What is a Causal Fermion System?

$\renewcommand{\H}{\mathscr{H}} \newcommand{\F}{\mathscr{F}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\Lin}{\mathrm{L}}$ As mentioned in the introduction, the Hilbert space $\H$ sets the stage on top of which the structures of the causal fermion system are built. The space of all possible events over $\H$ is given by a particular subset $\F$ (defined below) of the bounded linear operators $\Lin(\H)$ acting on the Hilbert Space. We will refer to $\F$ as the set of all possible events (of the physical system of interest). Now for us to be eventually able to formulate an action principle over the set $\F$ we need to introduce one more structure. Classically, a physical action is an integral over the spacetime manifold. In this set up so far, we do not have any sort of background spacetime structure. Therefore, if we want to define an action over the set $\F$ as an integral, we have to make sense of what “an integral over $\F$” means. The mathematical object needed for this purpose is a measure $\rho$ (more precisely, a Borel measure) on $\F$. You can think of a measure $\rho$ as a way to assign a “size” or “volume” to the elements in $\F$. In practical terms, it is exactly what allows us to give meaning to the notion of an integral over the set $\F$ (independent of the actual structure of $\F$). We will call $\rho$ the “universal measure”.
This measure plays a central role in the CFS framework. If you want to gain a deeper understanding of the mathematical structure of the CFS framework, it is necessary to have at least some familiarity with the basic concepts of measure theory. If you just care about the physics in the continuum limit, you can mostly do without.
Let us collect these structures in a formal definition for more clarity:

Definition (Causal fermion system) Let ${(\H, \la .|\,. \ra_\H)}$ be a Hilbert space. Given a parameter $n \in \mathbb{N}$ (“spin dimension”) we let  ${\F}$ be the set of all $x \in \Lin(\H) $  with the properties:

        •  $x$ is self-adjoint and has finite rank
        • $x$ has at most $n$ positive and at most $n$ negative eigenvalues.  

Finally, we let $\rho$ be a measure on $\F$ (“universal measure”).

A technical motivation for the choice of the subset $\F$ can be found  here. The crucial conceptional point is that the set of regular events (self-adjoint operators with exactly $n$ positive and $n$ negative eigenvalues) is a manifold.
Some interesting observations about $\F$: Clearly, the operators in $\F$ are of a very special form (finite rank, bounds on number of positive and negative eigenvalues). Nevertheless, the algebra generated by $\F$ lies dense in $\Lin(\H)$. This means that we can construct every operator in $\Lin(\H)$ (in particularly any self-adjoint operator) from elements in $\F$ by approximation with linear combinations of products of operators in $\F$. This is worth mentioning because, traditionally, observables of a quantum mechanical systems are considered to be selfadjoint linear operators on the Hilbert space that characterizes the physical system.
From now on, when we talk about a “causal fermion system” we always refer to a triple ${(\rho, \F, \H)}$. It is worth mentioning that the universal measure $\rho$ is not necessarily supported on all of $\F$. The elements of $\F$ where the measure is non-vanishing are said to be the events which are realized in a particular physical system. The collection of all events that are realized in a physical system, i.e. the support of the measure $\rho$ in $\F$ is called the “spacetime” ${M:= \text{supp} \rho}$.
Note, at this point in the discussion, you have to take the suggestive names of the CFS structures at face value. We hope the motivation why these structures are labelled the way they are, will become clear in the discussion of the continuum limit.