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

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 → mathbb{R}, a new "generalized observable" f(A,B) is obtained. Moreover, we show that this new functional calculus has some rather remarkable properties.