# The Theory of Causal Fermion Systems

## Positive Functionals in Spacetime

If $\rho$ is a minimizer of the causal action principle, the fact that second variations are non-negative gives rise to positive functionals in spacetime. As is worked out systematically in [positive17], these positivity results can be stated that for any compactly supported jet $\mathfrak{u}$, the following two inequalities hold,

\begin{gather}
\int_M \nabla^2 \ell|_x(\mathfrak{u},\mathfrak{u})\: d\rho(x) \geq 0 \\
\int_M d\rho(x) \int_M d\rho(y) \:\nabla_{1,\mathfrak{u}} \nabla_{2,\mathfrak{u}} \L(x,y)
+ \int_M \nabla^2 \ell|_x(\mathfrak{u},\mathfrak{u})\: d\rho(x) \geq 0 \:.
\end{gather}

The last inequality can be written in shorter form with the Laplacian as

$\la \mathfrak{u}, \Delta \mathfrak{u} \ra \geq 0 \:.$

This positivity of the Laplacian is important for the existence theory for the linearized field equations.

## A Positive Surface Layer Integral

The non-negativity of second variations also gives rise to a positive surface layer integral. Indeed, for any solution $\mathfrak{v}$ of the linearized field equations and any compact subset $\Omega \subset M$,

$-\int_\Omega d\rho(x) \int_{M \setminus \Omega} d\rho(y) \:\nabla_{1,\mathfrak{v}} \nabla_{2,\mathfrak{v}} \L(x,y) \geq 0 \:.$

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