Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads

Preface

Contents

A survey on recent developments in mathematical theory of quantum particles interacting with a quantum field is presented.

We discuss the main results on Hudson–Parthasarathy quantum stochastic differential equations and with and operators on the initial space. We prove the existence and uniqueness theorems and study the conditions on the and for the solutions to be a process of isometries or coisometries. As an application we show the dilation of two classes of quantum Markov semigroup one arising in quantum optics and the other in the construction of a quantum diffusion process.

After giving short reviews on free independence, random matrices and free entropy, we introduce the free relative entropy for compactly supported probability measures on the real line based on a large deviation for the empirical eigenvalue distribution of a relevant random matrix, and show the perturbation theory for probability measures via free relative entropy. Next, we survey three interpolation/deformation theories of free group factors: interpolated free group factors introduced independently by Dykema and by Rădulescu, free Araki-Woods factors in Shlyakhtenko’s functor and von Neumann algebras in Bożejko and Speicher’s q-functor. We finally consider q-deformed Araki-Woods algebras combining Shlyakhtenko’s and Bożejko and Speicher’s functors.

Recent developments in quantum white noise calculus are outlined on the basis of various characterization theorems for operator symbols. A quantum stochastic integral and a quantum stochastic differential equation are generalized to a white noise integral and a white noise differential equation, respectively. Unique existence of a solution is shown for a certain class of white noise differential equations. Their regularity properties are examined by means of operator symbols. Higher powers of quantum white noises, which are pathological objects in traditional theories, emerge naturally and are studied within the quantum white noise calculus. Cauchy problems associated with the infinite dimensional Laplacians are discussed by means of transformation groups on white noise functions.

We extend to polynomials in several variables the Accardi–Bozejko canonical isomorphism between one-mode interacting Fock spaces and orthogonal polynomials in one variable. This gives a constructive rule to write down easily the quantum decomposition, as a sum of creation, annihilation and number operators, of an arbitrary vector valued random variable with moments of any order. In the multi-mode case not all interacting Fock spaces are canonically isomorphic to spaces of orthogonal polynomials. We characterize those which enjoy this property in terms of a sequence of quadratic commutation relations among finite dimensional matrices.

In their constructions of system of quantum stochastic differential equations, mathematicians and/or several physicists interpret that the function of random force operator is to preserve the canonical commutation relation in time, i.e., to secure the unitarity of time evolution generator even for dissipative systems. If this is the case, it means physically that the origin of dissipation is attributed to quantum non-commutativity (quantumness). The mechanism that the mathematician’s approaches rest on will be investigated from the unified view point of Non-Equilibrium Thermo Field Dynamics (NETFD) which is a canonical operator formalism of quantum systems in far-from-equilibrium state including the system of quantum stochastic equations.

It is argued that the conventional formalism of quantum mechanics is insufficient for the description of quantum events, such as spontaneous decays say, and the new experimental phenomena related to individual quantum measurements. Although they all have received an adequate mathematical treatment in the phenomenological information dynamics such as quantum stochastics of open systems with counting or continuous observations, the measurement problem in these models is simply lifted to the apparatus level. Recent development in infinite-dimensional analysis of boundary-value problems for quantum measurement theory has indicated a possibility for resolution of this interpretational crisis. It is done in the new framework of Eventum Mechanics, a time asymmetric event enhanced quantum mechanics, by divorcing the algebra of the dynamical generators and the algebra of the actually observable events. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of the past events with the potential future. This gives a dynamical solution, in the form of a boundary value problem for an infinite number of particles, of the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting with Schrödinger and Bohr. The information dynamics are obtained then by a conditioning as the dynamical filtering equations. We prove that the quantum stochastic model for the continuous-in-time measurements of the spontaneous events is equivalent to a Dirac type boundary-value problem for the second quantized input “offer waves from the future” in one extra dimension, and to a reduction of the algebra of the consistent histories of past events to an Abelian subalgebra for the “trajectories of the output particles.” This supports the wave-particle duality in the form of the thesis that everything in the future will be quantized waves, everything in the past will be the trajectories of recorded particles.

The notion of a tensor product with projections or with inclusions is defined. It is shown that the definition of stochastic independence relies on such a structure and that independence can be defined in an arbitrary category with a tensor product with inclusions or projections. In this context, the classifications of quantum stochastic independence by Muraki, Ben Ghorbal, and Schürmann become classifications of the tensor products with inclusions for the categories of algebraic probability spaces and non-unital algebraic probability spaces. The notion of a reduction of one independence to another is also introduced. As examples the reductions of Fermi independence and boolean, monotone, and anti-monotone independence to tensor independence are presented.

We study creation-annihilation processes associated with face maps on cellar complecies and develop a quantum probabilistic approach to the category of cellar complecies and cellar maps. We show that the join operation of cellar complecies induces a process on the symmetric toy Fock space, due to its commutativity. We investigate a chain complex defined by a set of binary trees, of which face maps define a large graph with multiple edges in contrast to the case of Cayley and distance regular graphs. As a result we obtain a one-mode interacting Fock space with Jacobi parameters ω_k = 2(2k + 1)(k + 2).

White noise theory has well developed in stochastic analysis, and it is now the suitable time to expect further developments in line with the original idea for the study of stochastic processes. To be more concrete, we take random complex systems and wish to analyze them by following the steps: Reduction ⇒ Synthesis ⇒ Analysis. To establish such a way of research systematically, we need to introduce new classes of random functions different from the Hilbert space which is familier for us. By doing so, it is illustrated that interesting questions can be discussed within our new setup.

All the characters of the infinite Weyl groups W_B∞ of type B_∞/C_∞ and W_D∞ of type D_∞ are given. Also all the characters of wreath product groups of arbitrary finite abelian groups with the infinite symmetric group are given. These groups contain inductive limits lim_n→∞ G(r, 1, n) of complex reflection groups. The proofs are outlined.

We report our recent study on noncommutative central limit theorem related to the symmetric groups and Young diagrams, which includes an extension of Kerov’s Gaussian limit for the Plancherel measure of the symmetric group [S. Kerov: C. R. Acad. Sci. Paris 316 (1993)]. Applying this result, we further investigate the central limit theorem for the adjacency operators on the symmetric groups [A. Hora: Commun. Math. Phys. 195 (1998)] from the viewpoint of convergence of mixed matrix elements of relevant operators.

No abstract received.

We explain here relationship between quasi-equivalence of (projective) unitary representations for infinite dimensional groups U(∞), SO(∞), and Sp(∞, R), and that of quasifree states of CAR and CCR algebras.

The Lévy Laplacian is formulated as an operator acting in a direct integral space of white noise functions. From Cauchy processes an infinite dimensional stochastic process is constructed, of which the generator is the Lévy Laplacian.

Systematic methods to treat mathematically and numerically non-commutative operator functions are discussed with focus on exponential operators. Quantum analysis is also briefly reviewed to explain a general strategy for constructing such systematic methods.

A quick review of a series of our recent works on free product actions, partly in collaborations with Dimitri Shlyakhtenko and with Fumio Hiai, is given. We also work out type III theoretic subfactor analysis on subfactors associated with free product actions of SU_q(2). This part can be read as a supplementary appendix to those works.

This paper will be devoted to the study of the s-free convolution introduced by Bożejko. The s-free Fock space is constructed, which provides a concrete example of general constructions of the interacting Fock spaces. The s-free Gaussian random variables will be given by the position operators, that is, the sum of the s-creation and the s-annihilation operators, on the s-free Fock space together with the vacuum expectation. Family of non-commutative random variables will be also constructed, which have the distributions given by the s-free Poisson measures.

Memorandum: The titles of the past RIMS workshops

Author Index