Quantum stochastic calculus for the uniform measure and Boolean convolution
Quantum stochastic calculus on full Fock space
We present a new version of integration of time-adapted processes with respect to creation, annihilation and conservation processes on the full Fock space. Among the new features, in the first place, there is a new formulation of adaptedness which is both simpler and more general than the known ones. The new adaptedness allows for processes which are not restricted to be elements of some norm closure of the ∗-algebra which is generated by the basic creation processes.
Quantum stochastic convolution cocycles -algebraic and C*-algebraic
We summarise recent results concerning quantum stochastic convolution cocycles in two contexts-purely algebraic and C*-algebraic. In each case the class of cocycles arising as the solution of a quantum stochastic differential equation is characterised and the form taken by the stochastic generator of a *-homomorphic cocycle is described. Throughout the paper a common viewpoint on the algebraic and C*-algebraic situations is emphasised; the final section treats the unifying example of convolution...
Quantum stochastic convolution cocycles I
Quantum stochastic differential equations driven by free noises and dilations of markovian semigroups
Quantum stochastic dynamics. I.
Quantum stochastic processes arising from the strong resolvent limits of the Schrödinger evolution in Fock space
By using F. A. Berezin's canonical transformation method [5], we derive a nonadapted quantum stochastic differential equation (QSDE) as an equation for the strong limit of the family of unitary groups satisfying the Schrödinger equation with singularly degenerating Hamiltonians in Fock space. Stochastic differentials of QSDE generate a nonadapted associative Ito multiplication table, and the coefficients of these differentials satisfy the formal unitarity conditions of the Hudson-Parthasarathy type...
Quantum stopping times and quasi-left continuity
Quasi-probability distribution of nonclassical states in interacting Fock space
In this paper we study approximate quasi-probability distribution functions of nonclassical states such as incoherent states, Kerr states, squeezed states and k-photon coherent states in interacting Fock space.
q-White noise and non-adapted stochastic integral
The q-white noise is studied as the time derivative of the q-Brownian motion. As an application of the q-white noise, a non-adapted (non-commutative) stochastic integral with respect to the q-Brownian motion is constructed.
Realisation of a class of Markov processes through unitary evolutions in Fock space
Reduced and extended weak coupling limit
The main aim of our lectures is to give a pedagogical introduction to various mathematical formalisms used to describe open quantum systems: completely positive semigroups, dilations of semigroups, quantum Langevin dynamics and the so-called Pauli-Fierz Hamiltonians. We explain two kinds of the weak coupling limit. Both of them show that Hamiltonian dynamics of a small quantum system interacting with a large resevoir can be approximated by simpler dynamics. The better known reduced weak coupling...
Remarks on t-transformations of measures and convolutions
Restrictions of CP-semigroups to maximal commutative subalgebras
We give a necessary and sufficient criterion for a normal CP-map on a von Neumann algebra to admit a restriction to a maximal commutative subalgebra. We apply this result to give a far reaching generalization of Rebolledo's sufficient criterion for the Lindblad generator of a Markov semigroup on ℬ(G).
Series of iterated quantum stochastic integrals
Splitting the conservation process into creation and annihilation parts
The aim of this paper is the study of a non-commutative decomposition of the conservation process in quantum stochastic calculus. The probabilistic interpretation of this decomposition uses time changes, in contrast to the spatial shifts used in the interpretation of the creation and annihilation operators on Fock space.
Stationary Quantum Markov processes as solutions of stochastic differential equations
From the operator algebraic approach to stationary (quantum) Markov processes there has emerged an axiomatic definition of quantum white noise. The role of Brownian motion is played by an additive cocycle with respect to its time evolution. In this report we describe some recent work, showing that this general structure already allows a rich theory of stochastic integration and stochastic differential equations. In particular, if a quantum Markov process is represented by a unitary cocycle, we can...
Stochastic bosonization for a Fermi system
Stochastic bosonization for an interacting Fermi system
Superfluidity in the stochastic limit.