Positive Functionals Induced by Minimizers of Causal Variational Principles

TL;DR

This paper develops positive functionals from second variations of causal variational principles, establishing stability criteria and Hilbert space structures for solutions in space-time.

Contribution

It introduces a method to derive positive functionals ensuring nonlinear stability of minimizers and constructs Hilbert space frameworks for jets in space-time.

Findings

Positive functionals guarantee nonlinear stability of minimizers.

Hilbert space structures are established on the space of jets.

A positive surface layer integral is derived for solutions of linearized equations.

Abstract

Considering second variations about a given minimizer of a causal variational principle, we derive positive functionals in space-time. It is shown that the strict positivity of these functionals ensures that the minimizer is nonlinearly stable within the class of compactly supported variations with local fragmentation. As applications, we endow the space of jets in space-time with Hilbert space structures and derive a positive surface layer integral on solutions of the linearized field equations.