Existence of Minimizers for Causal Variational Principles on Compact Subsets of Momentum Space in the Homogeneous Setting

TL;DR

This paper proves the existence of minimizers for causal variational principles on compact momentum space subsets using Prohorov's theorem, under certain constraints, in a homogeneous setting.

Contribution

It establishes the existence of minimizers for a class of measures in a new setting using a novel application of Prohorov's theorem.

Findings

Existence of minimizers proven under side conditions.

Method employs Prohorov's theorem for measure convergence.

Results applicable to homogeneous causal variational principles.

Abstract

We prove the existence of minimizers in the class of negative definite measures on compact subsets of momentum space in the homogeneous setting under several side conditions (constraints). The method is to employ Prohorov's theorem. Given a minimizing sequence of negative definite measures, we show that, under suitable side conditions, a unitarily equivalent subsequence thereof is bounded. By restricting attention to compact subsets, from Prohorov's theorem we deduce the existence of minimizers in the class of negative definite measures.