Infinite Dimensional Analysis, Quantum Probability and Related Topics

Several possible generalizations of the classical notion of markovianity are given for states defined on a von Neumann algebras generated on a triple of subalgebras. Their mutual relation is discussed in the particular case in which they mutually commute, and the generalization of the classical; time reversal theorem is proved. A structure theorem for a class of Markov chains is also proved.

A generalization of Muraki's notion of monotonic independence onto the case of partially ordered index set is given: algebras indexed by chains are monotonically independent, and algebras indexed by non-comparable elements are boolean independent. Examples of central limit theorem are shown in two cases. For the integral-points lattices ℕ^d the moments of the limit measure are related to the combinatorics of the finite heap-ordered labelled rooted trees (if d = 2). For the integral-points lattice ℕ × ℤ^d in Minkowski spacetime the limit measure is given by the recurrence of it's moments, which, for the case d = 1 is related to the inverse error function. Various formulas for computing mixed moments are shown to be related to the boolean-monotonic non-crossing pair partitions.

We consider a scaling limit of the Hamiltonian of the generalized spin-boson (GSB) model which is an abstract quantum field theoretical model of particles interacting with a Bose field. Applying it to a Hamiltonian of the field of the nuclear force with isospin, we obtain an effective potential of the interaction between nucleons. Also, we discuss an application to a Hamiltonian of a lattice spin system interacting with a Bose field and obtain a spin–spin interaction in the vacuum of the Bose field. An interaction model between a Fermi field and a Bose field yields an interaction in the vacuum of the Bose field.

Let U be a unitary operator acting on the Hilbert space , and α: {1, …, m} ↦ {1, …, k} a partition of the set {1, …, m}. We show that the ergodic average

A new Gel'fand triple exp(S)₁ ⊆ (L)² ⊆ exp(S)_-1 is constructed as extension of the known Kondratiev one (S)₁ ⊆ (L)² ⊆ (S)_-1. Expansion theorems for generalized stochastic processes considered as elements of the spaces and are derived. This series expansion is used for solving a class of evolution stochastic differential equations. The Wick product is developed on the spaces exp(S)_-1, and . The series expansion of generalized stochastic processes is used for solving a class of nonlinear stochastic differential equations by means of Wick products.

We study white noise Heisenberg equations giving rise to flows which are *-automorphisms of the observable algebra, but not necessarily inner automorphisms. We prove that the causally normally ordered form of these white noise Heisenberg equations are equivalent to Evans–Hudson flows. This gives in particular, the microscopic structure of the maps defining these flows, in terms of the original white noise derivations.