A Hamiltonian Formulation of Causal Variational Principles

TL;DR

This paper reveals a Hamiltonian structure underlying causal variational principles, extending the framework to non-smooth cases and deriving associated symplectic and linearized field equations, with applications to specific models.

Contribution

It introduces a Hamiltonian formulation for causal variational principles, including symplectic structures and linearized equations, generalizing previous approaches to non-smooth settings.

Findings

Solution space forms a symplectic Fréchet manifold under smoothness assumptions.

Surface layer integral defines an invariant symplectic form.

Explicit example on  with a measure supported on a lattice.

Abstract

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.