Remarks on the GNS Representation and the Geometry of Quantum States

TL;DR

This paper explores a geometric framework for non-relativistic quantum mechanics using GNS representations, revealing symplectic and Hermitian structures and introducing quantum bi-Hamiltonian systems.

Contribution

It introduces a geometric description of the algebraic approach to quantum mechanics via GNS representations, highlighting new Hamiltonian structures and quantum bi-Hamiltonian systems.

Findings

GNS representation provides symplectic and Hermitian realizations of quantum Poisson algebra

Different GNS representations lead to alternative Hamiltonian structures

Framework offers a natural setting for quantum bi-Hamiltonian systems

Abstract

It is shown how to introduce a geometric description of the algebraic approach to the non-relativistic quantum mechanics. It turns out that the GNS representation provides not only symplectic but also Hermitian realization of a `quantum Poisson algebra'. We discuss alternative Hamiltonian structures emerging out of different GNS representations which provide a natural setting for quantum bi-Hamiltonian systems.