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. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.


Show 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 | 10.05.2024