The Theory of Causal Fermion Systems
Positive Functionals
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Positive Functionals
In the mathematical setting of causal fermion systems, various quantities appear which have a definite sign.
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 $\Omega \subset M$,
\[ -\int_\Omega d\rho(x) \int_{M \setminus \Omega} d\rho(y) \:\nabla_{1,\mathfrak{v}} \nabla_{2,\mathfrak{v}} \L(x,y) \geq 0 \:. \]
Inequalities for Nonlinear Surface Layer Integrals
Other positive functionals arise that the action of a minimizing measure $\rho$ is smaller than the action of any other test measure $\hat{\rho}$. A fruitful method for obtaining the test measure is to start from a general measure $\tilde{\rho}$ and interchange parts of $\rho$ and $\tilde{\rho}$. More precisely, as proposed in [FK23], one chooses subsets $\Omega \subset M$ and $\tilde{\Omega} \subset \tilde{M}$ of the same volume and sets
\[ \hat{\rho} := \chi_{\tilde{\Omega}}\, \tilde{\rho} + \chi_{M \setminus \Omega}\, \rho \:. \]
The minimality of the action of $\rho$ gives rise to the inequality
\begin{align*}
& 2 \int_{\tilde{\Omega}} d\tilde{\rho}(x) \int_{M \setminus \Omega} d\rho(y)\: \L(x,y)
\leq 2 \int_{\Omega} d\rho(x) \int_{M \setminus \Omega} d\rho(y)\: \L(x,y) \\
&\quad – \int_{\tilde{\Omega}} d\tilde{\rho}(x) \int_{\tilde{\Omega}} d\tilde{\rho}(y)\: \L(x,y)
– \int_\Omega d\rho(x) \int_\Omega d\rho(y)\: \L(x,y)
\end{align*}
The left side contains a nonlinear surface layer integral, similar as used in the construction of the quantum state. The first summand on the right can be interpreted as the surface area of $\partial \Omega$. The two other summands, on the other hand, can be regarded as volume integrals over $\tilde{\Omega}$ and $\Omega$, respectively.
This method can be generalized and adapted in various ways, also to cases when $\tilde{\Omega}$ and $\Omega$ do not have the same volume. Moreover, the resulting inequality can be written in a particularly useful form if also the measure $\tilde{\rho}$ satisfies the EL equations. Details and applications to a positive quasi-local mass can be found in [FK23].
Felix Finster
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