Model Theory of Hilbert Spaces with a Discrete Group Action

TL;DR

This paper investigates the model theory of infinite-dimensional Hilbert spaces expanded by discrete group actions, revealing different logical properties depending on whether the group is finite or countably infinite.

Contribution

It characterizes the model-theoretic properties of Hilbert spaces expanded by discrete groups, including finite and infinite cases, with results on categoricity, stability, and perturbation.

Findings

Finite group actions lead to theories with quantifier elimination and stability.

Countably infinite group actions result in theories that are categorical up to perturbations.

Model completeness implies stability up to perturbations.

Abstract

In this paper we study expansions of infinite dimensional Hilbert spaces with a unitary representation of a discrete countable group. When the group is finite, we prove the theory of the corresponding expansion, regardless if it is existentially closed, has quantifier elimination, is $\aleph_0$-categorical, $\aleph_0$-stable and SFB. On the other hand, when the group involved is countably infinite, the theory of the Hilbert space expanded by the representation of this group is $\aleph_0$-categorical up to perturbations. Additionally, when the expansion is model complete, we prove that it is $\aleph_0$-stable up to perturbations.