The Theory of Causal Fermion Systems

Space of symmetric operators

• The object $\F \subset \text{L}(\H)$ is the “embedding space” for the spacetime in a causal fermion system. It consists of all symmetric operators on $\H$ of 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 principle.

Spacetime

• Spacetime $M$ of a causal fermion system is defined as the support of the measure $\rho$ (i.e., intuitively speaking, all points where the measure is non-zero),
  \[
  M := \text{supp}(\rho) \subset \F
  \]
  This totally different concept is motivated by the causal action principle which is by itself motivated from the Lagrangian principle in physics.

  In other words, the minimizing operators define a subset which is the spacetime of the causal fermion system.

Points in spacetime

• Since $M \subset \F$ the spacetime points $x\in M$ in a causal fermion system are symmetric linear operators of finite rank on $\H$. In the example of a classical spacetime described by a Lorentzian manifold, the points of this manifold can be identified with spacetime points of the causal fermion system via the local correlation map

→ Basic definitions
→ Minkowski space as a causal fermion system

On a Lorentzian manifold ${\mathcal{M}}$, a spinorial wave function is a section in the spinor bundle $S{\mathcal{M}}$, an associated vector bundle $S$ with local trivialization $SM \approx M \times C^4$. In the setting of causal fermion systems, on the other hand, we define spinors as elements of the spin space $S_xM = x(\H)$. Similar to a section of a bundle, a wave function associates to each spacetime point $x \in M$ a vector $\psi(x)$ in the corresponding spin space. Every vector in the Hilbert space is represented by a wave function, the so-called physical wave function.
→ Spin spaces and physical wave functions

Bundle / topological structures

• Since causal fermion sytems provide a framework for non-smooth geometries, one can ask how topological notions on a differentiable manifold generalize to causal fermion systems. 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 causal fermion systems via the so called Clifford subspace. It can be seen as a generalization of the space spanned by the usual Dirac matrices.

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 spaces 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
  \[
  D_{x,y} : S_y \rightarrow S_x
  \]
  as a unitary mapping from one spin space to another, even in the absence of regularity.

Metric connection

• In Lorentzian geometry, the metric connection is derived from the metric and vectors in the tangent space $TM$. But this requires the existence of such a metric with a regularity of at least $C^2$. Having defined the metric connection, one can subsequently introduce the spin connection on a spinor bundle.

  In the setting of causal fermion systems, one proceeds vice versa, without the need of a metric or any regularity. This 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 connection. In the end, one obtains an isometry which we call the metric connection
  \[
  \nabla_{x,y} : T_y \rightarrow T_x .
  \]

Curvature

• Having a spin or metric connection in a causal fermion system, one can define the curvature as usual as the holomony of the corresponding conneection,
  \[
  \mathfrak{R}(x,y,z) : T_x \rightarrow T_x \:.
  \]

→ Geometric structures