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The moment operators of phase space observables and their number margins. (English) Zbl 0924.47054

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.

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
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References:

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