Abstract:
Cirelli, Mani\`{a} and Pizzocchero generalized quantum mechanics by K\"{a}hler geometry. Furthermore they proved that any unital C$^{*}$-algebra is represented as a function algebra on the set of pure states with a noncommutative $*$-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$. The space ${\cal P}({\cal H})$ is an infinite dimensional K\"{a}hler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general K\"{a}hler manifold of nonzero constant holomorphic sectional curvature $c$. The Hilbert ball $B_{{\cal H}}$ is defined by the open unit ball in ${\cal H}$ and it is a K\"{a}hler manifold with $c<0$. We introduce the quantum mechanics on $B_{{\cal H}}$. As an application, we show the structure of the noncommutative function algebra on $B_{{\cal H}}$.

Abstract:
In order to enlarge the present arsenal of semiclassical toools we explicitly obtain here the Husimi distributions and Wehrl entropy within the context of deformed algebras built up on the basis of a new family of q-deformed coherent states, those of Quesne [J. Phys. A 35, 9213 (2002)]. We introduce also a generalization of the Wehrl entropy constructed with escort distributions. The two generalizations are investigated with emphasis on i) their behavior as a function of temperature and ii) the results obtained when the deformation-parameter tends to unity.

Abstract:
We initiate a mathematically rigorous study of Klein-Gordon position operators in single-particle relativistic quantum mechanics. Although not self-adjoint, these operators have real spectrum and enjoy a limited form of spectral decomposition. The associated C*-algebras are identifiable as crossed products. We also introduce a variety of non self-adjoint operator algebras associated with the Klein-Gordon position representation; these algebras are commutative and continuously (but not homeomorphically) embeddable in corresponding function algebras. Several open problems are indicated.

Abstract:
We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore can be viewed as identical. In particular, we are able to give an alternative and much simpler geometric proof of a result of E. Date, M. Jimbo, A. Kuniba, T. Miwa and M. Okado on the construction of bases of affine Lie algebra representations. At the same time, we give a simple parametrization of the irreducible components of Nakajima quiver varieties associated to infinite and cyclic quivers. We also define new varieties whose irreducible components are in one-to-one correspondence with bases of the highest weight representations of affine gl_{n+1}.

Abstract:
The probability operator for a generic non-equilibrium quantum system is derived. The corresponding stochastic, dissipative Schr\"odinger equation is also given. The dissipative and stochastic propagators are linked by the fluctuation-dissipation theorem that is derived from the unitary condition on the time propagator. The dissipative propagator is derived from thermodynamic force and entropy fluctuation operators that are in general non-linear.

Abstract:
The Jordan algebra structure of the bounded real quantum observables was recognized already in the early days of quantum mechanics. While there are plausible reasons for most parts of this structure, the existence of the distributive nonassociative multiplication operation is hard to justify from a physical or statistical point of view. Considering the non-Boolean extension of classical probabilities, presented in a recent paper, it is shown in this paper that such a multiplication operation can be derived from certain properties of the conditional probabilities and the observables, i.e., from postulates with a clear statistical interpretation. The well-known close relation between Jordan operator algebras and C*-algebras then provides the connection to the quantum-mechanical Hilbert space formalism, thus resulting in a novel axiomatic approach to general quantum mechanics that includes the types II and III von Neumann algebras.

Abstract:
These are the lecture notes for quantum and statistical mechanics courses that are given by DC at Ben-Gurion University. They are complementary to "Lecture Notes in Quantum Mechanics" [arXiv: quant-ph/0605180]. Some additional topics are covered, including: introduction to master equations; non-equilibrium processes; fluctuation theorems; linear response theory; adiabatic transport; the Kubo formalism; and the scattering approach to mesoscopics.