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Banach Manifold Structure and Infinite-Dimensional Analysis for Causal Fermion Systems
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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 Fr\'echet-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 H\"older continuous functions which are differentiable on expedient subspaces. These results are made applicable to causal fermion systems by proving that the causal Lagrangian is H\"older continuous. Moreover, H\"older continuity is analyzed for the integrated causal Lagrangian.
Forward citations
Cited by 3 Pith papers
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A Geometric Derivation of the Einstein Equations from the Causal Action Principle
Using osculating vacua, the authors derive Einstein's equations from the causal action principle, with the gravitational coupling identified as the square of the regularization length.
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The $\mathcal{L}$-Calculus for Causal Variational Principles: An Exterior Differential Calculus on Non-Smooth Spaces
An exterior differential calculus, based on Lagrangian-mollified weak derivatives and osculating vacua, is constructed for non-smooth causal variational principles, with cohomology, Stokes and Gauss theorems, and work...
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Causal Fermion Systems: Spacetime as the web of correlations of a many-body quantum system
In causal fermion systems, spacetime points are reinterpreted as bundles of correlations among occupied fermion states, and, for a broad class including the Minkowski vacuum, the causal action equals the variance of t...
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