Reviews in Mathematical Physics

The work is a brief review of articles by the author and his young colleagues to describe the classical statistical systems with the use of properties of integrals with respect to Poisson measure on configuration space. A variant of cell gas model which is an approximation of the continuous classical gas is proposed. Cell gas (CG) model for continuous classical system of interacting point particles is defined as follows. For a given partition of the space ℝ^d into an infinite set of mutually disjoint hypercubes with edges a, every point of a phase space of an infinite particle system is a configuration in which every hypercube contains no more than one particle. We present a structure of measurable sets of the configuration space and show that for strong superstable interaction the pressure and correlation functions of the system pointwise converge to the corresponding values of the conventional continuous system if the edges a of cubes tend to zero. We also define lattice gas (LG) model which approximates the CG model and thus provides a continuous transition LG model in the model of continuous gas.

The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic C*-algebras and to define, on each of these C*-algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included.

We present a rigorous and rather self-contained analysis of the Verdet constant in graphene-like materials. We apply the gauge-invariant magnetic perturbation theory to a nearest-neighbor tight-binding model and obtain a relatively simple and exactly computable formula for the Verdet constant, at all temperatures and all frequencies of sufficiently large absolute value. Moreover, for the standard nearest-neighbor tight-binding model of graphene we show that the transverse component of the conductivity tensor has an asymptotic Taylor expansion in the external magnetic field where all the coefficients of even powers are zero.