Generalized probabilities in statistical theories

In this review article we present different formal frameworks for the description of generalized probabilities in statistical theories. We discuss the particular cases of probabilities appearing in classical and quantum mechanics, possible generalizations of the approaches of A. N. Kolmogorov and R. T. Cox to non-commutative models, and the approach to generalized probabilities based on convex sets

Extremal length in higher dimensions and complex systolic inequalities

Extremal length is a classical tool in 1-dimensional complex analysis for building conformal invariants. We propose a higher-dimensional generalization for complex manifolds and provide some ideas on how to estimate and calculate it. We also show how to formulate certain natural geometric inequalities concerning moduli spaces in terms of a complex analogue of the classical Riemannian notion of systole.Comment: Contains improved presentation of complex systolic inequalities and comparisons with special Lagrangian geometry. Various other minor changes. To appear in Journal of Geometric Analysi

Spin picture of the one-dimensional Hubbard model: Two-fluid structure and phase dynamics

We propose a scheme for investigating the quantum dynamics of interacting electron models by means of time-dependent variational principle and spin coherent states of space lattice operators. We apply such a scheme to the one-dimensional hubbard model, and solve the resulting equations in different regimes. In particular, we find that at low densities the dynamics is mapped into two coupled nonlinear Schroedinger equations, whereas near half-filling the model is described by two coupled Josephson junction arrays. Focusing then to the case in which only the phases of the spin variables are dynamically active, we examine a number of different solutions corresponding to the excitations of few macroscopic modes. Based on fixed point equation of the simpler among them, we show that the standard one-band ground state phase space is found.Comment: 10 pages, 1 figure, to appear on Phys. Rev.

Mirror symmetry and quantization of abelian varieties

The paper consists of two sections. The first section provides a new definition of mirror symmetry of abelian varieties making sense also over $p$-adic fields. The second section introduces and studies quantized theta-functions with two-sided multipliers, which are functions on non-commutative tori. This is an extension of an earlier work by the author. In the Introduction and in the Appendix the constructions of this paper are put into a wider context.Comment: 24 pp., amstex file, no figure