Title: The Two-fold Role of Observables in Classical and Quantum Kinematics,
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 Kähler
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.