Elliptic methods for solving the linearized field equations of causal variational principles

The existence theory is developed for solutions of the inhomogeneous linearized field equations for causal variational principles. These equations are formulated weakly with an integral operator which is shown to be bounded and symmetric on a Hilbert space endowed with a suitably adapted weighted L...

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Vydáno v:	Calculus of variations and partial differential equations Ročník 61; číslo 4
Hlavní autoři:	Finster, Felix, Lottner, Magdalena
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2022 Springer Nature B.V
Témata:	Analysis Calculus Calculus of Variations and Optimal Control; Optimization Control Differential calculus Elliptic functions Fermions Function space Hilbert space Linearization Mathematical analysis Mathematical and Computational Physics Mathematics Mathematics and Statistics Operators (mathematics) Partial differential equations Systems Theory Theoretical Variational principles 46E35 49S05 35J50 49Q20 58C35
ISSN:	0944-2669, 1432-0835
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Abstract	The existence theory is developed for solutions of the inhomogeneous linearized field equations for causal variational principles. These equations are formulated weakly with an integral operator which is shown to be bounded and symmetric on a Hilbert space endowed with a suitably adapted weighted L 2 -scalar product. Guided by the procedure in the theory of linear elliptic partial differential equations, we use the spectral calculus to define Sobolev-type Hilbert spaces and invert the linearized field operator as an operator between such function spaces. The uniqueness of the resulting weak solutions is analyzed. Our constructions are illustrated in simple explicit examples. The connection to the causal action principle for static causal fermion systems is explained.
AbstractList	The existence theory is developed for solutions of the inhomogeneous linearized field equations for causal variational principles. These equations are formulated weakly with an integral operator which is shown to be bounded and symmetric on a Hilbert space endowed with a suitably adapted weighted L2-scalar product. Guided by the procedure in the theory of linear elliptic partial differential equations, we use the spectral calculus to define Sobolev-type Hilbert spaces and invert the linearized field operator as an operator between such function spaces. The uniqueness of the resulting weak solutions is analyzed. Our constructions are illustrated in simple explicit examples. The connection to the causal action principle for static causal fermion systems is explained. The existence theory is developed for solutions of the inhomogeneous linearized field equations for causal variational principles. These equations are formulated weakly with an integral operator which is shown to be bounded and symmetric on a Hilbert space endowed with a suitably adapted weighted L 2 -scalar product. Guided by the procedure in the theory of linear elliptic partial differential equations, we use the spectral calculus to define Sobolev-type Hilbert spaces and invert the linearized field operator as an operator between such function spaces. The uniqueness of the resulting weak solutions is analyzed. Our constructions are illustrated in simple explicit examples. The connection to the causal action principle for static causal fermion systems is explained. The existence theory is developed for solutions of the inhomogeneous linearized field equations for causal variational principles. These equations are formulated weakly with an integral operator which is shown to be bounded and symmetric on a Hilbert space endowed with a suitably adapted weighted $$L^2$$ L 2 -scalar product. Guided by the procedure in the theory of linear elliptic partial differential equations, we use the spectral calculus to define Sobolev-type Hilbert spaces and invert the linearized field operator as an operator between such function spaces. The uniqueness of the resulting weak solutions is analyzed. Our constructions are illustrated in simple explicit examples. The connection to the causal action principle for static causal fermion systems is explained.
ArticleNumber	133
Author	Lottner, Magdalena Finster, Felix
Author_xml	– sequence: 1 givenname: Felix orcidid: 0000-0002-9531-7742 surname: Finster fullname: Finster, Felix email: finster@ur.de organization: Fakultät für Mathematik, Universität Regensburg – sequence: 2 givenname: Magdalena surname: Lottner fullname: Lottner, Magdalena organization: Fakultät für Mathematik, Universität Regensburg
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References	Finster, F., Kleiner, J.: Causal fermion systems as a candidate for a unified physical theory. arXiv:1502.03587 [math-ph]. J. Phys.: Conf. Ser. 626, 012020 (2015) Helgason, S.: Groups and Geometric Analysis, Mathematical Surveys and Monographs, vol. 83. American Mathematical Society, Providence (2000). Integral geometry, invariant differential operators, and spherical functions, Corrected reprint of the 1984 original Finster, F.: Perturbation theory for critical points of causal variational principles. arXiv:1703.05059 [math-ph]. Adv. Theor. Math. Phys. 24, no. 3, 563–619 (2020) Finster, F., Schiefeneder, D.: On the support of minimizers of causal variational principles. arXiv:1012.1589 [math-ph]. Arch. Ration. Mech. Anal. 210(2), 321–364 (2013) Finster, F., Kleiner, J.: A Hamiltonian formulation of causal variational principles. arXiv:1612.07192 [math-ph]. Calc. Var. Partial Differential Equations 56:73(3), 33 (2017) LangSFundamentals of Differential Geometry, Graduate Texts in Mathematics1999New YorkSpringer10.1007/978-1-4612-0541-8 Finster, F., Kamran, N.: Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles. arXiv:1808.03177 [math-ph]. Pure Appl. Math. Q. 17(1), 55–140 (2021) Link to web platform on causal fermion systems. https://www.causal-fermion-system.com Dappiaggi, C., Finster, F.: Linearized fields for causal variational principles: existence theory and causal structure. arXiv:1811.10587 [math-ph]. Methods Appl. Anal. 27(1), 1–56 (2020) Finster, F.: The continuum limit of causal fermion systems. arXiv:1605.04742 [math-ph]. Fundamental Theories of Physics, vol. 186, Springer (2016) Finster, F.: Causal fermion systems: a primer for Lorentzian geometers. arXiv:1709.04781 [math-ph]. J. Phys.: Conf. Ser. 968, 012004 (2018) Finster, F., Platzer, A.: A positive mass theorem for static causal fermion systems. arXiv:1912.12995 [math-ph]. To appear in Adv. Theor. Math. Phys. (2022) Finster, F., Langer, C.: Causal variational principles in the σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-locally compact setting: Existence of minimizers. arXiv:2002.04412 [math-ph]. To appear in Adv. Calc. Var. (2021) BogachevVIMeasure Theory2007BerlinSpringer10.1007/978-3-540-34514-5 Finster, F., Lottner, M.: Banach manifold structure and infinite-dimensional analysis for causal fermion systems. arXiv:2101.11908 [math-ph]. Ann. Global Anal. Geom. 60(2), 313–354 (2021) Finster, F.: Causal variational principles on measure spaces. arXiv:0811.2666 [math-ph]. J. Reine Angew. Math. 646, 141–194 (2010) Finster, F., Jokel, M.: Causal fermion systems: An elementary introduction to physical ideas and mathematical concepts. arXiv:1908.08451 [math-ph]. Progress and Visions in Quantum Theory in View of Gravity (F. Finster, D. Giulini, J. Kleiner, and J. Tolksdorf, eds.), Birkhäuser Verlag, Basel, pp. 63–92 (2020) Finster, F., Treude, J.-H.: An Introductory Course on Causal Fermion Systems, in preparation. https://www.causal-fermion-system.com/intro-public.pdf Bäuml, L., Finster, F., von der Mosel, H., Schiefeneder, D.: Singular support of minimizers of the causal variational principle on the sphere. arXiv:1808.09754 [math.CA]. Calc. Var. Partial Differential Equations 58(6), 205 (2019) Finster, F.: Positive functionals induced by minimizers of causal variational principles. arXiv:1708.07817 [math-ph]. Vietnam J. Math. 47, 23–37 (2019) GlimmJJaffeAQuantum Physics, a Functional Integral Point of View19872New YorkSpringer0461.46051 Finster, F., Kindermann, S.: A gauge fixing procedure for causal fermion systems. arXiv:1908.08445 [math-ph], J. Math. Phys. 61(8), 082301 (2020) ReedMSimonBMethods of Modern Mathematical Physics. I, Functional Analysis19802New YorkAcademic Press Inc.0459.46001 2237_CR6 2237_CR12 S Lang (2237_CR21) 1999 VI Bogachev (2237_CR2) 2007 2237_CR7 2237_CR11 2237_CR22 2237_CR8 2237_CR10 2237_CR9 2237_CR20 2237_CR16 2237_CR3 2237_CR15 2237_CR4 2237_CR14 2237_CR5 2237_CR13 M Reed (2237_CR23) 1980 2237_CR1 J Glimm (2237_CR19) 1987 2237_CR18 2237_CR17
References_xml	– reference: Finster, F.: Perturbation theory for critical points of causal variational principles. arXiv:1703.05059 [math-ph]. Adv. Theor. Math. Phys. 24, no. 3, 563–619 (2020) – reference: Link to web platform on causal fermion systems. https://www.causal-fermion-system.com – reference: Dappiaggi, C., Finster, F.: Linearized fields for causal variational principles: existence theory and causal structure. arXiv:1811.10587 [math-ph]. Methods Appl. Anal. 27(1), 1–56 (2020) – reference: GlimmJJaffeAQuantum Physics, a Functional Integral Point of View19872New YorkSpringer0461.46051 – reference: Helgason, S.: Groups and Geometric Analysis, Mathematical Surveys and Monographs, vol. 83. American Mathematical Society, Providence (2000). Integral geometry, invariant differential operators, and spherical functions, Corrected reprint of the 1984 original – reference: Finster, F.: Causal fermion systems: a primer for Lorentzian geometers. arXiv:1709.04781 [math-ph]. J. Phys.: Conf. Ser. 968, 012004 (2018) – reference: Finster, F., Langer, C.: Causal variational principles in the σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-locally compact setting: Existence of minimizers. arXiv:2002.04412 [math-ph]. To appear in Adv. Calc. Var. (2021) – reference: Finster, F., Kindermann, S.: A gauge fixing procedure for causal fermion systems. arXiv:1908.08445 [math-ph], J. Math. Phys. 61(8), 082301 (2020) – reference: Finster, F., Schiefeneder, D.: On the support of minimizers of causal variational principles. arXiv:1012.1589 [math-ph]. Arch. Ration. Mech. Anal. 210(2), 321–364 (2013) – reference: Finster, F., Jokel, M.: Causal fermion systems: An elementary introduction to physical ideas and mathematical concepts. arXiv:1908.08451 [math-ph]. Progress and Visions in Quantum Theory in View of Gravity (F. Finster, D. Giulini, J. Kleiner, and J. Tolksdorf, eds.), Birkhäuser Verlag, Basel, pp. 63–92 (2020) – reference: Finster, F.: Causal variational principles on measure spaces. arXiv:0811.2666 [math-ph]. J. Reine Angew. Math. 646, 141–194 (2010) – reference: Finster, F., Treude, J.-H.: An Introductory Course on Causal Fermion Systems, in preparation. https://www.causal-fermion-system.com/intro-public.pdf – reference: Finster, F., Kamran, N.: Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles. arXiv:1808.03177 [math-ph]. Pure Appl. Math. Q. 17(1), 55–140 (2021) – reference: ReedMSimonBMethods of Modern Mathematical Physics. I, Functional Analysis19802New YorkAcademic Press Inc.0459.46001 – reference: Bäuml, L., Finster, F., von der Mosel, H., Schiefeneder, D.: Singular support of minimizers of the causal variational principle on the sphere. arXiv:1808.09754 [math.CA]. Calc. Var. Partial Differential Equations 58(6), 205 (2019) – reference: BogachevVIMeasure Theory2007BerlinSpringer10.1007/978-3-540-34514-5 – reference: Finster, F., Kleiner, J.: Causal fermion systems as a candidate for a unified physical theory. arXiv:1502.03587 [math-ph]. J. Phys.: Conf. Ser. 626, 012020 (2015) – reference: Finster, F.: Positive functionals induced by minimizers of causal variational principles. arXiv:1708.07817 [math-ph]. Vietnam J. Math. 47, 23–37 (2019) – reference: Finster, F., Platzer, A.: A positive mass theorem for static causal fermion systems. arXiv:1912.12995 [math-ph]. To appear in Adv. Theor. Math. Phys. (2022) – reference: Finster, F., Kleiner, J.: A Hamiltonian formulation of causal variational principles. arXiv:1612.07192 [math-ph]. Calc. Var. Partial Differential Equations 56:73(3), 33 (2017) – reference: Finster, F., Lottner, M.: Banach manifold structure and infinite-dimensional analysis for causal fermion systems. arXiv:2101.11908 [math-ph]. Ann. Global Anal. Geom. 60(2), 313–354 (2021) – reference: LangSFundamentals of Differential Geometry, Graduate Texts in Mathematics1999New YorkSpringer10.1007/978-1-4612-0541-8 – reference: Finster, F.: The continuum limit of causal fermion systems. arXiv:1605.04742 [math-ph]. Fundamental Theories of Physics, vol. 186, Springer (2016) – ident: 2237_CR10 doi: 10.4310/PAMQ.2021.v17.n1.a3 – volume-title: Methods of Modern Mathematical Physics. 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SubjectTerms	Analysis Calculus Calculus of Variations and Optimal Control; Optimization Control Differential calculus Elliptic functions Fermions Function space Hilbert space Linearization Mathematical analysis Mathematical and Computational Physics Mathematics Mathematics and Statistics Operators (mathematics) Partial differential equations Systems Theory Theoretical Variational principles
Title	Elliptic methods for solving the linearized field equations of causal variational principles
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