Cutting rules in non-equilibrium many-body theory

Quantum many-body theory is a tool for modeling the behaviour of systems of many interacting quantum particles. It breaks transitions of the many-particle system from one state to another down to the possible ways this transition can occur in terms of interactions between individual particles. These possible transitions are depicted using diagrams, and can be further broken down into diagrams depicting the basic interaction processes from which the full transition process is build of. Any set of possible interaction processes can then be chosen and applied as corrections to a non-interacting system, thus building an approximate model of the interacting system, that only allows transitions via the included processes. What can reasonably be included in this way is necessarily a tiny subset of the full complexity of the many-body system. Still, in practice quantum many-body theory can be applied successfully to many real-world cases, since often the interactions involved in a specific process are primarily of the simplest types. The variety of different approximations that quantum many-body theory allows raises the question of choosing the best option for a particular application. The choice of an approximation is important not only in order to include the interaction processes that contribute to the phenomenon under investigation, but also to retain relevant properties of the exact system. Certain approximations can, for example, violate conservation laws (of energy, particle number etc.). This thesis addresses in particular another important property that can be violated in approximations: the positivity of probabilities. A recipe to construct positive approximations, i.e. approximations that are guaranteed to give non-negative probabilities, has been previously developed for system in equilibrium at zero-temperature [1, 2]. This recipe is based on diagrammatic cutting-rules, which are used to cut diagrams depicting basic interaction processes further into so called scattering diagrams. Expressing an approximation in terms of scattering diagrams makes its positivity, or lack of it, apparent. Furthermore, this approach makes the physical content of the diagrams more clear, providing further aid in the choice of the correct approximation. In this thesis cutting rules that can be applied to systems in finite temperature are developed, and used to generalize the recipe for building positive approximations. This generalized recipe works not only for finite temperature systems, but also for systems that are perturbed to non-equilibrium state from an initial equilibrium. Several general results related to working with complicated diagrams are also derived.