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Spectral problems for the Weyl-ordered form of operators
1. N. Bohr, Nature (London) 121, 245 (1928);
1.also in Atomic Theory and the Description of Nature I (Cambridge University Press, Cambridge, England, 1961).
5. D. A. Dubin, M. A. Hennings, and T. B. Smith, Mathematical Aspects of Weyl Quantization and Phase (World Scientific, Singapore, 2000).
6. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982).
7. E. A. Galapon, “Time in quantum mechanics,” Lecture Notes in Physics (Springer-Verlag, Heidelberg, Berlin, 2010), p. 25–63.
8.Significant progress has been stimulated by the realization that self-adjointness actually imposes an unnecessary restriction on the concept that in general quantum observables are actually positive operator-valued measures (POVMs)(Ref. 6) that can transform covariantly under certain group actions Ref. 10. As is evident, this perspective pertains strongly to the experiment. Instead of being compelled to manage ideal quantum observables, or the self-adjoint operators in the standard quantization program, one can introduce operators associated with feasible quantum observables whose definition is based on the use of measurement processes, the simplest concrete example probably being the so-called heterodyne detection of e.m. phases (see e.g.: J. H. Shapiro and S. S. Wagner, IEEE J. Quantum Electron. QE-20, 803 (1984);
8.M. Paris, G. Landolfi, and G. Soliani, J. Phys. A: Math. Theor. 40, F531 (2007)). Notwithstanding this, dealing with the complementarity of dynamical variables without reference to POVMs has also proved to be fruitful (see, e.g., the discussion in Ref. 7).
15.It is worth recalling that Kijowski's arrival distribution for a free particle and the Aharonov-Bohm time operator have been formally linked to each other by means of positive operator-valued measures in the unconfined case (see: R. Giannitrapani, Int. J. Theor. Phys. 36, 1575 (1997))
15.and as limit of the distribution obtained by restricting the position domain to a finite real interval (see: E. A. Galapon, F. Delgado, J. Gonzalo Muga, and I. Egusquiza, Phys. Rev. A 72, 042107 (2005)).
16. P. Henrici, Applied and Computational Complex Analysis, Power Series Integration Conformal Mapping Location of Zero (Wiley, New York, 1974).
17.Another way to proceed might be: with the coefficients Dα, k to be determined.
20. M. Braun, Differential Equations and their applications. An Introduction to Applied Mathematics (Springer-Verlag, Berlin, 1975).
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