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

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.

Keywords:

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