Quantum Probability and Related Topics

The following sections are included:

• CONTENTS

We show that the investigation of the weak coupling limit (WCL) of the quantum electro-magentic (QEM) field naturally leads to the study of the free quantum stochastic calculus on a Hilbert module. A new type of Fock space, the so–called interacting Fock space also comes out from the limit.

An example of a quantum stochastic flow on a manifold is constructed in closed form out of two commuting smooth vector fields and a single canonical Brownian motion. This flow is shown to be of the class whose structure maps do not all preserve the initial algebra.

The following sections are included:

• Introduction

• Notations

• Integral representation

• Maassen-Meyer kernels

• Application to Wiener space endomorphisms

• References

The following sections are included:

• Introduction

• Notations

• Characterizations of integral representations

• Characterisations of kernel representations

• References

The idea of measurements continuous in time in quantum mechanics [7,8] led to the introduction of the notion of convolution semigroup of instruments [9–12] and to the study of dilations of such semigroups obtained by the use of quantum stochastic calculus [13–16]. In [17] we discussed the connections between these notions and that of stochastic process with independent increments over a *-bialgebra [18–23]. In the present paper we want to show how our results can be formulated in terms of particular Evans–Hudson quantum flows (EH–flows) [24–33] and of perturbations of them [34].

To replace the false functional generalisation [1] of Hudson and Parthasarathy's Ito formula [2] we exposite a universal pseudo-Poisson one, recently discovered within the *-Ito non-commutative algebraic approach in [3]. Our formula unifies the classical Ito formulas for Wiener and Poisson cases, extends [4] the chain rule , to the case of non-commuting stochastically differentiable operators X_i(t), and also admits the non-adapted generalization [5]…

The following sections are included:

• Introduction

• Notations and Preliminaries

• From kernel to operator

• From operator to kernel

• References

The existence and uniqueness of solutions of a certain class of quantum stochastic differential equations in Boson Fock space is established with the use of Tichonov's fixed point theorem.

The stability of the zero solution of the quantum stochastic differential equation dX(t)=A₁(t)X(t)dB(t)+A₂(t)X(t)dt, in the presence of Fock space white noise B, is defined and studied with the use of the Hudson-Parthasarathy quantum stochastic calculus. An explicit formula is obtained for the solution of the associated initial value problem in the case when A₁, A₂ are compact self-adjoint adapted processes. A variation of parameters formula is obtained in the case of constant A₂.

The following sections are included:

• Introduction

• Notations and preliminaries

• Some notions of markovianity

• A Markov equivalence theorem

• Aknowledgement

• Bibliography

Let be a *-algebra and let be a second *-algebra which is also a 2-sided -*-module so that there are actions , satisfying

We expose Chebotarev's sufficient conditions for the conservativity of a quantum dynamical semigroup with some extensions and simplifications needed in the applications to quantum stochastic calculus.

We prove a characterization theorem for weakly differentiable (contractive, isometric and unitary) Markovian cocycles in the Boson Fock space analogue to Stone's theorem on strongly continuous groups of unitary operators.

In a preceding research [4], [5], we characterized resonance for pure states by means of Shannon entropy of their spectral measures. In the present article we extent this investigation to general states ρ as considered in Quantum Probability [9].

Together with resonance phenomena we investigate the complementary behaviour, that is, the characterization of non resonant states. Namely, we establish that resonant states are not smooth in the sense of Kato [6] and derive main properties of smooth states.

We introduce quantum stochastic integrals with respect to creation and annihilation operators on a general symmetric Fock space. Conditions for existence and boundedness of these integrals are given. It is shown that for local operators from a quasilocal algebra the quantum stochastic integrals exist. In the special case of non-anticipating kernel processes on an interval [0, T] these operators coincide with the integrals introduced by Maassen.

The concept of statistical invariant, introduced in the recent years in the field of Quantum Probability, is examined from the viewpoint of de Finetti's coherence principle. It is argued that, in the Kolmogorovian case, the statistical invariants are nothing but particular coherence conditions for conditional probability assessments: the subjective theory provides the most suitable framework to study them with full generality. Then, using some results on conditional probabilities, a few examples of statistical invariants are examined.

In the setup of semifinite von Neumann algebras, we first present an additivity formula for the trace of operator logarithm, and second prove an inequality for the relative entropy under the hyperfiniteness assumption.

The following sections are included:

• Introduction and Discussion

• Classical Theory and C*-algebraic Reformulation
  • The classical theory of topological entropy
  • Motivations for the C*-algebraic formulation
  • C*-algebraic reformulation of the classical theory

• Quantum Topological Entropy for C*—dynamical Systems
  • The general theory for nuclear C*-algebras
  • AF algebra theory and first applications

• Concluding Remarks (3.35)

• Acknowledgements

• References

We consider the operator Feynman–Kac formula in which the unperturbed semigroup is got by contracting a quantum stochastic evolution, and ask when is the perturbation cocycle of exponential integral form.

The following sections are included:

• Introduction

• Notations and Preliminaries

• Integral-sum operators

• Uniqueness of the kernel

• References

We report on a generalization of groups and supergroups relevant to particles of braid statistics. This is based on the notion of a braided group, for which the ring of co-ordinate functions is acted upon by a braid group action Ψ, generalizing Ψ = ±1 familiar for super-groups. We explain the physical motivation, the current status of the programme, and some problems that must be solved for a systematic approach to braided increment processes.

We define a class of discrete dynamical systems, called Quantum Boltzmann Maps, and we introduce an appropriate perturbative expansion of such maps, in order to construct a model for the dynamics of N-level atoms, N ≥ 2, in a confined radiation field. In particular, we use the increase of the entropy under the maps to prove the convergence of the dynamics to the appropriate equilibrium positions, in the relevant phase spaces.

We consider a class of discrete dynamical systems, called Boltzmann Maps, both in classical and quantum probability. We show that many physical models used in nonequilibrium statistical physics can be expressed in terms of Boltzmann Maps, and we discuss how new models can be constructed in this framework. In particular, the increase of the entropy under the Boltzmann Maps is used to prove the convergence of the dynamics to the appropriate equilibrium positions, in the relevant phase spaces.

The weak coupling limit for a quantum system interacting with a free Bose or Fermi gas as well as the low density limit for the similar models with bilinear interactions are considered. In both cases the choice of collective squeezed vectors as reference states leads to the proper unitary limit dynamics driven by the quantum Brownian or Poisson processes respectively. The proof of the associated limit theorem is given for the case of weak coupling limit while the low density limit is treated in a separate paper.

We introduce the notion of ‘admissible’ partitions, outline its importance for non-commutative probability theory and start the examination of the lattice of admissible partitions by calculating the corresponding Möbius function.

On the basis of non-commutative central limit theorems, starting from a dynamical system in equilibrium the corresponding macroscopic system of field fluctuations at momentum k is constructed. A mathematical mechanism for the appearance of a macroscopic quasi particle structure is developed in the BCS-model. Due to the spontaneous breaking of symmetry below the critical temperature the resulting dispersion relation shows an energy gap at zero momentum. The BCS-model is used as a toy model for explaining the general theory.