# Talk by Valter Moretti

On January 13th, 2021, Prof. Dr. Valter Moretti (University of Trento) gave a talk about "An operational construction of the sum of two non-commuting observables in quantum theory and related constructions" as part of the research seminar Analysis of the FernUniversität in Hagen.

## Abstract

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.

## Video of the talk

Liza Schonlau | 19.01.2023