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arXiv · 1407.7606

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

Richard DeJonghe·Kimberly Frey·Tom Imbo
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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.

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Richard DeJonghe, Kimberly Frey, Tom Imbo. 2014-07-29. A Generalization of the Functional Calculus of Observables and Notion of Joint Measurability to the Case of Non-commuting Observables. https://doi.org/10.1088/1751-8113/48/14/145302

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