# The Theory of Causal Fermion Systems

## Generalizations and Special Cases

The causal action principle for causal fermion systems has too rich a structure and is too difficult for analyzing it in one step in full generality. Instead, it is preferable to approach the problem step by step by analyzing simplifications, modifications or generalizations of the setting which capture particular aspects of the full problem. Proceeding in this way also gives a better understanding of the physical meaning of the different structures of a causal fermion system and of the interaction as described by the causal action principle. Here is an outline of the different settings considered so far.

## Prescribing the Eigenvalues

Clearly, the trace constraint and the boundedness constraint complicate the analysis. Therefore, it might be a good idea to consider a simplified setting where these constraints are not needed. This can be accomplished most easily by prescribing the eigenvalues of the operators in $\F$. This method was first proposed in [continuum08, Section 2] in a slightly different formulation. Given $n \in \N$, we choose real numbers $\nu_1, \ldots, \nu_{2n}$ with

$\nu_1 \leq \cdots \leq \nu_{n} \;\leq\; 0 \;\leq\; \nu_{n+1} \leq \ldots \leq \nu_{2n} .$

We let $\F$ be the set of all symmetric operators on $\H$ of rank $2n$ whose eigenvalues (counted with multiplicities) coincide with $\nu_1, \ldots, \nu_{2n}$. If $\H$ is finite-dimensional, the set $\F$ is compact. This is the reason why it is sensible to minimize the causal action

$\Sact = \int_\F d\rho(x) \int_\F d\rho(y)\: \L(x,y)$

under variations of the universal measure in the class of all regular Borel measures on $\F$, keeping only the volume constraint, which for simplicity we implement by restricting attention to normalized measures.

## The Causal Variational Principle on the Sphere

The simplest interesting case is obtained by choosing the spin dimension $n=1$ and the Hilbert space $\H = \C^2$. We denote the eigenvalues $\nu_1$ and $\nu_2$ by

$\nu_{1\!/\!2} = 1 \pm \tau \qquad \text{with} \qquad \tau>1 \:.$

Then $\F$ consists of all Hermitian $2 \times 2$-matrices $F$ which have eigenvalues $\nu_1$ and $\nu_2$. Such a matrix can be represented with the help of Pauli matrices as

$\F = \big\{ F = \tau\: \vec{x} \vec{\sigma} + \1 \quad \text{with} \quad \vec{x} \in S^2 \subset \R^3 \big\} \:.$

Thus the set $\F$ can be identified with the unit sphere $S^2$.

The causal variational principle on the sphere was first introduced in [continuum08, Section 2]. The numerical study in [support10, Section 2] indicates that for large $\tau>\sqrt{2}$, minimizing measures are weighted counting measures supported at a finite number of points on the sphere. Preliminary results in proving this finding are:

• In [support10, Section 3.3] it is shown that the support of a minimizing measure has an empty interior.
• In [sphere18] it is shown that in the case $\tau>\sqrt{6}$ the Hausdorff dimension of the minimizing measure is at most $6/7$.

These results indicate that the causal action gives rise to a “discreteness” of spacetime and of the interaction (for details see [rrev11, Sections 4 and 5]).

## Causal Variational Principles in the Compact Setting

Generalizing the causal variational principle on the sphere, one can replace $\F$ by a smooth compact manifold of dimension $m \geq 1$. Moreover, the Lagrangian is a given non-negative function $\L : \F \times \F \rightarrow \R^+_0$ with the following properties:

1. $\L$ is symmetric: $\L(x,y) = \L(y,x)$ for all $x,y \in \F$.
2. $\L$ is lower semi-continuous, i.e. for all sequences $x_n \rightarrow x$ and $y_{n’} \rightarrow y$,

$\L(x,y) \leq \liminf_{n,n’ \rightarrow \infty} \L(x_n, y_{n’})\:.$

The causal variational principle in the compact setting is to minimize the corresponding action $\Sact$ under variations of the measure $\rho$ in the class of regular Borel measures, keeping the total volume $\rho(\F)$ fixed.

This variational principle was first considered in [support10, Section 1.2]. It is the preferable choice for studying phenomena for which the detailed form of the Lagrangian as well as the constraints of the causal action principle are irrelevant.

Given a minimizing measure $\rho$, the Lagrangian induces on spacetime $M:= \text{supp}\, \rho$ a causal structure. Namely, two spacetime points $x,y \in M$ are said to be timelike and spacelike separated if $\L(x,y)>0$ and $\L(x,y)=0$, respectively. But of course, spinorial wave functions are missing in this setting.

In the compact setting, the following results have been proven:

## Causal Variational Principles in the Non-Compact Setting

In the compact setting, spacetime $M$ clearly is a compact subset of $\F$. This is not suitable for describing situations when spacetime has an asymptotic future or past, when it has an asymptotic end at spatial infinity, or when spacetime has singularities (like at the big bang or inside a black hole). For studying such situations, it is preferable to work in the so-called non-compact setting as introduced in [jet16, Section 2.1].

In the non-compact setting, $\F$ is a non-compact smooth manifold of dimension $m \geq 1$. One chooses the Lagrangian $\L$ again as a non-negative function on $\F \times \F$ with the above properties (i) and (ii). In this setting, it is not sensible to work with normalized measures. Instead, the total volume $\rho(\F)$ is infinite. In order to introduce a well-defined variational principle, we let $\tilde{\rho}$ be another regular Borel measure on $\F$ which has the same total volume in the sense that it satisfies the conditions

$\big|\tilde{\rho} – \rho \big|(\F)$       and       $\big( \tilde{\rho} – \rho \big) (\F) = 0$

(where $|.|$ denotes the total variation of a signed measure). We then consider the difference of the actions defined by

\begin{align*}
\big( &\Sact(\tilde{\rho}) – \Sact(\rho) \big) = \int_\F d(\tilde{\rho} – \rho)(x) \int_\F d\rho(y)\: \L(x,y) \\
&\quad + \int_\F d\rho(x) \int_\F d(\tilde{\rho} – \rho)(y)\: \L(x,y)
+ \int_\F d(\tilde{\rho} – \rho)(x) \int_\F d(\tilde{\rho} – \rho)(y)\: \L(x,y) \:.
\end{align*}

The measure~$\rho$ is said to be a minimizer of the causal action if this difference is non-negative for all $\tilde{\rho}$ with the same total volume,

$\big( \Sact(\tilde{\rho}) – \Sact(\rho) \big) \geq 0 \:.$

In the non-compact setting, surface layer integrals have been studied in [noether15], [jet16], [osi18]. Moreover, the connection to bosonic Fock spaces is worked out in [FK18]. The existence of minimizers is proved in [noncompact20].

## Static Causal Fermion Systems

Static causal fermion systems were introduced in [pmt] as a simplification useful for describing time-independent physical systems. In order to describe the time translation symmetry, one lets $(U_t)_{t \in \R}$ be a strongly continuous group of unitary transformations on $\H$. A causal fermion system $(\H, \F, \rho)$ is static with respect to $(U_t)_{t \in \R}$ if it has the following properties:

1. Spacetime $M:= \text{supp}\, \rho \subset \F$ is a topological product, i.e. $M = \R \times N$.
2. The one-parameter group $(U_t)_{t \in \R}$ leaves the universal measure invariant, i.e.

$\rho\big( U_t \,\Omega\, U_t^{-1} \big) = \rho(\Omega)$         for all $\rho$-measurable $\Omega \subset \F$.

Under these assumptions, the universal measure and the spacetime points can be decomposed as

$d\rho = dt\: d\mu$       and       $x = (t,\mathbf{x})$ with $t \in \R$, $\mathbf{x} \in \mathscr{G}$,

where ${\mathscr{G}}:= {\mathscr{F}}/\R$ is the quotient obtained by dividing out the symmetry. If $\rho$ is a minimizing measure, then $\mu$ is a minimizer of the static causal action

$\Sact_{\text{s}} = \int_{\mathscr{G}} d\mu(\mathbf{x}) \int_{\mathscr{G}} d\mu(\mathbf{y}) \:\L(\mathbf{x}, \mathbf{y})$
where
$\L(\mathbf{x}, \mathbf{y}) := \int_{-\infty}^\infty \L \big((0,\mathbf{x}), (t,\mathbf{y}) \big) \: dt$

The measure $\mu$ describes space of a static spacetime together with all structures therein. From the mathematical point of view, the static setting makes the connection to the theory of elliptic PDEs.

## Homogeneous Causal Fermion Systems

Similar as just described for the group of time translations, one can also divide out more general symmetry groups (for the general setup of equivariant variational principle see [lagrange, Section 4]. A particular important case is to consider the group of translations in $\R^4$ and to assume that the group action is proper, transitive and has no fixed points. Then space-time $M$ can be identified with $\R^4$. Moreover, after suitable identifications of the spin spaces, the kernel of the fermionic projector $P(x,y)$ depends only on the difference vector $\xi := y-x$. Taking its Fourier representation,

$P(x,y) = \int_{\R^4} \frac{d^4k}{(2 \pi)^4}\: \hat{P}(k)\: e^{-i k (y-x)} \:,$

one can analyze the system and the causal action more explicitly in momentum space. This procedure has been used in many calculations (see for example [reg, vacstab, action]). A detailed introduction and analysis of the homgeneous causal action principle in a compact domain in momentum space is given in [FFL22].

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