Banach Manifold Structure and Infinite-Dimensional Analysis for Causal Fermion Systems

Abstract

A mathematical framework is developed for the analysis of causal fermion systems in the infinite-dimensional setting. It is shown that the regular spacetime point operators form a Banach manifold endowed with a canonical Fréchet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating derivatives of functions on Banach spaces which are differentiable only in certain directions. A chain rule is proven for Hölder continuous functions which are differentiable on expedient subspaces. These results are made applicable to causal fermion systems by proving that the causal Lagrangian is Hölder continuous. Moreover, Hölder continuity is analyzed for the integrated causal Lagrangian.

arXiv comment38 pages, LaTeX, minor improvements (published version)

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Cited by in corpus (11)

• Fermionic Fock Spaces and Quantum States for Causal Fermion Systems
• The Linear Dynamics of Wave Functions in Causal Fermion Systems
• A Mechanism of Baryogenesis for Causal Fermion Systems
• Modified Measures as an Effective Theory for Causal Fermion Systems
• A Positive Mass Theorem for Static Causal Fermion Systems
• A Notion of Entropy for Causal Fermion Systems
• Elliptic Methods for Solving the Linearized Field Equations of Causal Variational Principles
• A Positive Quasilocal Mass for Causal Variational Principles
• Linear Bosonic Quantum Field Theories Arising from Causal Variational Principles
• Holographic Mixing and Fock Space Dynamics of Causal Fermion Systems
• A Canonical Construction of the Extended Hilbert Space for Causal Fermion Systems