Infinite Dimensional Analysis, Quantum Probability and Related Topics

Let , be the conditionally free product of unital free *-algebras , where ϕ_l, ψ_l are states on , l∈I. We construct a sequence of noncommutative probability spaces , m∈N, where and , m∈N, , and the states , ϕ_l are Boolean extensions of ϕ_l, ψ_l, l∈I, respectively. We define unital *-homomorphisms such that converges pointwise to *_l∈I(ϕ_l,ψ_l). Thus, the variables j^(m)(w), where w is a word in , converge in law to the conditionally free variables. The sequence of noncommutative probability spaces , where and Φ^(m) is the restriction of to , is called a hierarchy of freeness. Since all finite joint correlations for known examples of independence can be obtained from tensor products of appropriate *-algebras, this approach can be viewed as a unification of independence. Finally, we show how to make the m-fold free product into a cocommutative *-bialgebra associated with m-freeness.

A Hamiltonian of an interaction system between N-particles and a massive scalar field is considered. The Hamiltonian with an ultraviolet cutoff is defined as a self-adjoint operator acting in a Hilbert space. Renormalized Hamiltonians are defined by subtracting renormalization terms from the Hamiltonian. It is shown that N-body Schrödinger Hamiltonians can be derived from taking a weak coupling limit and removing the ultraviolet cutoff simultaneously for the renormalized Hamiltonians. In particular, in the case where the space dimension equals three, the Yukawa potential appears in the N-body Schrödinger Hamiltonian. It is also shown that, in the case where the space dimensions are one or two, infimum of the spectra of the renormalized Hamiltonians converge to those of the N-body Schrödinger Hamiltonians.

In this paper we shall discuss a C₀-group generated by the Lévy Laplacian and operators approximating that group in the strong topology of the space of generalized white noise functionals.

The problem of obtaining the quantum stochastic limit for a system interacting with a reservoir is investigated when both are considered as quantum fields. The pre-limit fields are as encountered in standard quantum field theory and in particular we do not resort to the simplifying responseless interaction for the reservoir field. In the separation of time scales between the unperturbed (slow) and interaction (fast) dynamics we find a new quantum stochastic noise emerging which describes multi-particle scatterings and has extended free statistics. We give explicit diagrammatic rules to compute the 2n-point correlators of the noise.

A nonlinear and stochastic analysis of free Bose field is established in the framework of white noise calculus. Wick algebra structure is introduced in the space of generalized operators generated by quantum white noise, some fundamental properties of the calculus based on the Wick algebra are investigated. As applications, quantum stochastic integrals and quantum stochastic differential equations are treated from the viewpoint of Wick calculus.

We prove that, in the stochastic limit of the Anderson model only the non-crossing diagrams survive for the transition amplitude from the first excited state of the free Hamiltonian to the first excited state of the interacting Hamiltonian. This confirms a conjecture of Migdal (1958) and Abrikosov, Gorkov, Dzyaloshinski (1975). From this we deduce a closed (nonlinear) Schwinger–Dyson type equation for the limit transition amplitude whose solution can be found and gives the explicit dependence of this amplitude on the momentum of the excited state.

It is well-known^3,5 that Brownian motion and Poisson process arise naturally from the canonical commutation relations (CCR) of free field operators in a boson Fock space. Eyre and Hudson² have recently shown how to construct fields of operators in a boson Fock space obeying super Lie commutation relations. We establish the essential self-adjointness of their real and imaginary parts on the domain ∊, the linear manifold generated by all the exponential (coherent) vectors and determine a family of Markov processes which they give rise to in a natural manner. These Markov processes yield examples of Evans–Hudson flows^3,5 and Azéma-like martingales.^1,4,6