Causal Variational Principles on Measure Spaces

TL;DR

This paper introduces causal variational principles on measure spaces, establishing existence results and exploring properties like non-uniqueness of minimizers, with applications to indefinite inner product spaces.

Contribution

It develops a new class of variational principles that incorporate causality on measure spaces, providing foundational existence results and analyzing minimizer properties.

Findings

Existence of minimizers under the introduced principles

Minimizers may not be unique in general

Counterexamples highlight compactness issues

Abstract

We introduce a class of variational principles on measure spaces which are causal in the sense that they generate a relation on pairs of points, giving rise to a distinction between spacelike and timelike separation. General existence results are proved. It is shown in examples that minimizers need not be unique. Counter examples to compactness are discussed. The existence results are applied to variational principles formulated in indefinite inner product spaces.