A Hamiltonian Formulation of Causal Variational Principles

Causal variational principles, which are the analytic core of the physical theory of causal fermion systems, are found to have an underlying Hamiltonian structure, giving a formulation of the dynamics in terms of physical fields in space-time. After generalizing causal variational principles to a class of lower semi-continuous Lagrangians on a smooth, possibly non-compact manifold, the corresponding Euler-Lagrange equations are derived. In the first part, it is shown under additional smoothness assumptions that the space of solutions of the Euler-Lagrange equations has the structure of a symplectic Fr\'echet manifold. The symplectic form is constructed as a surface layer integral which is shown to be invariant under the time evolution. In the second part, the results and methods are extended to the non-smooth setting. The physical fields correspond to variations of the universal measure described infinitesimally by one-jets. Evaluating the Euler-Lagrange equations weakly, we derive linearized field equations for these jets. In the final part, our constructions and results are illustrated in a detailed example on $\mathbb{R}^{1,1} \times S^1$ where a local minimizer is given by a measure supported on a two-dimensional lattice.