"The Parametric Representation of an Open Quantum System" (Tesi di Dottorato in Fisica)

In this thesis work, we developed an exact approach, dubbed parametric representation, to describe any open quantum system. The description associates to the couple "open system-environment" a set of pure states, parametrized by a variable representing the environmental degrees of freedom, whose occurrence is ruled by a probability distribution defined over the space containing such variable. The parametric representation acquire a surplus value when the environmental degrees of freedom are mapped into a continuous variable, in particular when univocally obtained through an algorithm that starts from the identification of the relevant dynamical group for the environment to produce the set of generalized coherent states, therefore implying that such variable is a point in an accordingly defined environmental phase space. As a first outcome, the usage of coherent states yields the possibility to straightforwardly obtain the classical limit of the environment; this in turn means to define such a limit without affecting the quantum character of the open system: the formalism yields, from a composite system, a closed but not isolated one, where the parameters appearing in the local Hamiltonian are related to the environmental and original global system configuration. Moreover, the state of the open system assumes in parametric representation a natural interpretation in terms of vector fiber bundles, so that a relevant part of the work has been devoted to the presentation of various aspects of differential geometry necessary to understand the construction. Thanks to such premises, the parametric representation eventually establishes a strict relationship between the entanglement pertaining to the original composite state and the geometric phase proper to the derived semiclassical description, as extensively presented in the application of the formalism to the physical situation of the spin-star with frustration.