general relativity, we introduce local Lorentz transformations as an additional gauge symmetry. We show how reducing by this symmetry after 3+1 decomposition leads to the usual symplectic approach to general relativity.