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We review some of the difficulties previously encountered in defining the phase operator for finite quantum systems. We then propose the Wigner-Weyl quantization of the angle function q> on phase space as the phase operator, and derive a closed expression for its matrix elements with respect to the Hermite functions. We also determine the quantization of e l

W~ prese!1t a ne~ ~es~ription of time translations in the C·-algebraic formulation of statistical me-chanIcs. This descflI?tlO.n IS based on weaker assumptions than the hitherto accepted ones, due to Haag, Hugenh?!tz, a~d Wmnmk (HH~). [Commun. Math. Phys. 5, 215 (1967)]. It is shown that these weaker a.ssumptl(~ns shlliead to the prmclp~l results ~f HHW for Gibbs states and, further, that our assumptions, unlike those of HHW, are valid for the Ideal Bose gas and strong-coupling BCS models.

We consider various systematic ways of defining unbounded operator valued integrals of complex functions with respect to (mostly) positive operator measures and positive sesquilinear form measures, and investigate their relationships to each other in view of the extension theory of symmetric operators. We demonstrate the associated mathematical subtleties with a physically relevant example involving moment operators of the momentum observable of a particle confined to move on a bounded intervalPeer reviewe

A convolution algebra is a normal sequence space ^ that is closed under the convolution product. Let E be a nuclear Frechet locally convex space, and let P={_p r