Infinite Dimensional Analysis, Quantum Probability and Related Topics

We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of the Schrödinger equations in Fock space: the strong resolvent limit is unitarily equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitarily equivalent to the symmetric (or Stratonovich) form of QSDE.

We also prove that QSDE is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps in phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.

The central limit problem for algebraic probability spaces associated with the Haagerup states on the free group with countably many generators leads to a new form of statistical independence in which the singleton condition is not satisfied. This circumstance allows us to obtain nonsymmetric distributions from the central limit theorems deduced from this notion of independence. In the particular case of the Haagerup states, the role of the Gaussian law is played by the Ullman distribution. The limit process is explicitly realized on the finite temperature Boltzmannian Fock space. The role of entangled ergodic theorems in the proof of the central limit theorems is discussed.

Regarding the adjacency matrix of a graph as a random variable in the framework of algebraic or noncommutative probability, we discuss a central limit theorem in which the size of a graph grows in several patterns. Various limit distributions are observed for some Cayley graphs and some distance-regular graphs. To obtain the central limit theorem of this type, we make combinatorial analysis of mixed moments of noncommutative random variables on one hand, and asymptotic analysis of spectral structure of the graph on the other hand.

A new type of interacting Boltzmannian Fock space, emerging from the stochastic limit of the Anderson model, is investigated. We describe the structure of the space and the form, assumed in this case, by the principles of factorization and of total connection. Using these principles, the vacuum expectation of any product of creation and annihilation operators can be calculated. By means of these results, for any test function, a system of diffrence equations satisfied by the moments of the field operator and an integral equation satisfied by their generating function is deduced. In many interesting cases this equation is solved and the vacuum distribution function of the field operator (even its density) is explicitly determined. This evidentiates a new phenomenon which cannot take place in the usual Fock spaces (and did not appear in the simplest examples of interacting Fock spaces): by taking different test functions, the vacuum distribution of the field operator does not change only parametrically, but radically. In particular we find the semi-circle, the reciprocal-semi-circle (or Arcsine), the double-beta,…, and many other distributions.

The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite-dimensional superspaces, and construct super-analogs of the classical function spaces with a reproducing kernel — including the Bargmann–Fock representation — and of the Wiener–Segal representation. The latter representation requires the investigation of Wick ordering on Z₂-graded algebras. As application we derive a Mehler formula for the Ornstein–Uhlenbeck semigroup on the Fock space.

The non-parametric version of Information Geometry has been developed in recent years. The first basic result was the construction of the manifold structure on ℳ_μ, the maximal statistical models associated to an arbitrary measure μ (see Ref. 48). Using this construction we first show in this paper that the pretangent and the tangent bundles on ℳ_μ are the natural domains for the mixture connection and for its dual, the exponential connection. Second we show how to define a generalized Amari embedding A^Φ:ℳ_μ→S^Φ from the Exponential Statistical Manifold (ESM) ℳ_μ to the unit sphere S^Φ of an arbitrary Orlicz space L^Φ. Finally we show that, in the non-parametric case, the α-connections ∇^α(α∈(-1,1)) must be defined on a suitable α-bundle ℱ^α over ℳ_μ and that the bundle-connection pair (ℱ^α, ∇^α) is simply (isomorphic to) the pull-back of the Amari embedding A^α: ℳ_μ→S^2/1-α were the unit sphere S^2/1-αcL^2/1-α is equipped with the natural connection.

Free coherent states for a system with p degrees of freedom are defined. An isomorphism between the space free coherent states and the space of distributions on p-adic disk is constructed.