de Finetti-type theorems on quasi-local algebras and infinite Fermi tensor products

TL;DR

This paper establishes de Finetti-type theorems for quasi-local algebras and infinite Fermi tensor products, characterizing invariant states and their extreme points, with implications for the structure of states in quantum statistical mechanics.

Contribution

It introduces a de Finetti theorem for infinite Fermi tensor products and analyzes invariant states under local permutation actions on quasi-local algebras.

Findings

Invariant states are automatically even under local actions.

Extreme invariant states are strongly clustering and are infinite products of a single even state.

Infinite products of factorial even states remain factorial.

Abstract

Local actions of $\mathbb{P}_\mathbb{N}$, the group of finite permutations on $\mathbb{N}$, on quasi-local algebras are defined and proved to be $\mathbb{P}_\mathbb{N}$-abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of $C^*$-algebras, which include the CAR algebra, are then addressed as particular examples of quasi-local algebras acted upon $\mathbb{P}_\mathbb{N}$ in a natural way. Extreme invariant states are characterized as infinite products of a single even state, and a de Finetti theorem is established. Finally, infinite products of factorial even states are shown to be factorial by applying a twisted version of the tensor product commutation theorem, which is also derived here.