The Theory of Causal Fermion Systems

Positive Functionals

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,

\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 \:.

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 $\Omega \subset M$,

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

The fermionic signature operator