An operational construction of the sum of two non-commuting observables in quantum theory and related constructions

Nicolò Drago, Sonia Mazzucchi, Valter Moretti
September 24, 2019
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 the composed observable $aA+bB$ ($a,b\in \mathbb{R}$) 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 generally unbounded selfadjoint operators $A$ and $B$, whose spectral measures may not commute, and a wide class of functions $f: \mathbb{R} \to \mathbb{C}$. We prove that, in the bounded case the Jordan product 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.
open access link
%%% contains utf-8, see: http://inspirehep.net/info/faq/general#utf8 %%% add \usepackage[utf8]{inputenc} to your latex preamble @article{Drago:2019hpc, author = "Drago, Nicolò and Mazzucchi, Sonia and Moretti, Valter", title = "{An operational construction of the sum of two non-commuting observables in quantum theory and related constructions}", year = "2019", eprint = "1909.10974", archivePrefix = "arXiv", primaryClass = "math-ph", SLACcitation = "%%CITATION = ARXIV:1909.10974;%%" }

Keywords: 
Observable, spectral theory, Projection-valued Measure, Trotter Formula, Feynman integral