Quantum Probability and Related Topics

The following sections are included:

• GUIDELINES TO CONTRIBUTORS

• CONTENTS

The following sections are included:

• Scientific career

• Scientific work

• Acknowledgments

• List of publications

• References

In the present paper we begin to apply the technique of the stochastic limit in quantum field theory to quantum chromodynamics. We start from the Hamiltonian of the quantum Yang-Mills theory with volume and ultraviolet cutoffs. We introduce several approximations inspired by QED and we prove the existence of the weak coupling limit for an appropriate set of collective states. The stochastic differential equations for the limit wave operator is deduced. In the polynomial nonlinearities the difference between pointwise multiplications and tensor multiplications is underlined. This difference is at the root of the many (in fact 16) different Brownian motions that arise in the limit. After completion of the present paper a preprint by K. Wilson et al. [Wil ] appeared where a similar program is formulated. The techniques (and the language) of the two papers are however quite different.

An algebra characterization of deterministic and stochastic parallel transport, in terms of imprimitivity systems, is given. This leads to the notion of projective parallel transports, which seems not yet been considered in differential geometry. Given this characterization, the corresponding infinitesimal characterization can be achieved, both in the deterministic and in the stochastic case, by standard techniques.

Fermion stochastic flows on ℤ₂ - graded * - algebras A are obtained via conjugation by unitary operator valued processes which satisfy stochastic differential equations with unbounded coefficients driven by fermion noise. The case where A is a subalgebra of a C*–algebra is examined in some detail.

The following sections are included:

• Introduction

• Criteria for Conservativity

• Examples

• References

By using the Hudson-Parthasarathy quantum stochastic calculus, we explicitly compute the control process that minimizes the quantum stochastic analogue of the quadratic performance (cost) functional of classical stochastic control theory. The solution to the associated (and for the first time stochastic) Riccati equation is obtained.

Two different approaches for computing the ground state fluctuations of discrete quantum mean field systems are presented. Both exploit the characteristic property of mean field Hamiltonians, namely invariance under permutation of the sites. We first investigate directly the spectral decomposition of the local Hamiltonians. The second approach consists in studying "finite size corrections" to the theorems of Støormer -de Finetti that describe permutation invariant states of infinite systems. We investigate in particular the example of the strongly coupled BCS model in a transverse external field.

The following sections are included:

• Introduction

• AKNOWLEDGEMENT

• BIBLIOGRAPHY

The following sections are included:

• Introduction

• Preliminaries and notations

• Expected processes

• A structure theorem for strongly markovian processes

• Minimal strongly markovian processes

• Acknowledgement

• Bibliography

This paper is a review of path integral representations for semigroups H_ŕs {exp - t H_r/ħ}_{t≥0} where are relativistic quantum Hamiltonians. We consider three different cases: in the first one H_r is a relativistic Schrödinger operator, in the second is the Hamiltonian associated to Klein-Gordon equation and in the third is that coming from the Dirac one. The paths are trajectories of diffusion processes and, for the Dirac case, they differ from previous constructions based on jump Markov processes. All formulas describe one relativistic particle in an external static electromagnetic field therefore they are relativistic versions of the Feymann-Kac representation for Schrödinger semigroups.

We give a general scheme for constructing classical Markov processes as quantum flows in Fock space by conjugations with respect to a unitary operator valued adapted process satisfying a quantum stochastic differential equation. As an application we construct multidimensional homogeneous diffusion processes with strictly elliptic infinitesimal generator with smooth coefficients.

First we give a short review of the notion of quantum Markov chains. After introducing the boson Fock space in a way adapted to the language of counting measures we will define two operators and being very closely related to the Malliavin derivative and Skorohod integral. With the aid of these operators we define a transition expectation that relates to each initial normal state on the Fock space a homogenous quantum Markov chain. We will characterize the chain of position distributions related to such a quantum Markov chain. So in this paper we illustrate the more general theory of quantum Markov chain on a special class of examples, and we give certain relations to point process theory.

The paper is a survey of some recent results concerning the limit theorems in L₂-space over a von Neumann algebra with a faithful normal state. Different kinds of convergence in L₂ which are noncommutative analogues of the classical almost everywhere convergence are considered. Typical results are formulated and some of them with proofs.

The following sections are included:

• Introduction

• P-adic valued Gaussian distribution on the finite dimensional spaces

• The space L₂ (Q_p, γ_p (dx)) of p-adic valued functions square integrable with respect to the Gaussian distribution

• Gaussian distribution on

• Space

• Generalized p-adic valued white noise functionals

• Chaotic representation of square integrable and generalized functionals of white noise

• Acknowledgments

• References

Reversely to the usual procedure, we construct symmetric multiple Poisson integrals by the help of Charlier polynomials. So we get a chaos decomposition of the space of square integrable functionals of a general Poisson process and an isomorphism between the Poisson space and the symmetric Fock space.

We give a review of the properties of quantum characteristic exponents and show that this recently introduced notion can be considered as an indicator of chaos for non-commutative dynamical systems.

The following sections are included:

• Introduction

• Cumulants

• Generalized master equations for stochastic differential equations

• The Kraichnan equation

• The Random Frequency-Modulated Oscillator (Anderson-Kubo Model)

• A last remark

• Figure Captions

• Acknowledgements

• References

Based on Hida's white noise calculus, we develop a general theory of operators on white noise functionals. It is proved that every continuous operator admits a canonical expression in terms of creation and annihilation operators at points with kernel distributions (Fock expansion). During the discussion we establish an analytic characterization of operator symbols. Then we discuss the infinite dimensional rotation group, Laplacians and Kuo's Fourier transform within our framework. In particular, Laplacians (the Gross Laplacian Δ_G and the number operator N) and Kuo's Fourier transform are characterized by means of their invariance properties.

We prove a central limit theorem and an invariance principle under quite general assumptions and show how various known central limit theorems and invariance principles arise as special cases of our theorem. We also indicate that the limiting states have in many cases an inherent structure, which allows to represent them with the help of creation and annihilation operators on some kind of Fock space.

Let E(x), for x in a 2d-dimensional phase space, be an irreducible Weyl system, and a convex function with Φ(0) = 0. We discuss the maximum of with respect to unit vectors φ, ψ. According to a result of Lieb, when Φ(t) = t^p with 1 < p < ∞, the maximum is attained if and only if φ and ψ are coherent states with respect to the same quadratic form. We show that this statement is not correct for more general convex functions Φ.

We introduce the notion of a generalized CCR–algebra (GCCR–algebra for short), and we show that certain limit processes in the theory of quantum meanfield systems lead to an algebra of this type, the "algebra of fluctuation operators", in a natural manner. We are able to interpret the generating subset of unitaries as exponentials of fluctuation operators. Moreover we show that the quasifree time evolution on the algebra of fluctuation operators corresponding to a given time evolution of the one–particle system is itself a generalized limit dynamics similar to that one on the intensive observables. The GNS–representation of the algebra corresponding to a product state will give the CCR–algebra of the normal fluctuations of the meanfield system as introduced in [GVV1] and generalized in [GV], [GVV2].

The following sections are included:

• QP-PQ