The Theory of Causal Fermion Systems

Motivation for the Choice of F

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In order to motivate the definition of the set of possible events in a physical system $\F$ we begin by looking at the mathematical foundations of the theories that we aim to unite. The basis of Quantum Mechanics are (bounded linear) operators on a Hilbert space, while (differentiable) manifolds lie at the basis of General Relativity. If we want to unify General Relativity and Quantum Mechanics, it seems natural to search for a suitable definition of an operator manifold, thus combining the basic concepts from both sides. This is exactly what the definition of $\F$ provides. In fact, the subset of regular events in $\F$ (those with exactly $n$ positive and $n$ negative eigenvalues) has the structure of a manifold.

In the following we will try to convey a rough idea of the interesting properties of $\F$ that make it a good starting point to formulate a theory, beyond just the fact that it is a manifold. We begin our explanation with the definition of a more familiar structure, namely the tangent bundle of the differentiable manifold $\mathcal{M}$
$T\mathcal{M}= \bigcup_{x\in \mathcal{M}}\{x\} \times T_x\mathcal{M}$,
where $T_{x}{\mathcal{M}}$ denotes the tangent space of $\mathcal{M}$ at the point $x$. In particular, $T_{x}M$ is a real vector space with the same dimension as the base manifold $\mathcal{M}$. Thus an element of $T\mathcal{M}$ can be thought of as a pair $( x , v )$, where $x$ is a point in $\mathcal{M}$ and $v$ is a tangent vector to $\mathcal{M}$ at $x$. There is a natural projection
\[ \pi: T\mathcal{M}\rightarrow\mathcal{M} \]

defined by $\pi(x,v)=x$. This projection maps each tangent space $T_x\mathcal{M}$ to the base point $x$. The tangent bundle comes equipped with a natural topology. With this topology, the tangent bundle of a manifold is the prototypical example of a vector bundle (a fiber bundle whose fibers are vector spaces). A section of $T\mathcal{M}$ is a vector field on $\mathcal{M}$, and the dual bundle to $T\mathcal{M}$ is the cotangent bundle, which is the disjoint union of the cotangent spaces of $\mathcal{M}$.

When we introduce a metric $g$, we get a natural isomorphism between the tangent bundle and the cotangent bundle. The metric itself is a bilinear form on the tangent bundle with smooth dependence on the base point $x$. Now with that little recap of the set up of a (tangent/vector) bundle, lets have a look at the definition of $\F$.

Definition. (Space of all possible events)
Let ${\F}$ be set of all $x \in \Lin(\H)$  with the following properties:

  • $x$ is self-adjoint and has finite rank
  • $x$ has at most $n$ positive and at most $n$ negative eigenvalues

For most applications, it suffices to restrict attention to the regular events in $\F$ define as follows:

Definition. (Regular events in $\F$)
A regular event in $\F$ is an operator $x\in \F$ such that $rank (x)=2n$.

We note that $\F$ is the closure of the regular events in $\Lin(\H)$ (in the sup-norm topology).
Now let us introduce the following notation
$S_x:=x(\mathscr{H}) \subset \mathscr{H}$      subspace of dimension $\leq 2n$
$\pi_x : \mathscr{H} \rightarrow \mathscr{H}$             orthogonal projection to $S_x$,
where $\pi_x$ is the orthogonal projection operator to the image of the operator $x$. To recap in detail what is going on here, let us take a moment to appreciate the elements we just introduced: $S_x$ is a closed subspace of a Hilbert Space, and by that property it is a Hilbert space itself. Therefore, in particular it is a complex vector space endowed with the scalar product $\langle u| v \rangle_{S_x}:= \langle u| v \rangle_{\mathscr{H}}$ for all $u,v \in S_x \subset \H$. Furthermore, by the nature of a Hilbert space, it is its own dual space. Moreover, we can introduce a sesquilinear form ${\prec}u | v {\succ}_x$ on $S_x$, referred to as the spin inner product,
\[ {\prec}u | v {\succ}_x = \langle u | x v \rangle_{S_x} \:, \]
which (for regular $x$) is an indefinite inner product of signature $(n,n)$.
To summarize, every regular event in $\F$ comes with a similar structure as a point in a (tangent/spin) bundle endowed with a metric (isomorphism to its dual space). The crucial difference is that $S_x$ does not yield a unique projection because there exist regular events in $x,y \in \F$ such that $S_x=S_y$. A point $x$ in a tangent bundle $T\mathcal{M}$ comes with a $d$-dimensional real vector space $T_x$ which has a natural identification with its dual space $T^*_x$ through the metric, which itself is a bilinear form of signature $(s, d-s)$. Likewise, a regular element $x$ in the space of all possible events $\F$ comes with a $2n$ dimensional complex vector space $S_x$ which has a natural identification with its dual space $S^*_x$ through the Hilbert space scalar product inherited from $\mathcal{H}$ and features a bilinear form of signature $(n,n)$ defined through the operator $x$.
As already mentioned, the set $\F$ is the closure of the set of all regular events in $\F$ in the topology induced by the operator norm of $\Lin(\mathcal{H})$. In the same topology, the spin inner product $\prec \psi | \phi \succ_x := -\langle \psi | x \phi\rangle_{\mathscr{H}}$ with $\psi, \phi \in S_x \subset \H$ is continuous. Spacetime $M= \text{supp}\rho$ is a subset of $\F$ and hence the choice of $\F$ together with $\rho$ gives us a generalization of the structure of a spinor bundle (for details see [topology14]). A substantial difference is that in the setting of causal fermion systems, all the fibers $S_x$ are subspaces of the same Hilbert space $\H$ (see also the mathematics section → spin spaces and the physical wave functions).
We finally remark that the choice of the spin dimension $n$ constrains the possible dimension of the spacetime manifold which one can obtain in a suitable limiting case from a causal fermion system (roughly speaking, choosing $n$ corresponds to choosing the spacetime dimension in General Relativity).