A Generalization of the Functional Calculus of Observables and Notion of Joint Measurability to the Case of Non-commuting Observables

Abstract

For any pair of bounded observables $A$ and $B$ with pure point spectra, we construct an associated "joint observable" which gives rise to a notion of a joint (projective) measurement of $A$ and $B$, and which conforms to the intuition that one can measure non-commuting observables simultaneously, provided one is willing to give up arbitrary precision. As an application, we show how our notion of a joint observable naturally allows for a construction of a "functional calculus," so that for any pair of observables $A$ and $B$ as above, and any (Borel measurable) function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, a new "generalized observable" $f(A,B)$ is obtained. Moreover, we show that this new functional calculus has some rather remarkable properties.