# A Hamiltonian formulation of causal variational principles

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## 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 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.

### Mathematics Subject Classification

49Q20 49S05 58C35 58Z05 49K21 49K27 53D30 28C99 83C47## Notes

### Acknowledgements

We would like to thank Niky Kamran and Olaf Müller for helpful discussions on jet spaces and Fréchet manifolds as well as Jordan Payette and the referee for valuable comments. We are grateful to the Center of Mathematical Sciences and Applications at Harvard University for hospitality and support. J.K. gratefully acknowledges support by the “Studienstiftung des deutschen Volkes.”

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