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.

32 pages, LaTeX, many small improvements (published version). arXiv admin note: text overlap with arXiv:1912.12995

References in corpus (5)

• Causal Fermion Systems: An Elementary Introduction to Physical Ideas and Mathematical Concepts
• Banach Manifold Structure and Infinite-Dimensional Analysis for Causal Fermion Systems
• A Gauge Fixing Procedure for Causal Fermion Systems
• Causal Variational Principles in the $\sigma$-Locally Compact Setting: Existence of Minimizers
• A Positive Mass Theorem for Static Causal Fermion Systems

Cited by in corpus (2)

• Linear Bosonic Quantum Field Theories Arising from Causal Variational Principles
• A Positive Quasilocal Mass for Causal Variational Principles