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 $$L^2$$ 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.

Modified measures as an effective theory for causal fermion systems

“…This corresponds to our physical conception that the transitivity of the causal relations could be violated both on the cosmological scale (there might be closed timelike curves) and on the microscopic scale (there seems no compelling reason why the causal relations should be transitive down to the Planck scale). However, in [14,40] causal cone structures were constructed, which do give rise to transitive causal relations (for details see in particular [14, section 4.1]). All these causal relations coincide on length scales which are much larger than the Planck length.…”