On the Structure of Minimizers of Causal Variational Principles in the Non-Compact and Equivariant Settings

TL;DR

This paper analyzes the structure of minimizers in causal variational principles within non-compact, symmetric settings, deriving Euler-Lagrange equations and examining second variations to understand their properties.

Contribution

It introduces a novel approach to derive Euler-Lagrange equations and analyze second variations for causal variational principles in complex, non-compact, and symmetric contexts.

Findings

Minimizers are supported on the intersection of a hyperplane and a level set of a homogeneous function.

The quadratic part of the action is represented by a positive semi-definite operator.

A modified Lagrange multiplier method is used for variational principles on convex sets.

Abstract

We derive the Euler-Lagrange equations for minimizers of causal variational principles in the non-compact setting with constraints, possibly prescribing symmetries. Considering first variations, we show that the minimizing measure is supported on the intersection of a hyperplane with a level set of a function which is homogeneous of degree two. Moreover, we perform second variations to obtain that the compact operator representing the quadratic part of the action is positive semi-definite. The key ingredient for the proof is a subtle adaptation of the Lagrange multiplier method to variational principles on convex sets.