# The Theory of Causal Fermion Systems

## Video

$\newcommand{\H}{\mathscr{H}} \newcommand{\F}{\mathscr{F}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\Lin}{\mathrm{L}} \newcommand{\N}{\mathbb{N}} \newcommand{\C}{\mathbb{C}} \newcommand{\Sact}{\mathscr{S}} \newcommand{\L}{\mathscr{L}} \newcommand{\T}{\mathscr{T}} \DeclareMathOperator{\tr}{tr}$

Definition. Given a separable complex Hilbert space $\mathscr{H}$ with scalar product $\la .|. \ra_\H$ and a parameter $n \in \N$ (the spin dimension), we let $\F \subset \Lin(\H)$ be the set of all self-adjoint operators on $\H$ of finite rank, which (counting multiplicities) have at most $n$ positive and at most $n$ negative eigenvalues.
On $\F$ we are given a positive measure $\rho$ (defined on a $\sigma$-algebra of subsets of $\F$), the so-called
universal measure. We refer to $(\H, \F, \rho)$ as a causal fermion system.

In order to single out the physically admissible causal fermion systems, one must formulate physical equations. To this end, we impose that the universal measure should be a minimizer of the causal action principle, which we now introduce. For any $x, y \in \F$, the product $x y$ is an operator of rank at most $2n$. We denote its non-trivial eigenvalues counting algebraic multiplicities by $\lambda^{xy}_1, \ldots, \lambda^{xy}_{2n} \in \C$ (more specifically, denoting the rank of $xy$ by $k \leq 2n$, we choose $\lambda^{xy}_1, \ldots, \lambda^{xy}_{k}$ as all the non-zero eigenvalues and set $\lambda^{xy}_{k+1}, \ldots, \lambda^{xy}_{2n}=0$). We introduce the spectral weight $| \,.\, |$ of an operator as the sum of the absolute values of its eigenvalues. In particular, the spectral weights of the operator products $xy$ and $(xy)^2$ are defined by

$|xy| = \sum_{i=1}^{2n} \big| \lambda^{xy}_i \big| \qquad \text{and} \qquad \big| (xy)^2 \big| = \sum_{i=1}^{2n} \big| \lambda^{xy}_i \big|^2 \:.$

We introduce the Lagrangian $\L$ and the causal action $\Sact$ by

\begin{align}
\L(x,y) &= \big| (xy)^2 \big| – \frac{1}{2n}\: |xy|^2 \\
\Sact(\rho) &= \iint_{\F \times \F} \L(x,y)\: d\rho(x)\, d\rho(y) \:.
\end{align}

The causal action principle is to minimize $\Sact$ by varying the universal measure under the following constraints,

volume constraint:                                               $\rho(\F) = \text{const}$
trace constraint:                                      $\displaystyle \int_\F \tr(x)\: d\rho(x) = \text{const}$
boundedness constraint:       $\displaystyle \T(\rho) := \iint_{\F \times \F} |xy|^2\: d\rho(x)\, d\rho(y) \leq C$,

where $C$ is a given parameter (and $\tr$ denotes the trace of a linear operator on $\H$).

→ generalizations and special cases
→ existence theory
→ Euler-Lagrange equations
→ Example: Describing Minkowski space as a causal fermion system

## Causal Structure

$\newcommand{\H}{\mathscr{H}} \newcommand{\F}{\mathscr{F}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\Lin}{\mathrm{Lin}} \newcommand{\N}{\mathbb{N}} \newcommand{\C}{\mathbb{C}} \newcommand{\Sact}{\mathscr{S}} \newcommand{\L}{\mathscr{L}} \newcommand{\T}{\mathscr{T}} \DeclareMathOperator{\tr}{tr}$
We define spacetime as the support of the universal measure,

spacetime $M:= \text{supp} \,\rho$

The fact that the eigenvalues of the above operator products are in general complex gives rise to the following “spectral” definition of the causal structure.

Definition (causal structure). Two points $x, y \in M$ are called spacelike separated if all the $\lambda^{xy}_j$ have the same absolute value. They are said to be timelike separated if the $\lambda^{xy}_j$ are all real and do not all have the same absolute value. In all other cases (i.e if the $\lambda^{xy}_j$ are not all real and do not all have the same absolute value), the points $x$ and $y$ are said to be lightlike separated.

## Other Inherent Structures

Apart from the causal structure, a causal fermion system gives rise to many additional structures. These structures are all inherent in the sense that use information already encoded in the causal fermion system.