# The Theory of Causal Fermion Systems

## Manifolds of Operators

### Prerequisites

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## Manifolds of Operators

The definition of a causal fermion system involves the subset $\F \subset \Lin(\H)$ all all self-adjoint operators on $\H$ of finite rank, which (counting multiplicities) have at most $n$ positive and at most $n$ negative eigenvalues. A point $x \in \F$ is called *regular* if it has maximal rank $2n$ (and thus exactly $n$ positive and exactly $n$ negative eigenvalues). The set of all regular points of $\F$ is denoted by $\F^\text{reg}$. The set $\F^\text{reg}$ has the structure of a manifold, making it possible to apply concepts and methods of differential geometry and analysis on manifolds. We distinguish two cases:

- The finite-dimensional case $\dim \H< \infty$
- The infinite-dimensional case $\dim \H=\infty$

## The Finite-Dimensional Case

In [FKi19] it is proven that in the case $\dim H =:f < \infty$, the set $\F^\text{reg}$ is a non-compact manifold of dimension

\[ \dim \F^\text{reg} = 4n f – 4 n^2 \:. \]

This manifold is endowed with a Riemannian metric inherited from the Hilbert-Schmidt scalar product,

\[ g_x \::\: T_x\F^\text{reg} \times T_x\F^\text{reg} \rightarrow \R \:,\qquad g_x(u,v) = \tr(u v) \:. \]

It is useful to use wave functions as charts, leading to so-called *wave charts*. A specific class of such charts, referred to as *symmetric wave charts,* can be used to fix the local gauge freedom up to global gauge transformations (→ local gauge princple in underlying physical principles).

## The Infinite-Dimensional Case

In [FL21] the infinite-dimensional case is studied. It is show that $\F^\text{reg}$ is an infinite-dimensional Banach manifold, endowed with a Fréchet-smooth Riemannian metric inherited from the Hilbert-Schmidt scalar product.

For the analysis of the causal action principle it is useful to work with the *expedient differential calculus* designed for treating derivatives of Hölder continuous functions on Banach spaces which are differentiable only in certain directions.

Linearized fields can be described by vector fields on $T\F^\text{reg}$ along $M$.

### Felix Finster

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