Lahti, Pekka; Mączyński, Maciej; Ylinen, Kari The moment operators of phase space observables and their number margins. (English) Zbl 0924.47054 Rep. Math. Phys. 41, No. 3, 319-331 (1998). Summary: We show that the moment operators of any of the phase space observables \[ {\mathcal B}(\mathbb{C})\ni Z\mapsto A_{| s\rangle}(Z):= {1\over\pi} \int_Z D_z| s\rangle\langle s| D^*_z d\lambda(z)\in{\mathcal L}({\mathcal H}) \] associated with the number states \(| s\rangle\) are the powers of the lowering operator. We also determine all the moment operators of the number margins of these observables. They are integer valued polynomials of the number operator, the degree of the polynomial being the degree of the moment and the integer coefficients depending on \(| s\rangle\). In the Appendix we develop the necessary tools for integrating unbounded functions with respect to operator measures. Cited in 11 Documents MSC: 47N50 Applications of operator theory in the physical sciences 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 46G10 Vector-valued measures and integration 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.) 47A57 Linear operator methods in interpolation, moment and extension problems Keywords:moment operators; phase space observables; number states; integrating unbounded functions with respect to operator measures PDFBibTeX XMLCite \textit{P. Lahti} et al., Rep. Math. Phys. 41, No. 3, 319--331 (1998; Zbl 0924.47054) Full Text: DOI References: [1] Davies, E. B., Quantum Theory of Open Systems (1976), Academic Press: Academic Press London · Zbl 0388.46044 [2] Holevo, A. S., Probabilistic and Statistical Aspects of Quantum Theory (1982), North-Holland: North-Holland Amsterdam · Zbl 0497.46053 [3] Grabowski, M., New observables in quantum optics and entropy, (Busch, P.; Lahti, P.; Mittelstaedt, P., Symposium on the Foundations of Modern Physics 1993 (1993), World Scientific: World Scientific Singapore), 182-191 [4] Riordan, J., Combinatorial Identities (1968), Wiley: Wiley New York · Zbl 0194.00502 [5] Dunford, N.; Schwartz, J. T., Linear Operators, Part I: General Theory (1958), Interscience Publishers: Interscience Publishers New York [6] Berberian, S. K., Notes on Spectral Theory (1966), D. Van Nostrand Co., Inc: D. Van Nostrand Co., Inc Princeton · Zbl 0138.39104 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.