Cutting rules in nonequilibrium manybody theory
Quantum manybody theory is a tool for modeling the behaviour of systems
of many interacting quantum particles. It breaks transitions of the manyparticle
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 noninteracting 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 manybody system. Still, in practice quantum manybody theory
can be applied successfully to many realworld cases, since often the interactions
involved in a specific process are primarily of the simplest types.
The variety of different approximations that quantum manybody 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 nonnegative probabilities, has been previously developed for
system in equilibrium at zerotemperature [1, 2]. This recipe is based on diagrammatic
cuttingrules, 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 nonequilibrium state from an initial
equilibrium. Several general results related to working with complicated diagrams
are also derived.
...
Publisher
Jyväskylän yliopistoISBN
9789513992026ISSN Search the Publication Forum
24899003Contains publications
 Artikkeli I: Hyrkäs, M., Karlsson, D., & van Leeuwen, R. (2019). Contour calculus for manyparticle functions. Journal of Physics A: Mathematical and Theoretical, 52(21), Article 215303. DOI: 10.1088/17518121/ab165d . JYX: jyx.jyu.fi/handle/123456789/82879
 Artikkeli II: Hyrkäs, M., Karlsson, D., & van Leeuwen, R. (2019). Diagrammatic Expansion for Positive Spectral Functions in the SteadyState Limit. Physica Status Solidi. B: Basic Research, 256(7), 1800615. DOI: 10.1002/pssb.201800615. JYX: jyx.jyu.fi/handle/123456789/67347
 Artikkeli III: Hyrkäs, M., Karlsson, D., & van Leeuwen, R. (2022). Cutting rules and positivity in finite temperature manybody theory. Journal of Physics A : Mathematical and Theoretical, 55(33), Article 335301. DOI: 10.1088/17518121/ac802d . JYX: jyx.jyu.fi/handle/123456789/82804
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