# The Theory of Causal Fermion Systems

## Causal Cone Structures

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The causal structure of a causal fermion system gives rise to a distinction between timelike and spacelike separation. Moreover, there a notion of time direction is induced by the functional

${\mathscr{C}} : M \times M \rightarrow \R$,      ${\mathscr{C}}(x, y) := i \tr \big( y\,x \,\pi_y\, \pi_x – x\,y\,\pi_x \,\pi_y \big)$

(where $\pi_x : \H \rightarrow S_x$ is the orthogonal projection to the spin space). Namely, this functional is anti-symmetric in its argument, making it possible to say that $y$ lies in the future (or past) of $x$ if the two points are timlike separated and if ${\mathscr{C}}(x, y)>0$ (or ${\mathscr{C}}(x, y)<0$, respectively). The disadvantage of these definitions is that the resulting relation “lies in the future of” is not necessarily transitive. This corresponds to our physical conception that the transitivity of the causal relations could be violated both on the cosmological scale (there might be closed timelike curves) and on the microscopic scale (there seems no compelling reason why the causal relations should be transitive down to the Planck scale). Nevertheless, it is desirable to have transitive causal relations at least for certain classes of causal fermion systems.

Such relations were constructed in [linhyp18, FK22] based on properties of the solution space of the linearized field equations. More precisely, assuming that the causal fermion system is compactly hyperbolic to every spacetime point $x \in M$ one can associate the future cone $J^\vee(x)$ and the open future cone $I^\vee(x)$. The relation “lies in the open future cone” is transitive, meaning that

$y \in I^\vee(x)$ and $z \in I^\vee(y)$    $\Longrightarrow$      $z \in I^\vee(x)$.

The resulting cone structure is similar to that of a causal set (see also → relation to other theories). However, in contrast to causal sets, in the Theory of Causal Fermion Systems, the transitive causal relations are a derived structure.