Recent zbMATH articles in MSC 46L65https://www.zbmath.org/atom/cc/46L652021-04-16T16:22:00+00:00WerkzeugAn operational construction of the sum of two non-commuting observables in quantum theory and related constructions.https://www.zbmath.org/1456.810102021-04-16T16:22:00+00:00"Drago, NicolĂ˛"https://www.zbmath.org/authors/?q=ai:drago.nicolo"Mazzucchi, Sonia"https://www.zbmath.org/authors/?q=ai:mazzucchi.sonia"Moretti, Valter"https://www.zbmath.org/authors/?q=ai:moretti.valterSummary: 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 self-adjoint operators \(A\) and \(B\), whose spectral measures may not commute, and a wide class of functions \(f: \mathbb{R}\rightarrow\mathbb{C} \). 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.A product expansion for Toeplitz operators on the Fock space.https://www.zbmath.org/1456.470072021-04-16T16:22:00+00:00"Hagger, Raffael"https://www.zbmath.org/authors/?q=ai:hagger.raffaelSummary: We study the asymptotic expansion of the product of two Toeplitz operators on the Fock space. In comparison to earlier results, we require significantly fewer derivatives and get the expansion to arbitrary order. This, in particular, improves a result of Borthwick related to Toeplitz quantization [\textit{D. Borthwick}, Contemp. Math. 214, 23--37 (1998; Zbl 0903.58013)]. In addition, we derive an intertwining identity between the Berezin star product and the sharp product.