The Theory of Causal Fermion Systems

Given a causal fermion system $(\H, \F, \rho)$ and an open set $\Omega \subset M$ of spacetime, one forms the vector space of all spacetime operators smeared out by integration over ontinuous functions,

\[ L_\Omega  := \Big\{ \int_M\: f(x)\, x\: d\rho(x) \: \Big| \: f\in C_0^0(\Omega,\C) \Big\} \:. \]

Then one takes the *-algebra generated by these smeared spacetime operators,

${\mathfrak{A}}_\Omega := \la L_\Omega \ra$ .

This algebra was introduced and analyzed in [FO20]. It was shown to have the following properties:

1. The smeared spacetime point operators do not satisfy the local commutation relations. The commutator is in general non-zero even if the supports are spacelike separated.
2. In the example of the regularized Minkowski vacuum, a time-slice axiom holds.
3. Again in the example of the regularized Minkowski vacuum, the causal structure is encoded in the algebra via the singular behavior of operator products when the regularization length $\varepsilon$ tends to zero.

The last point gives a connection to the events, tries and histories (ETH) formulation of quantum theory (see [FFOP20] or the explanations on the relation to other theories). Namely, if $\Omega$ lies in the causal future of a spacetime point $x$, then $x$ lies in the center of the algebra ${\mathfrak{A}}_\Omega$, up to an error involving a positive power of the regularization length.