The Theory of Causal Fermion Systems

Existence Theory for Linearized Field Equations

Existence Theory in the Time-Dependent Setting

$ \renewcommand{\H}{\mathscr{H}} \newcommand{\Lin}{\mathrm{L}} \newcommand{\F}{{\mathscr{F}}} \newcommand{\Sact}{{\mathcal{S}}} \newcommand{\T}{{\mathcal{T}}} \renewcommand{\L}{{\mathcal{L}}} \newcommand{\N}{\mathbb{N}} \newcommand{\C}{\mathbb{C}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\R}{\mathbb{R}} \newcommand{\1}{\mathbb{1}} \DeclareMathOperator{\tr}{tr} $ The existence theory for the linearized field equations is developed in [linhyp18]. The method is inspired by energy estimates for symmetric hyperbolic systems as used in the theory of hyperbolic partial differential equations. The methods are based on positivity properties of so-called softened surface layer integrals of the form\[ \int_U d\rho(x)\: \eta_t(x) \int_U d\rho(y)\: \big(1-\eta_t(y)\big)\: \Big( \nabla_{1,\mathfrak{v}}^2 – \nabla_{2,\mathfrak{v}}^2 \Big) \L(x,y) \:, \]where $\eta_t$ is a smooth cutoff function in time, and $U \subset M$ is a so-called lens-shaped region. The time function and the lens-shaped region must satisfy hyperbolicity conditions. Under these assumptions, one obtains global existence of solutions of the Cauchy problem and finite propgation speed. The solutions are unique up to jets in the orthogonal complement of the test jets.

Existence Theory in the Static Setting

For static causal fermion systems, the existence theory is inspired by methods of elliptic partial differential equations. The starting point are the positive functionals obtained from second variations of the causal action.

surface layer integrals