Elliptic Methods for Solving the Linearized Field Equations of Causal Variational Principles

TL;DR

This paper develops an existence theory for solutions to linearized field equations in causal variational principles, employing elliptic methods and spectral calculus to define Sobolev spaces and analyze solutions.

Contribution

It introduces elliptic methods and spectral calculus to solve linearized field equations within causal variational principles, providing a new analytical framework.

Findings

Bounded and symmetric integral operator on a Hilbert space

Defined Sobolev-type Hilbert spaces for the equations

Illustrated methods with explicit examples

Abstract

The existence theory is developed for solutions of the inhomogeneous linearized field equations for causal variational principles. These equations are formulated weakly with an integral operator which is shown to be bounded and symmetric on a Hilbert space endowed with a suitably adapted weighted $L^2$-scalar product. Guided by the procedure in the theory of linear elliptic partial differential equations, we use the spectral calculus to define Sobolev-type Hilbert spaces and invert the linearized field operator as an operator between such function spaces. The uniqueness of the resulting weak solutions is analyzed. Our constructions are illustrated in simple explicit examples. The connection to the causal action principle for static causal fermion systems is explained.