A generalized Hubbard-Stratonovich transform is proposed and it is used to show that the thermodynamic partition function of a generic many-body quantum lattice model can be expressed in terms of traces of single-particle evolution operators governed by the dynamic Hubbard-Stratonovich fields. In the second part of the paper, we further extend the Lie-algebraic approach to a wide class of interacting many-particle lattice models. This represents quantum-to-classical correspondence. It is shown that deterministic dynamics of a closed classical dynamical system occurs as action of a symmetry group on a classical manifold and is driven by the same dual generators as in the corresponding quantum problem. These dual Schroedinger-Bloch equations are derived and analyzed on a number of specific examples. The evolution operator is uniquely determined by the time-dependent dual generators that satisfy a system of differential equations, dubbed here dual Schroedinger-Bloch equations, which represent more » a viable alternative to the conventional Schroedinger formulation. The first part of the paper utilizes a geometric Hilbert-space-invariant formulation of unitary time evolution, where a quantum Hamiltonian is viewed as a trajectory in an abstract Lie algebra, while the sought-after evolution operator is a trajectory in a dynamic group, generated by the algebra via exponentiation. We propose a Lie-algebraic duality approach to analyze nonequilibrium evolution of closed dynamical systems and thermodynamics of interacting quantum lattice models (formulated in terms of Hubbard-Stratonovich dynamical systems). Part II considers the additional structures of differential forms and finitely generated quantum Lie algebras - it is devoted to the construction of the Cartan calculus, based on an undeformed Cartan identity. We study this in detail for quasitriangular Hopf algebras, giving the determinant and orthogonality relation for the ``reflection`` matrix. This construction allows the translation of undeformed matrix expressions into their less obvious quantum algebraic counter parts. The pure braid group is introduced as the commutant of U and thereby bicovariant vector fields, casimirs and metrics. Using a generalized semi-direct product construction we combine the dual Hopf algebras A of functions and U of left-invariant vector fields into one fully bicovariant algebra of differential operators. The material is organized in two parts: Part I studies vector fields on quantum groups, emphasizing Hopf algebraic structures, but also introducing a ``quantum geometric`` construction. The construction of a tangent bundle on a more » quantum group manifold and an BRST type approach to quantum group gauge theory are given as further examples of applications. A Cartan calculus that combines functions, forms, Lie derivatives and inner derivations along general vector fields into one big algebra is constructed for quantum groups and then extended to quantum planes. A generalization of unitary time evolution is proposed and studied for a simple 2-level system, leading to non-conservation of microscopic entropy, a phenomenon new to quantum mechanics. The topic of this thesis is the development of a versatile and geometrically motivated differential calculus on non-commutative or quantum spaces, providing powerful but easy-to-use mathematical tools for applications in physics and related sciences.
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