Twisted duality of the CAR-Algebra

TL;DR

This paper provides a comprehensive proof of the twisted duality property for the CAR-Algebra in any Fock representation, utilizing modular theory and operator techniques, and offers explicit formulas for the modular operator.

Contribution

It presents a complete proof of twisted duality for the CAR-Algebra using modular theory and operator techniques, including explicit formulas for the modular operator.

Findings

Proof of twisted duality for CAR-Algebra in all Fock representations

Explicit formula for the modular operator's graph

Application of the formula to fermionic free nets and double cones

Abstract

We give a complete proof of the twisted duality property M(q)'= Z M(q^\perp) Z* of the (self-dual) CAR-Algebra in any Fock representation. The proof is based on the natural Halmos decomposition of the (reference) Hilbert space when two suitable closed subspaces have been distinguished. We use modular theory and techniques developed by Kato concerning pairs of projections in some essential steps of the proof. As a byproduct of the proof we obtain an explicit and simple formula for the graph of the modular operator. This formula can be also applied to fermionic free nets, hence giving a formula of the modular operator for any double cone.