Talk by Valter Moretti

The existence of a real linear-space structure on the set of observables of a quantum system – i.e., the requirement that the linear combination of two generally non-commuting observables $ A,B $ is an observable as well – is a fundamental postulate of the quantum theory yet before introducing any structure of algebra.

However, it is by no means clear how to choose the measuring instrument of a general observable of the form $ aA + bB (a, b \in \mathbb{R}) $ if such measuring instruments are given for the addends observables $ A $ and $ B $ when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of $ f(aA + bB) $ out of the spectral measures of $ A $ and $ B $.

We present such a construction with a formula which is valid for general unbounded selfadjoint operators $ A $ and $ B $, whose spectral measures may not commute, and a wide class of functions $ f : \mathbb{R} \to \mathbb{C} $ . In the bounded case, we prove that the Jordan product of $ A $ and $ B $ (and suitably symmetrized polynomials of $ A $ and $ B $) can be constructed with the same procedure out of the spectral measures of $ A $ and $ B $. The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman-Kac formula.