The Two-fold Role of Observables in Classical and Quantum Kinematics

TL;DR

This paper explores the dual role of observables in classical and quantum mechanics, revealing how their algebraic and geometric structures underpin the fundamental differences and similarities between the two frameworks.

Contribution

It provides a unified algebraic and geometric analysis of observables, clarifying their dual roles and the key distinctions in their interrelation in classical versus quantum kinematics.

Findings

Algebraic structures reflect the dual role of observables.

Geometric reformulation shows classical and quantum observables behave similarly in transformation roles.

The difference lies in how the two roles of observables are interconnected.

Abstract

Observables have a dual nature in both classical and quantum kinematics: they are at the same time \emph{quantities}, allowing to separate states by means of their numerical values, and \emph{generators of transformations}, establishing relations between different states. In this work, we show how this two-fold role of observables constitutes a key feature in the conceptual analysis of classical and quantum kinematics, shedding a new light on the distinguishing feature of the quantum at the kinematical level. We first take a look at the algebraic description of both classical and quantum observables in terms of Jordan-Lie algebras and show how the two algebraic structures are the precise mathematical manifestation of the two-fold role of observables. Then, we turn to the geometric reformulation of quantum kinematics in terms of K\"ahler manifolds. A key achievement of this reformulation is to show that the two-fold role of observables is the constitutive ingredient defining what an observable is. Moreover, it points to the fact that, from the restricted point of view of the transformational role of observables, classical and quantum kinematics behave in exactly the same way. Finally, we present Landsman's general framework of Poisson spaces with transition probability, which highlights with unmatched clarity that the crucial difference between the two kinematics lies in the way the two roles of observables are related to each other.