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Operator Deformations in Quantum Measurement Theory

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Abstract

In this paper, we develop a rigorous observable- and symmetry generator-related framework for quantum measurement theory by applying operator deformation techniques previously used in noncommutative quantum field theory. This enables the conventional observables (represented by unbounded operators) to play a role also in the more general setting. In addition, it gives a way of explicitly calculating the so-called instrument of the measurement process.

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References

  1. Aschieri, P., Dimitrijević, M., Kulish, P., Lizzi, F., Wess, J.: Noncommutative spacetimes: symmetries in noncommutative geometry and field theory. Lect. Notes Phys. 774 (2009)

  2. Bowes D., Hannabuss K.C.: Weyl quantization and star products. J. Geom. Phys. 22, 319–348 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. Buchholz D., Lechner G., Summers S.: Warped convolutions, Rieffel deformations and the construction of quantum field theories. Commun. Math. Phys. 304, 95–123 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Buchholz, D., Summers, S.: Warped convolutions: a novel tool in the construction of quantum field theories. In: Quantum Field Theory and Beyond, pp. 107–121. World Scientific, Singapore. arXiv:0806.0349v1 (2008)

  5. Busch P., Lahti P.J., Mittelstaedt P.: The Quantum Theory of Measurement – Second Revised Edition. Springer, Berlin (1996)

    Google Scholar 

  6. Davies E.B., Lewis J.T.: An operational approach to quantum probability. Commun. Math. Phys. 18, 239 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  7. Emch G.C.: Algebraic Methods in Statistical Mechanics and QFT. Wiley, New York (1972)

    Google Scholar 

  8. Estrada R., Garcia-Bondía , Várilly J.C.: On asymptotic expansions of twisted products. J. Math. Phys. 30, 2789 (1989)

  9. Fiore, G., Wess, J.: On “full” twisted Poincaré symmetry and QFT on Moyal–Weyl spaces. arXiv:hep-th/0701078v3 (2007)

  10. Fox, D.W.: Spectral measures and separation of variables. J. Res. Natl. Bureau Stand. Sect. B Math. Sci. 80B(3), 347–351 (1976)

    Google Scholar 

  11. Fredenhagen K.: On the modular structure of local algebras of observables. Commun. Math. Phys. 97, 79–89 (1985)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Haag R.: Local Quantum Physics. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  13. Harada, R., Ojima, I.: A unified scheme of measurement and amplification processes based on micro-macro duality – Stern–Gerlach experiment as a typical example. arXiv:0810.3400v1 (2008)

  14. Horváty, P.A.: Non-commutative mechanics in mathematical and in condensed matter physics. Symmetry Integrability Geom. Methods Appl. 2. arXiv:cond-mat/0609571v2 (2006)

  15. Lechner, G., Waldmann, S.: Strict deformation quantization of locally convex algebras and modules. arXiv:1109.5950 (2011)

  16. Much, A.: Quantum mechanical effects from deformation theory. arXiv:1307.2609 (2013)

  17. Ojima, I.: Micro-macro duality in quantum physics. arXiv:math-ph/0502038v1 (2005)

  18. Ojima, I.: Micro-macro duality and emergence of macroscopic levels. Talk at the International Symposium, QBIC (2007)

  19. Ozawa M.: Quantum measuring processes of continuous observables. J. Math. Phys. 25, 79 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  20. Podsedkowska H.: Correlations in a general theory of quantum measurement. Open Syst. Inf. Dyn. 14, 445–458 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Pool J.C.T.: Mathematical aspects of the Weyl correspondence. J. Math. Phys. 7, 66 (1966). doi:10.1063/1.1704817

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Rieffel, M.A.: Deformation quantization for actions of \({\mathbb{R}^d}\) . Mem. Am. Math. Soc. 106(506) (1993)

  23. Tanimoto, Y.: Construction of wedge-local nets of observables through Longo-Witten endomorphisms. arXiv:1107.2629v2 (2012)

  24. Von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955)

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Correspondence to Andreas Andersson.

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Andersson, A. Operator Deformations in Quantum Measurement Theory. Lett Math Phys 104, 415–430 (2014). https://doi.org/10.1007/s11005-013-0672-z

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  • DOI: https://doi.org/10.1007/s11005-013-0672-z

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