# The Theory of Causal Fermion Systems

## Operator Algebras

### Prerequisites

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### Related Topics

## Operator Algebras

Causal fermion systems give rise to operator algebras in two essentially different ways:

## Algebras Generated by Spacetime Point Operators

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:

- 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. - In the example of the regularized Minkowski vacuum, a time-slice axiom holds.
- 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 e*vents, 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.

## The Algebra of Field Operators

Operator algebras also arise in [FK21] when making the connection to quantum field theory. In this context, one associates operators to the solutions of the linearized field equations (the *bosonic operators*) and of the dynamical wave equation (the *fermionic operators*). These operators satisfy *canonical commutation* respectively *anti-commutation relations*. The unital *-algebra generated by these operators is denoted by $\mathcal{A}$. It corresponds to the *algebra of observables* as considered in the algebraic formulation of quantum field theory. In this formulation, the quantum state $\omega^t$ is a positive linear mapping from $\mathcal{A}$ to the complex numbers. Fock spaces arise when constructing representations of this algebra.

### Felix Finster

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