After briefly reviewing some basic aspects of quantum
probability theory, especially questions surrounding a
celebrated theorem of Gleason, we turn to analogues of
quantum probabilities studied in symplectic geometry
known as quasi-states, which are functions on Lie algebras
which are linear on abelian sub-algebras.

A prototypical (and well-studied) example is the so called
Maslov index. We will discuss the existence of non-linear quasi-states on
various families of finite-dimensional Lie algebras. No fluency
in the language of Lie algebras will be assumed.

Joint work with Tobias Hartnick (Technion).