# The Theory of Causal Fermion Systems

## General Questions

What is the input, and what the output of the theory?

•

Why do we need yet another theory which does not make any new physical predictions?

• The theory of causal fermion systems aims at making physical predictions.  The path from first principles to predictions is long and takes time to develop. Nevertheless, there are first steps towards physical predictions.
→ phenomenology

## Geometry

Space of symmetric operators

• This object $\F \subset \text{L}(\H)$ is the “embedding space” for the spacetime in a CFS. It consists of self-adjoint operators on $\H$ with finite rank and at most $n$ positive and $n$ negative eigenvalues (counting multiplicities).

The motivation is that those operators are used in the causal lagrangian which then determine the universal measure as a minimizer of the causal action principal.

Spacetime

• In the context of QFT the underlying spacetime is a Lorentzian manifold $M$.
Compared to that we define the spacetime as the support of the so called universal measure $\rho$
$M := \text{supp}(\rho) \subset \F$
This totally different concept is motivated by the causal action principal which is by itself motivated from the Lagrangian principal in physics.

In other words, the minimizing operators define a subset which is the underlying manifold in a CFS.

Points in spacetime

• On a Lorentzian manifold the spacetime points are four vectors often denoted by $\tilde{x}^\mu = (t,\vec{x})^T$. Since $M \subset \F$ the spacetime
points $x\in M$ in a CFS are finite self-adjoint linear operators on $\H$. By the local correlation operator $F$ both can be identified $x \equiv \tilde{x}$.

Spinors

• Classically spinors $\psi \in \Gamma (SM)$ are sections of an associated vector bundle $SM$ with local trivialization $SM \approx M \times C^4$.
In the setting of CFS we define spinors as elements of the spin space $S_xM = x(\H)$ (see definition spin space)

Wave functions

• The interpretation of a wave function is similar to the in quantum mechanics. We define it as a function
which associates a spinor to every spacetime point $x \in M$
$\psi : M \rightarrow \H \quad \text{with} \quad \psi(x) \in S_xM \quad \forall x \in M$

Bundle / topological structures

• Since CFS provide a framework for non-smooth geometries one can ask how topological notions on a differentiable manifold generalize to CFS.
Since $M$ is a topological space by definition (subset topology from the operator norm on $\F$) one can create the structure of a sheaf by attaching
to each $x$ the corresponding spin space $S_x$. This makes it possible to describe the topology by sheaf cohomology.

Additionally, if one assumes that all space-time points are regular this gives rise to a topological vector bundle.

Clifford algebra

• In the context of classical spin geometry one works with the Clifford algebra which is used to define the Spin algebra which then is used to define the spinor space as an associated vector bundle via Clifford multiplication. This notion is generalized in CFS via the so called Clifford subspaces. It can be seen as a generalization of the space spanned by the usual Dirac matrices.

Weyl spinors

• In spin geometry the notion of Weyl spinors arises as eigenspaces of the (oriented) volume $\omega$ element with $\omega^2 = 1$.
In CFS one can introduce the Euclidean sign operator $s_x$ with $s_x^2 = \1$ whose eigenspaces correspond to the eigenvalues $\pm 1$.
This leads to an orthogonal decomposition of the spin space
$S_x = S_x^+ \oplus S_x^- \quad \forall \, x .$

Spin connection

• It is possible to generalize the notion of a spin connection from a smooth spin manifold to one with lower regularity. Like in the classical
setting a connection should relate two different spin space with each other. Doing so, one needs to define a closed chain
$A_{xy} : S_x \rightarrow S_x$
which then can be used to define a spin connection
$D_{x,y} : S_y \rightarrow S_x$
even in the absence of regularity.

Metric connection

• Normally the metric connection is derived from the metric and vectors in the tangent space $TM$. But this requires the
existence of such a metric and at least a regularity of two. After that having the metric connection one can define the spin connection on a spinor bundle.

In CFS this setting is vice versa and one does not need a metric or regularity at all.
That can be done by defining Splice maps $U_z^{(y|x)}$ relating two different Clifford subspaces at the same spacetime point $z$
and combing it with the spin connections. In the end one obtains an isometry which we call the metric connections
$\nabla_{x,y} : T_y \rightarrow T_x .$

Curvature

• Having a spin or metric connection in a CFS one can define the curvature as usual as the holomony of the corresponding correction
$\mathfrak{R}(x,y,z) : T_x \rightarrow T_x \, .$

## Analysis

What is the Hilbert space?

• Apriori the Hilbert space is abstract. However, in applications the Hilbert space may be thought of as (a subspace) of the solutions of the Dirac equation. For example in Minkowski space the physical wave function representation of the Hilbert space is precisely given by the Dirac sea (see https://causal-fermion-system.com/theory/math/minkowski-space-as-cfs/).

How do you construct a causal fermion system from a given spacetime?

• The starting point is hereby the solution space of the dirac equation in the physical spacetime. This immediately provides the Hilbert space $\H$ of the corresponding causal fermion system as well as ${\mathcal{F}}$.
The next essential ingredient is the so called “local correlation operator” $R_\varepsilon$ which enables us to evaluate the solutions of the Dirac equation pointwise by mapping them to continuous functions (the regularization parameter \vareps shall stay fixed for this brief introduction). $R_\varepsilon$ then defines a bilinear form on $\H$ by
$b_x(u,v):= -\overline{R_\varepsilon(u)(x)}R_\varepsilon(v)(x) .$
Applying Fréchet-Riesz (with respect to the ordinary inner product on $\H$) we may represent $b_x$ as a linear operator $F_\varepsilon(x)$. This operator is then contained in $\F$ of the abstract causal fermion system.
It also gives a measure on the abstract causal fermion system by the push-forward of the volume measure of the physical spacetime under $F_\varepsilon$. Then the support of this measure – i.e. the abstract space time – corresponds to the physical space time we started from.
For details and further reading see [cfs16, p.22-27].

Why is it called “causal fermion system”?

## Objects and Structures

How is the causal structure defined? And what properties does it have?

• The word causal in causal fermion systems is due to the fact that the set $\mathcal{F}$ is endowed with a certain causal structure. This is at the core of the causal fermion systems framework, as we will be briefly discuss.

Recall that, in the context of causal fermion systems, a spacetime $M$ is the support of the universal measure $\rho$, so its points are bounded self-adjoint operators of finite rank. In order to define causality relations between arbitrary points $x,y \in M$, the product operator $xy$ plays a central role. In particular, the causal structure on $M$ depends on the eigenvalues of the product operator. Two points $x,y \in M$ are called timelike separated if all the eigenvalues are real and do not have the same the absolute value. Furthermore, they are called spacelike separated if all the eigenvalues have the same absolute value. Finally, if none of the previous conditions is satisfied, they are called lightlike separated.

There are a number of important comments to be made on this causal structure. First of all, it can be proven that the previously defined causal relations agree with the usual causal relations in Lorentzian Geometry in suitable limiting cases. Secondly, spacelike separated points do not contribute to the Lagrangian (and, hence, do not play a role when deriving physical equations from the causal action principle). This is analogous to General Relativity, in which spacelike separated points cannot influence each other. Finally, an ordering of timelike separated points can be introduced and allows to distinguish a direction of time on $M$. However, in stark contrast with Lorentzian Geometry (considering the analogous ordering “$\ll$” relation), this ordering of timelike separated points is not necessarily transitive.