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.
Highlights
The theory of causal fermion systems is a recent approach to fundamental physics
Spacetime and all objects therein are described by a measure on a set F of linear operators of rank at most 2n on a Hilbert space (H, ⟨. .⟩H)
The physical equations are formulated via the so-called causal action principle, a nonlinear variational principle where an action S is minimized under variations of the measure
Summary
The theory of causal fermion systems is a recent approach to fundamental physics (see the basics in Sect. 2, the reviews [11, 12, 16], the textbook [10] or the website [1]). In view of the importance of the examples and physical applications, it is a task of growing significance to analyze causal fermion systems systematically in the infinite-dimensional setting It is the objective of this paper to put this analysis on a sound mathematical basis. Extending methods and results in [15] to the infinite-dimensional setting, we endow the set of all regular points of F with the structure of a Banach manifold (see Definition 3.1 and Theorem 3.4) To this end, we construct an atlas formed of so-called symmetric wave charts (see Definition 3.3). 1 2n−1 for all ỹ ∈ U (where 2n is the maximal rank of the operators in F ) Relying on these results, we can generalize the jet formalism as introduced in [17] for causal variational principles to the infinite-dimensional setting The example of causal fermion systems in Minkowski space will be worked out separately in [25]
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