The Theory of Causal Fermion Systems

Manifolds of Operators

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

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 $M$.

Operator algebras