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Spectral Resolutions and Quantum Observables

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Abstract

An n-dimensional quantum observable in quantum structures is a kind of a σ-homomorphism defined on the Borel σ-algebra of \(\mathbb R^{n}\) with values in a monotone σ-complete effect algebra or in a σ-complete MV-algebra. It defines an n-dimensional spectral resolution that is a mapping from \(\mathbb R^{n}\) into the quantum structure which is a monotone, left-continuous mapping with non-negative increments and which is going to 0 if one variable goes to \(-\infty \) and it goes to 1 if all variables go to \(+\infty \). The basic question is to show when an n-dimensional spectral resolution entails an n-dimensional quantum observable. We show cases when this is possible and we apply the result to existence of three different kinds of joint observables.

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Acknowledgements

The authors are very indebted to anonymous referees for their careful reading and suggestions which helped us to improve the presentation of the paper.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73, Bratislava, Slovakia

    Anatolij Dvurečenskij

  2. Depart. Algebra Geom., Palacký University, 17. listopadu 12, CZ-771 46, Olomouc, Czech Republic

    Anatolij Dvurečenskij & Dominik Lachman

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  1. Anatolij Dvurečenskij
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  2. Dominik Lachman
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Correspondence to Anatolij Dvurečenskij.

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The paper has been supported by the grant of the Slovak Research and Development Agency under contract APVV-16-0073 and the grant VEGA No. 2/0069/16 SAV (the first author), and by grant CZ.02.2.69/0.0/0.0/16-027/0008482 SPP 8197200115 (the second one)

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Dvurečenskij, A., Lachman, D. Spectral Resolutions and Quantum Observables. Int J Theor Phys 59, 2362–2383 (2020). https://doi.org/10.1007/s10773-020-04507-z

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