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 cl...

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Vydáno v:Calculus of variations and partial differential equations Ročník 56; číslo 3; s. 1 - 33
Hlavní autoři: Finster, Felix, Kleiner, Johannes
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2017
Springer Nature B.V
Témata:
Analysis
Calculus of Variations and Optimal Control; Optimization
Construction
Control
Euler-Lagrange equation
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Smoothness
Surface layers
Systems Theory
Theoretical
Variational principles
53D30
49K21
49S05
28C99
83C47
58C35
58Z05
49K27
49Q20
ISSN:0944-2669, 1432-0835
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Popis
Shrnutí: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 R 1 , 1 × S 1 where a local minimizer is given by a measure supported on a two-dimensional lattice.
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content type line 14
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-017-1153-5