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Diagram technique for models with internal Lie-group dynamics

Nanosystems and strongly correlated systems can possess a more complicated internal Lie-group dynamics in comparison with the Lie-group dynamics of Bose and Fermi systems described by the Heisenberg algebra and superalgebra, respectively. In order to investigate properties of such quantum systems, we represent operators of quantum systems by differential operators over a commutative algebra of regular functionals and develop a new diagram technique on the basis of the expansion of the generating functional for the temperature Green functions. The differential representation makes it possible to generalize functional equations and the diagram technique for the case of quantum systems on topologically nontrivial manifolds by the substitution of the generating functional on a sheaf of function rings on a nontrivial manifold for the generating functional of a constant sheaf of functions. Nontrivial cohomologies of the manifold, on which the quantum system is acted, lead to the existence of additional excitations. We consider the self-consistent-field approximation and the approximation of effective Green functions and interactions. Poles of the matrix of effective interactions and Green functions determine quasi-particle excitations of the quantum system. For special cases of models the diagram expansion is simplified. In particular, if the internal dynamics is determined by the Heisenberg algebra (superalgebra), the diagram expansion reduces to Feynman's diagrams for Bose (Fermi) quantum systems. We consider the reduction of the developed diagram technique and excitations for the case of the spin system with an uniaxial anisotropy.