The Theory of Causal Fermion Systems

Geometric Structures

Prerequisites

Basic definitions
Spin spaces and physical wave functions
The fermionic projector

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The Euler-Lagrange equations

Related Topics

The continuum limit
Connection to General Relativity

Geometric Structures

Causal fermion systems give rise to various types of geometric structures. On the most fundamental level, one has the structures of a Lorentzian quantum geometry as first explored in [lqg11]. Here one has the following basic structures:

A more direct connection to the familiar objects of differential topology and geometry has been obtained more recently based on the concept of osculating vacua.

The Spin Connection

The kernel of the fermionic projector $P(x,y) : S_y \rightarrow S_x$ relates different spacetime points to each other. In [lqg11] it is shown that, using a polar decomposition of $P(x,y)$, one can construct a unitary mapping $D_{x,y} \::\: S_y \rightarrow S_x$, referred to as the spin connection. More precisely, the spin connection has the form

\[ D_{x,y} = e^{i \varphi_{xy}\, v_{xy}}\: A_{xy}^{-\frac{1}{2}}\: P(x,y) \:, \]

where $A_{xy}:= P(x,y)\, P(y,x) : S_x \rightarrow S_x$ is the closed chain, $v_{xy}$  is the directional sign operator, and $\varphi_{xy}$ is a suitably chosen real number. The conditions needed for the spin connection to be well-defined are subsumed in the notion that $x$ and $y$ must be spin connectable.

Curvature

Curvature is defined via the holonomy of the spin connection,

\[ \mathfrak{R}(x,y,z) = D_{x,y}\:D_{y,z}\: D_{z,x} \::\: S_x \longrightarrow S_x \:. \]

In [lqg11] the following inherent structures are defined for a general causal fermion system:

  • Clifford subspaces and Clifford multiplication
  • tangent space
  • a metric connection $\nabla_{x,y}$, which is shown to be compatible with Clifford multiplication and the spin connection.

Moreover, in [lqg11] it is shown that in a specific limiting case, these objects go over to the corresponding objects of spin geometry in a globally hyperbolic spacetime.

For more details we also refer to the introduction and survey in [nrstg17] and to the talk on geometric structures at the 2018 spring school on causal fermion systems.

→ The Euler-Lagrange equations

Felix Finster

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