A Hamiltonian formulation of causal variational principles

作者：Finster Felix^*; Kleiner Johannes

来源：Calculus of Variations and Partial Differential Equations, 2017, 56(3): 73.

DOI：10.1007/s00526-017-1153-5

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 Frechet 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 R-1,R-1 x S-1 where a local minimizer is given by a measure supported on a two-dimensional lattice.

• 出版日期2017-6