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Roberts' proposal of a rigged Hilbert space Φ⊂G⊂Φ× for a certain class of quantum systems is reinvestigated and developed in order to exhibit various properties of this kind of rigged Hilbert spaces which might be of interest for the application of this formalism to special quantum systems. It is shown that on the basis of this proposal one also obtains a satisfactory solution for a rigged Hilbert space for composite systems. Another part is concerned with topological properties of the so-called eigenoperators γ(t) belonging to an A-eigen-integral decomposition of Φ with respect to a self-adjoint operator A on G. We derive a representation of γ(t) in terms of the generalized eigenvectors of A and in the same context give a rough topological characterization of these eigenvectors.

We give an explicit stochastic Hamiltonian model of discontinuous unitary evolution for quantum spontaneous jumps like in a system of atoms in quantum optics, or in a system of quantum particles that interacts singularly with "bubbles" which admit a continual counting observation. This model allows to watch a quantum trajectory in a photodetector or in a cloud chamber by spontaneous localisations of the momentums of the scattered photons or bubbles. Thus, the continuous reduction and spontaneous localization theory is obtained from a Hamiltonian singular interaction as a result of quantum filtering, i.e., a sequential time continuous conditioning of an input quantum process by the output measurement data. We show that in the case of indistinguishable particles or atoms the a posteriori dynamics is mixing, giving rise to an irreversible Boltzmann-type reduction equation. The latter coincides with the nonstochastic Schrödinger equation only in the mean field approximation, whereas the central limit yelds Gaussian mixing fluctuations described by a quantum state reduction equation of diffusive type.1

For an isolated macrosystem classical state parameters ζ(t) are introduced inside a quantum mechanical treatment. By a suitable mathematical representation of the actual preparation procedure in the time interval [T, t0] a statistical operator is constructed as a solution of the Liouville von Neumann equation, exhibiting at time t the state parameters ζ(t ′ ), t0 ≤ t ′ ≤ t, and preparation parameters related to times T ≤ t ′ ≤ t0. Relation with Zubarev's non-equilibrium statistical operator is discussed. A mechanism for memory loss is investigated and time evolution by a semigroup is obtained for a restricted set of relevant observables, slowly varying on a suitable time scale.

No abstract