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A Hamiltonian formulation of causal variational principles. (English) Zbl 1375.49060

Summary: 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 Fréchet 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.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49S05 Variational principles of physics
58C35 Integration on manifolds; measures on manifolds
58Z05 Applications of global analysis to the sciences
49K21 Optimality conditions for problems involving relations other than differential equations
49K27 Optimality conditions for problems in abstract spaces
53D30 Symplectic structures of moduli spaces
28C99 Set functions and measures on spaces with additional structure
83C47 Methods of quantum field theory in general relativity and gravitational theory
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