# The Theory of Causal Fermion Systems

## The Causal Structure

### Prerequisites

One of the insights in the development of the Theory of Causal Fermion Systems was the fact that one can introduce a basis independent causal structure on $\F$. This is crucial if we hope to recover the well known Lorentzian spacetime structures in an appropriate limit.

A causal structure gives relations between any two points $x, y \in \F$. Since $x$ and $y$ are now operators, we will make use of their operator product $x \!\cdot\! y \in \Lin(H)$ to define their causal relations. It is important to note that $\F$ is not a group under operator composition. Indeed, for points $x,y \in \F$, their product $x \!\cdot\! y$ is not necessarily in $\F$. Nevertheless, this operator product still has finite rank $\leq 2n$ and hence it has a finite number of non-trivial eigenvalues ${\lambda^{xy}_1, \ldots, \lambda^{xy}_{2n}}$, which are in general complex. Note here, as mentioned above, we aim to define a causal structure in a way that is independent of the choice of basis in the underlying Hilbert space. Working with the eigenvalues is the most natural way to realize this objective. This also ensure that the mathematical structures to be compatible with ideas underlying General Relativity.

We can now define a causal structure on $\F$, using the eigenvalues of the operator product of $x,y$  as follows:

Definition (Causal Structure)
The points $x,y \in \F$ are called
• spacelike separated if $|\lambda^{xy}_j|=|\lambda^{xy}_k|$ for all $j,k=1,\ldots, 2n$
• timelike separated if $\lambda^{xy}_1, \ldots, \lambda^{xy}_{2n}$ are all real and $|\lambda^{xy}_j| \neq |\lambda^{xy}_k|$ for some $j,k$
• lightlike separated otherwise.

Therefore, the CFS framework allows us to define consistent pairwise causality relations without referring to any form of metric. In the following, we will call two points $x$ and $y$ causally separated if they are either timelike or lightlike separated. Now, to get a full causal structure we need in addition an ordering of the causal relation. There are different ways to introduce such an ordering in the sense of a time direction. In any case we need to define an anti-symmetric functional

{\mathscr{C}} :  \F \times \F \rightarrow \R

which allows us to define a time direction in the following way

Definition (Time Direction)
For causally separated points $x,y \in M$,
• $y$ lies in the future of $x$ if ${\mathscr{C}}(x,y)>0$
• $y$ lies in the past of $x$ if ${\mathscr{C}}(x,y)<0$

In general, the so-defined causal ordering is not necessarily transitive. Hence if $z$ is in the future of $y$ and $y$ is in the future of $x$, we cannot necessarily infer that $z$ is in the future of $x$. This is in accordance with the observation that the transitivity of causal relations might be violated on the cosmological scale (there might exist closed timelike curves) and on the microscopic scale (there does not appear to be a compelling reason why causal relations ought to be transitive at very tiny scales, down to the Planck scale). Note further that for a pair of points $x$ and $y$ which are timelike or lightlike separated, a causal ordering might not exist (because ${\mathscr{C}}(x,y)=0$). The most notable example is obtained by choosing $x=y$: A generic point $x\in \F$ is timelike separated from itself ($x^2$ is self-adjoint and thus has real eigenvalues; for a generic $x$, these eigenvalues are not all equal), but  ${\mathscr{C}}(x,y)$ vanishes, meaning that there is no causal order. At present, it is unclear whether functionals of the above form can be used to make interesting global statements on causal ordering. See [linhyp18, Section 4] for recent progress with respect to that question and here for a sharper formulation of a possible research question.

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