Reviews in Mathematical Physics

We construct Euclidean random fields X over by convoluting generalized white noise F with some integral kernels G, as X=G*F. We study properties of Schwinger (or moment) functions of X. In particular, we give a general equivalent formulation of the cluster property in terms of truncated Schwinger functions which we then apply to the above fields. We present a partial negative result on the reflection positivity of convoluted generalized white noise.

Furthermore, by representing the kernels G_α of the pseudo-differential operators for α∈(0, 1) and m₀>0 as Laplace transforms we perform the analytic continuation of the (truncated) Schwinger functions of X=G_α*F, obtaining the corresponding (truncated) Wightman distributions on Minkowski space which satisfy the relativistic postulates on invariance, spectral property, locality and cluster property.

Finally we give some remarks on scattering theory for these models.

Motivated from the chemical potential theory, we study quantum statistical thermodynamics in AFC*-systems generalizing usual one-dimensional quantum lattice systems. Our systems are C*-algebras which have a localization of finite-dimensional subalgebras indexed by finite intervals of Z and an automorphism γ acting as a right shift on the localization. Model examples are supplied from derived towers (string algebras) for type II₁ factor-subfactor pairs. Given a (γ-invariant) interaction and a specific tracial state, we formulate the Gibbs conditions and the variational principle for (γ-invariant) states on , and investigate the relationship among these conditions and the KMS condition for the time evolution generated by the interaction. Special attention is paid to C*-systems of gauge invariance (typical model in the chemical potential theory) and to C*-systems considered as quantum random walks on discrete groups. The CNT-dynamical entropy for the shift automorphism γ is also discussed.

Object of the study is the operator H=H₀(h, µ)+V in L (R^d), d≥2, where H₀(h, μ) is the Schrödinger operator with a magnetic field of intensity μ≥0 and the Planck constant h∈(0, h₀]. The electric (real-valued) potential V=V(x) is assumed to be asymptotically homogeneous of order −β, β≥0 as x→0. One obtains asymptotic formulae with remainder estimates as h→0, μh≤C for the trace M_s=tr{ɸg_s(H)} where , s∈[0, 1]. Due to the condition μh≤C the leading term of M_s does not depend on μ. It depends on the relation between the parameters d, s and β. There are five regions, in which either leading terms or remainder estimates have different form. In one of these regions M_s admits a two-term asymptotics. In this case, for an asymptotically Coulomb potential the second term coincides with the well-known Scott correction term.