The Theory of Causal Fermion Systems

The Causal Action Principle

$
\renewcommand{\H}{\mathscr{H}}
\newcommand{\F}{\mathscr{F}}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\tr}{\mbox{tr}}
$
The core of every modern physical theory is an action principle that selects the physically relevant configurations as being critical points of a suitable action functional. In General Relativity, for example, among all possible Lorentzian metrics on a manifold, only those are considered to be physical that are critical points of the Einstein-Hilbert action, and hence satisfy Einstein’s field equations. Quantum Field Theory, on the other hand, is based on an action defined as a spacetime integral of a Lagrangian.

In the context of causal fermion systems, we want to formulate an action principle over the set of linear operators $\F$. A crucial point here is that we wish to formulate it in a way that is independent of the choice of bases to make the mathematical structures compatible with the principles underlying general relativity.

Similar as already explained in the formulation of the causal relations on $\F$, we will again make use of the eigenvalues of the operator product $xy$. In order to ensure continuity of the Lagrangian in operators in $\F$ we will make use of the spectral weight of the operators where the non-trivial eigenvalues $\lambda_1^{xy}, \ldots, \lambda_{2n}^{xy}$ are counted with their algebraic multiplicities.

$\displaystyle |xy|=\sum_{i=1}^{2n} |\lambda_i^{xy}|$      and      $\displaystyle |(xy)^2|=\sum_{i=1}^{2n} |\lambda_i^{xy}|^2$.

Just a quick reminder: For a finite rank operator the algebraic multiplicity of an eigenvalue is the order of the factor in the characteristic polynomial. This does not necessarily agree with the geometric multiplicity, i.e. the dimension of the eigenspace to a particular eigenvalue.

To make a long story short, we will just present the causal action principle at the heart of the Causal Fermion Systems framework and explain the significance of its various constituent parts in detail afterward.

Lagrangian      $\displaystyle {{\mathcal{L}}(x,y)} =  |(xy)^2| -\frac{1}{2n}|xy|^2=\frac{1}{4n} \sum_{i,j=1}^{2n} \big(
|\lambda^{xy}_i| – |\lambda^{xy}_j| \big)^2 \;\geq\; 0$
Causal action            $\displaystyle \iint_{\F \times \F} {\mathcal{L}}(x,y)\: d\rho(x)\: d\rho(y) \;\in\; [0, \infty]$

It is worth noting that the Lagrangian in the second formulation is intimately related to the definition of causality in the causal fermion system. In particular, one sees immediately that spacelike separated points do not contribute to the Lagrangian. A physical system is then given by a measure $\rho$ which minimizes the causal action ${\mathcal{S}}$. Therefore, we are looking for conditions on the variation of the measure $\rho$ when we try to derive the Euler Lagrange equations for causal fermion systems. For the variational problem to be well-posed, we need to take the following constraints into account:

volume constraint:                                  $\rho(\F) = \text{const}$
trace constraint:                              $\displaystyle \int_\F \tr(x)\: d\rho(x) = \text{const}$
boundedness constraint:       $\displaystyle \iint_{\F \times \F} |xy|^2\: d\rho(x)\, d\rho(y) \leq {C}$

The need for the volume constraint is pretty clear: if we remove that constraint there will always be a trivial minimizer with measure zero (as the Lagrangian is strictly positive). The situation for the trace constraint is similar. If $x$ is in $\F$, then so is $\lambda x$. Without the trace constraint we could always reduce the action by rescaling all operators in the support of $\rho$ with $\lambda<1$ this can be done indefinitely, and hence if the constraint were removed there would be trivial minimizers. The boundedness constraint, on the other hand, has no such obvious motivation. In practice, it shows that the constant ${C}$ is tightly related to the regularization scale $\varepsilon$. There exist preliminary ideas that suggest that for a sensible physical interpretation, one might need to interpret the boundedness constraint and the Lagrangian together, but this is subject of ongoing research (see the research question on the interpretation of the causal action principle). For the time being, one just needs to accept that this additional constraint is required for the well-posedness of the variational problem.

A measure $\rho$ which minimizes the causal action under the above constraints satisfies corresponding Euler-Lagrange equations. For a detailed discussion see the mathematics sections
The Euler-Lagrange equations
The linearized field equations

In the Theory of Causal Fermion systems, the Euler-Lagrange equations play the role of the usual physical equations (like Maxwell’s equations, Einstein’s equations or the Yang-Mills equations). The precise connection is obtained in the so-called continuum limit. Moreover, these Euler-Lagrange equations are also expected to describe the dynamics of quantum fields, as is outlined in the mathematics section
Connection to quantum field theory