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

TL;DR

This paper develops a Banach manifold framework and a specialized differential calculus for analyzing causal fermion systems in infinite-dimensional spaces, establishing regularity properties of the causal Lagrangian.

Contribution

It introduces a Banach manifold structure and expedient differential calculus tailored for causal fermion systems, enabling analysis of derivatives and continuity in infinite dimensions.

Findings

Banach manifold structure for spacetime point operators

Introduction of expedient differential calculus

Proven H"older continuity of the causal Lagrangian

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 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.