Burkhard Kümmerer

q-Gaussian Processes: Non-commutative and Classical Aspects

Abstract: We examine, for −1<q<1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) – where the at fulfill the q-commutation relations for some covariance function – equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possesses a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB].

Markov dilations on W∗-algebras

Abstract In this paper a systematic study of Markov dilations is begun for completely positive operators on W ∗ -algebras which leave a faithful normal state invariant. It is shown that a minimal Markov dilation preserves important properties of the underlying completely positive operator. Afterwards some results are proved concerning the construction of dilations which lead to Markov dilations for large classes of operators. Finally some of the ideas developed here are applied to the study of a simple example over the 2 × 2 matrices.

Stochastic integration on the Cuntz algebra O

Abstract We develop a theory of non-commutative stochastic integration with respect to the creation and annihilation process on the full Fock space over L2( R ). Our theory largely parallels the theories of non-commutative stochastic Ito integration on Boson and Fermion Fock space as developed by R. Hudson and K. R. Parthasarathy. It provides the first example of a non-commutative stochastic calculus which does not depend on the quantum mechanical commutation or anticommutation relations, but it is based on the theory of reduced free products of C∗-algebras by D. Voiculescu. This theory shows that the creation and annihilation processes on the full Fock space over L2( R ), which generate the Cuntz algebra O∞, can be interpreted as a generalized Brownian motion. We should stress the fact that in contrast to the other theories of stochastic integration our integrals converge in the C∗-norm on O∞, i.e., uniformly rather than in some state-dependent strong operator topology or (non-commutative) L2-norm.

The essentially commutative dilations of dynamical semigroups onMn

AbstractFor identity and trace preserving one-parameter semigroups {Tt}t≧0 on then×n-matricesMn we obtain a complete description of their “essentially commutative” dilations, i.e., dilations, which can be constructed on a tensor product ofMn by a commutativeW*-algebra.We show that the existence of an essentially commutative dilation forTt is equivalent to the existence of a convolution semigroup of probability measures ρt on the group Aut(Mn) of automorphisms onMn such that

A SCATTERING THEORY FOR MARKOV CHAINS

T_t = \smallint _{Aut\left( {M_n } \right)} \alpha d\rho _t \left( \alpha \right)

Quantum Markov Processes and Applications in Physics

, and this condition is then characterised in terms of the generator ofTt. There is a one-to-one correspondence between essentially commutative Markov dilations, weak*-continuous convolution semigroups of probability measures and certain forms of the generator ofTt. In particular, certain dynamical semigroups which do not satisfy the detailed balance condition are shown to admit a dilation. This provides the first example of a dilation for such a semigroup.

A Markov dilation of a non-quasifree Bloch evolution

These notes give an introduction to some aspects of quantum Markov processes. Quantum Markov processes come into play whenever a mathematical description of irreversible time behaviour of quantum systems is aimed at. Indeed, there is hardly a book on quantum optics without having at least a chapter on quantum Markov processes. However, it is not always easy to recognize the basic concepts of probability theory in families of creation and annihilation operators on Fock space. Therefore, in these lecture notes much emphasis is put on explaining the intuition behind the mathematical machinery of classical and quantum probability. The lectures start with describing how probabilistic intuition is cast into the mathematical language of classical probability (Sects. 4.1-4.3). Later on, we show how this formulation can be extended such as to incorporate the Hilbert space formulation of quantum mechanics (Sects. 4.4,4.5). Quantum Markov processes are constructed and discussed in Sects. 4.6,4.7, and we add some further discussions and examples in Sects. 4.8-4.11.

Purfication of quantum trajectories

In the operator algebraic formulation of probability theory Markov processes typically appear as perturbations of Bernoulli processes. We develop a scattering theory for this situation. This theory applies to the isomorphism problem between Markov processes and Bernoulli shifts as well as to the description of open quantum systems.