UC Berkeley

In Chapter 2, the multisymplectic formalism of field theories developed over the last fifty years is extended to deal with manifolds that have boundaries. In particular, a multisymplectic framework for first-order covariant Hamiltonian field theories on manifolds with boundaries is developed. This work is a geometric fulfillment of Fock's formulation of field theories as it appears in recent work by Cattaneo, Mnev and Reshetikhin (2014). This framework leads to a geometric understanding of conventional choices for boundary conditions and relates them to the moment map of the gauge group of the theory.

It is also shown that the natural interpretation of the Euler-Lagrange equations as an evolution system near the boundary leads to a presymplectic Hamiltonian system in an extended phase space containing the natural configuration and momenta fields at the boundary together with extra degrees of freedom corresponding to the transversal components at the boundary of the momenta fields of the theory. The consistency conditions for evolution at the boundary are analyzed and the reduced phase space of the system is shown to be a symplectic manifold with a distinguished isotropic submanifold corresponding to the boundary data of the solutions of Euler-Lagrange equations. This setting makes it possible to define well-posed boundary conditions, and provides the adequate setting for the canonical quantization of the system.

The notions of the theory are tested against three significant examples: scalar fields, Poisson sigma-model and Yang-Mills theories.

In Chapter 3, inspired by problems encountered in the geometrical treatment of Yang-Mills theories and Palatini's gravity, a covariant formulation of Hamiltonian dynamical systems as a Hamiltonian field theory of dimension 1+0 on a manifold with boundary is developed. After a precise statement of Hamilton's variational principle in this context, the geometrical properties of the space of solutions of the Euler-Lagrange equations of the theory are analyzed. A sufficient condition is obtained that guarantees that the set of solutions of the Euler-Lagrange equations at the boundary of the manifold, fill a Lagrangian submanifold of the space of fields at the boundary. Finally a theory of constraints is introduced that mimics the constraints arising in Palatini's gravity.

In Chapter 4, a covariant Hamiltonian description of Palatini's gravity on manifolds with boundary is presented. Palatini's gravity appears as a gauge theory satisfying a constraint in a certain topological limit. This approach allows the consideration of non-trivial topological situations.

The multisymplectic framework for first-order covariant Hamiltonian field theories on manifolds with boundary, developed in Chapter 2, enables analysis of the system at the boundary. The reduced phase space of the system is determined to be a symplectic manifold with a distinguished isotropic submanifold corresponding to the boundary data of the solutions of the Euler-Lagrange equations.