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

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.◮ Partial and jet derivatives with an index i ∈ {1, 2}, as for example in (2.14), only act on the respective variable of the function L. This implies, for example, that the derivatives commute,1 This assumption is convenient, because then the restricted EL equations (2.12) imply that ℓ vanishes identically on M .