Causal Variational Principles in the Infinite-Dimensional Setting: Existence of Minimizers

TL;DR

This paper develops a method to construct measures that minimize causal variational principles in infinite-dimensional, non-locally compact spaces, deriving Euler-Lagrange equations and exploring topological implications.

Contribution

It introduces a finite-dimensional approximation approach to establish existence of minimizers in infinite-dimensional Banach spaces under causal variational principles.

Findings

Existence of minimizers constructed via finite-dimensional approximations.

Derivation of Euler-Lagrange equations for these minimizers.

Extension to Lagrangians vanishing in entropy and topological analysis of spacetime.

Abstract

We provide a method for constructing (possibly non-trivial) measures on non-locally compact Polish subspaces of infinite-dimensional separable Banach spaces which, under suitable assumptions, are minimizers of causal variational principles in the non-locally compact setting. Moreover, for non-trivial minimizers the corresponding Euler-Lagrange equations are derived. The method is to exhaust the underlying Banach space by finite-dimensional subspaces and to prove existence of minimizers of the causal variational principle restricted to these finite-dimensional subsets of the Polish space under suitable assumptions on the Lagrangian. This gives rise to a corresponding sequence of minimizers. Restricting the resulting sequence to countably many compact subsets of the Polish space, by considering the resulting diagonal sequence we are able to construct a regular measure on the Borel algebra over the whole topological space. For continuous Lagrangians of bounded range it can be shown that, under suitable assumptions, the obtained measure is a (possibly non-trivial) minimizer under variations of compact support. Under additional assumptions, we prove that the constructed measure is a minimizer under variations of finite volume and solves the corresponding Euler-Lagrange equations. Afterwards, we extend our results to continuous Lagrangians vanishing in entropy. Finally, assuming that the obtained measure is locally finite, topological properties of spacetime are worked out and a connection to dimension theory is established.