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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| Published in: | Calculus of variations and partial differential equations Vol. 56; no. 3; pp. 1 - 33 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Springer Berlin Heidelberg
01.06.2017
Springer Nature B.V |
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| ISSN: | 0944-2669, 1432-0835 |
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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
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where a local minimizer is given by a measure supported on a two-dimensional lattice. |
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| AbstractList | 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 R1,1×S1 where a local minimizer is given by a measure supported on a two-dimensional lattice. 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. |
| ArticleNumber | 73 |
| Author | Kleiner, Johannes Finster, Felix |
| Author_xml | – sequence: 1 givenname: Felix surname: Finster fullname: Finster, Felix email: finster@ur.de organization: Fakultät für Mathematik, Universität Regensburg – sequence: 2 givenname: Johannes surname: Kleiner fullname: Kleiner, Johannes organization: Fakultät für Mathematik, Universität Regensburg |
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| References | Finster (CR5) 2010; 646 CR4 CR3 Finster, Kleiner (CR10) 2015; 626 CR6 CR9 CR15 Bernard, Finster (CR1) 2014; 7 CR13 Bogachev (CR2) 2007 Finster, Grotz (CR7) 2012; 16 Halmos (CR14) 1974 Saunders (CR17) 1989 Finster, Grotz, Schiefeneder, Finster, Müller, Nardmann, Tolksdorf, Zeidler (CR8) 2012 Finster, Kleiner (CR11) 2016; 55:35 Finster, Schiefeneder (CR12) 2013; 210 Rudin (CR16) 1987 F Finster (1153_CR5) 2010; 646 DJ Saunders (1153_CR17) 1989 1153_CR15 F Finster (1153_CR12) 2013; 210 1153_CR13 1153_CR6 F Finster (1153_CR11) 2016; 55:35 PR Halmos (1153_CR14) 1974 1153_CR4 W Rudin (1153_CR16) 1987 1153_CR3 VI Bogachev (1153_CR2) 2007 F Finster (1153_CR7) 2012; 16 Y Bernard (1153_CR1) 2014; 7 F Finster (1153_CR8) 2012 F Finster (1153_CR10) 2015; 626 1153_CR9 |
| References_xml | – year: 1989 ident: CR17 publication-title: The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series doi: 10.1017/CBO9780511526411 – start-page: 157 year: 2012 end-page: 182 ident: CR8 article-title: Causal fermion systems: a quantum space-time emerging from an action principle publication-title: Quantum Field Theory and Gravity doi: 10.1007/978-3-0348-0043-3_9 – year: 1974 ident: CR14 publication-title: Measure Theory – ident: CR3 – ident: CR4 – ident: CR15 – year: 1987 ident: CR16 publication-title: Real and Complex Analysis – ident: CR13 – ident: CR9 – volume: 55:35 start-page: 41 issue: 2 year: 2016 ident: CR11 article-title: Noether-like theorems for causal variational principles publication-title: Calc. Var. Partial Differ. Equ. – ident: CR6 – volume: 646 start-page: 141 year: 2010 end-page: 194 ident: CR5 article-title: Causal variational principles on measure spaces publication-title: J. Reine Angew. Math. – volume: 7 start-page: 27 issue: 1 year: 2014 end-page: 57 ident: CR1 article-title: On the structure of minimizers of causal variational principles in the non-compact and equivariant settings publication-title: Adv. Calc. Var doi: 10.1515/acv-2012-0109 – volume: 16 start-page: 1197 issue: 4 year: 2012 end-page: 1290 ident: CR7 article-title: A Lorentzian quantum geometry publication-title: Adv. Theor. Math. Phys. doi: 10.4310/ATMP.2012.v16.n4.a3 – volume: 626 start-page: 012020 year: 2015 ident: CR10 article-title: Causal fermion systems as a candidate for a unified physical theory publication-title: J. Phys. Conf. Ser. doi: 10.1088/1742-6596/626/1/012020 – year: 2007 ident: CR2 publication-title: Measure Theory doi: 10.1007/978-3-540-34514-5 – volume: 210 start-page: 321 issue: 2 year: 2013 end-page: 364 ident: CR12 article-title: On the support of minimizers of causal variational principles publication-title: Arch. Ration. Mech. Anal. doi: 10.1007/s00205-013-0649-1 – volume: 7 start-page: 27 issue: 1 year: 2014 ident: 1153_CR1 publication-title: Adv. Calc. Var doi: 10.1515/acv-2012-0109 – volume-title: Measure Theory year: 2007 ident: 1153_CR2 doi: 10.1007/978-3-540-34514-5 – volume-title: The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series year: 1989 ident: 1153_CR17 doi: 10.1017/CBO9780511526411 – volume: 210 start-page: 321 issue: 2 year: 2013 ident: 1153_CR12 publication-title: Arch. Ration. Mech. Anal. doi: 10.1007/s00205-013-0649-1 – volume-title: Measure Theory year: 1974 ident: 1153_CR14 – ident: 1153_CR6 doi: 10.1007/978-3-319-42067-7 – volume: 626 start-page: 012020 year: 2015 ident: 1153_CR10 publication-title: J. Phys. Conf. Ser. doi: 10.1088/1742-6596/626/1/012020 – ident: 1153_CR13 – volume-title: Real and Complex Analysis year: 1987 ident: 1153_CR16 – volume: 55:35 start-page: 41 issue: 2 year: 2016 ident: 1153_CR11 publication-title: Calc. Var. Partial Differ. Equ. – ident: 1153_CR3 – ident: 1153_CR9 – ident: 1153_CR4 – start-page: 157 volume-title: Quantum Field Theory and Gravity year: 2012 ident: 1153_CR8 doi: 10.1007/978-3-0348-0043-3_9 – ident: 1153_CR15 doi: 10.1090/surv/083 – volume: 646 start-page: 141 year: 2010 ident: 1153_CR5 publication-title: J. Reine Angew. Math. – volume: 16 start-page: 1197 issue: 4 year: 2012 ident: 1153_CR7 publication-title: Adv. Theor. Math. Phys. doi: 10.4310/ATMP.2012.v16.n4.a3 |
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| SubjectTerms | 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 |
| Title | A Hamiltonian formulation of causal variational principles |
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