# Perturbation Theory for Critical Points of Causal Variational Principles

## TL;DR

This paper develops a perturbation theory for critical points of causal variational principles, analyzing measure perturbations and their expansions, with applications to causal fermion systems and continuum limits.

## Contribution

It introduces a novel perturbation framework for causal variational principles, extending to convex combinations and higher order expansions, with applications to causal fermion systems.

## Key findings

- Perturbation expansions for measures and their fluctuations are derived.
- The methods apply to the causal action principle in causal fermion systems.
- Continuum limit and microscopic mixing effects are recovered in specific cases.

## Abstract

The perturbation theory for critical points of causal variational principles is developed. We first analyze the class of perturbations obtained by multiplying the universal measure by a weight function and taking the push-forward under a diffeomorphism. Then the constructions are extended to convex combinations of such measures, leading to perturbation expansions for the mean and the fluctuation of the measure, both being coupled in higher order perturbation theory. It is explained how our methods and results apply to the causal action principle for causal fermion systems. It is shown how the perturbation expansion in the continuum limit and the effect of microscopic mixing are recovered in specific limiting cases.