The Theory of Causal Fermion Systems

The Continuum Limit

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\renewcommand{\H}{\mathscr{H}}
\newcommand{\F}{\mathscr{F}}
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In contrast to General Relativity and classical field theories, where we obtain differential equations for the fields of interest, the CFS–Euler–Lagrange equations do not unveil their content so easily.

Before we can elaborate on how one can tease out information from the CFS–Euler–Lagrange equations in practice, we need to introduce another concept. Namely, the idea of a physical wave function associated to a state in the Hilbert space $\mathcal{H}$.

\begin{equation}
\psi_i(x) := \pi_x (\,v_i) \quad \text{with} \quad v_i \in \H \:.
\end{equation}

Here $\pi_x$ is the projection of the vector $v_i$ to the image $S_x=x(\H)$ of the operator associated with the event $x$,

\[ \pi_x: \mathcal{H}\rightarrow S_x \:,\qquad
v_i \mapsto \psi_i(x) \:. \]
 

This notion of physical wave functions gives us a representation of the Hilbert space by wave functions in the spacetime $M$ (recall that $M$ is the support of the measure $\rho$ on $\F$). Hence for any basis $v_i$ of the Hilbert space, we get a collection of physical wave functions in spacetime $\psi_i(x)$ with values in $S_x$. (This wave function can be thought of similar to a section of a spinor bundle over the spacetime manifold, although to be clear this is mostly to be taken as an analogy, because $M$ does not need to be a manifold, and the vector spaces $S_x$ are not necessarily linearly independent. For more details see the discussion here )

With the notion of physical wave functions at hand, what one would like to do in principle  is the following procedure:

Find a minimizing measure of the causal action
$\Downarrow$
Determine all physical wave functions for a basis of $\mathcal{H}$
$\Downarrow$
Find a spacetime and matter configuration that gives an approximative effective description for the causal structure of the minimizer and gives rise to equations satisfied by the physical wave functions.
 
In practice, this procedure is too hard a task because the space of all minimizing measures (let us call this $\mathcal{B}$) is extremely large and, at present, little is known about it. This is the point where the analyzsis of the continuum limit comes in. At present, our best shot at extracting information from the causal action principle is to perform the steps in reverse:
 
      Choose a generic spacetime and configuration of matter fields of interest (which do not need to obey the usual physical equations).
$\Downarrow$
Select a set of functions in the spacetime at hand (obeying certain interesting conditions) as a physical wave function representation of a basis of  $\mathcal{H}$
$\Downarrow$
Find a way to represent this configuration as a causal fermion system
$\Downarrow$
Verify whether this system is a minimizer of the causal action principle in an appropriate sense
$\Downarrow$

Obtain restrictions on your spacetime and matter fields from the requirement that the configuration has to be a minimizer of the causal action principle

This program has been carried out in detail for the case of Minkowski space with different matter field configurations. The results of this analysis are outlined here.

We will sketch how this recipe is put to work in practice for the simplest case, namely the Minkowski vacuum.

Having specified spacetime and the matter fields (as being the vacuum), we must choose the Hilbert space $\H$. In what follows, we denote the spacetime manifold (here Minkowski space) by $\mathcal{M}$. It turns out that the  negative energy solutions to the Dirac equation

\begin{equation}
(i \gamma^j \partial_j -m) \psi(x)=0
\end{equation}
 

are a good choice. Here $\gamma^j$ (with $j=0,\ldots, 3$) denote the Dirac matrices and $\partial_j$ the partial derivatives. More precisely, the negative-energy solutions that lie in $L^2$ for every spacelike hypersurface $\Sigma$ in the sense that the scalar product

\begin{equation}
(\psi|\phi)_\mathcal{H} := \int_\Sigma \overline{\psi} \gamma^0\phi d^3x
\end{equation}
 

is well-defined and finite. Next we need a map from the spacetime manifold $\mathcal{M}$ into $\F$. This is achived by the local correlation operator $F(x)$ defined by

\begin{equation}
(\psi | F(x)\phi)_\mathcal{H} := -\overline{\psi(x)}\phi(x) \qquad \forall \psi,\phi \in \mathcal{H} \:.
\end{equation}
 

For a point in $3+1$-dimensional Minkowski space it is then clear that the operator $F(x)$ has exactly two positive and two negative eigenvalues for every point $x\in \mathcal{M}$. Hence $F(x) \in \F$ on the choosen Hilbert space $\mathcal{H}$ (where we chose the spin dimension $n=2$). Varying $x$, we obtain the local correlation map

\[ F: \mathcal{M} \rightarrow \F \:,\qquad x \mapsto F(x) \:, \]
 

which maps the spacetime manifold into $\F$. We can now use this map to construct a measure $\rho$ on $\F$ as the push-forward of the volume measure $d\mu=d^4x$ of Minkowsi space, i.e.

\begin{equation}
\rho(\Omega):= \mu(F^-1(\Omega))
\end{equation}
 

where $\Omega$ is any subset of $\F$. It turns out that the support of $\rho$ coincides with the image of the mapping $F$. We thus obtain a mapping from Minkowski space to the spacetime $M := \text{supp} \rho$ of the causal fermion system,

\[ F : \mathcal{M} \rightarrow M \subset\F \:,\qquad
x \mapsto F(x) \:. \]

The above construction gives a causal fermion system $(\H, \F, \rho)$ which describes the Minkowski vacuum.

Before explaining in which sense this causal fermion system is a minimizer of the causal action, we must append one important point which for ease of presentation we ignored above: The Dirac solutions $\psi, \phi \in \H$ are only square integrable, but they are not continuous. Therefore, the right side in the defining equation for the local correlation operator is ill-defined. In order to to fix this problem, we need to introduce a regularization to smooth out the functions on a length scale $\varepsilon$,

\begin{equation}
\psi^\varepsilon(x):= \mathcal{R}^\varepsilon [\psi](x)
\end{equation}

where $\mathcal{R}^\varepsilon$ is referred to as the regularization operator. The easiest example is a convolution with a smooth kernel, i.e.

\[ \mathcal{R}_{conv}^\varepsilon [\psi](x):= \int_\Sigma \psi(t,\vec{y})\: \eta(\vec{x}-\vec{y})\: d^3y \]
 
with a smooth test function $\eta$. Since we make use of the physical wave functions to define the map $F$, the regularization will effect the measure $\rho$. Indeed, using the regularized physical wave functions, we get a regularized map $F^\varepsilon$ defined via the equation
 
\[ (\psi | F^\varepsilon(x) \phi)_\mathcal{H} = -\overline{\psi^\varepsilon(x)}\phi^\varepsilon(x) \qquad \forall \psi,\phi \in \mathcal{H} \:, \]
 

which gives rise to a regularized measure $\rho^\varepsilon$ and a corresponding spacetime $M^\varepsilon$. The resulting causal fermion systems are referred to as regularized Dirac sea vacua.

In order to show that these regularized Dirac sea vacua are physically sensible systems within the Theory of Causal Fermion systems, one must show and specify in which sense they are minimizers of the causal action princple. This is indeed accomplished in the analysis of the continuum limit, where it is shown that the Euler Lagrange equations are satisfied in the limiting regime when the regularization length $\varepsilon$ tends to zero.

Another important result for regularized Dirac sea vacua is that, asymptotically as $\varepsilon \searrow 0$, these causal fermion systems encode the complete geometric information of Minkowski space. Even more, the causal fermion systems give rise to generalized notions of spinors, wave functions  spin connection which go over to the usual notions as $\varepsilon \searrow 0$. This so-called correspondence is worked out in [cfs16, Section 1.2] and [lqg11, Theorem 4.7]. More details can be found in the mathematics sections:
Minowski space as a causal fermion system
Geometric structures
The continuum limit