Identification of the theory of multidimensional orthogonal polynomials with the theory of symmetric interacting Fock spaces with finite dimensional one particle space

The identification mentioned in the title allows a formulation of the multidi mensional Favard Lemma different from the ones currently used in the literature and which exactly parallels the original one dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of Hermitean matri ces of the same dimension. Moreover, in this identification, the multidimensional extension of the compatibility condition for the positive Jacobi sequence becomes the condition which guarantees the existence of the creator in an interacting Fock space. The above result opens the way to the program of a purely algebraic clas sification of probability measures on $mathbb{R}^d$ with finite moments of any order. In this classification the usual Boson Fock space over $mathbb{C}^d$ is characterized by the fact that the positive Jacobi sequence is made up of identity matrices and the real Jacobi sequences are identically zero. The quantum decomposition of classical real valued random variables with all moments is one of the main ingredients in the proof.

$C^*$-non-linear second quantization

Recently, we have constructed a non{linear (polynomial) extension of the 1-mode Heisenberg group and the corresponding Fock and Weyl representations. The transition from the 1-mode case to the current algebra level, in which the operators are indexed by elements of an appropriate test function space (second quantization), can be done at Lie algebra level. A way to bypass the difficulties of constructing a (non trivial) Hilbert space representation is to try and construct directly a $C^*$-algebra rep- resentation and then to look for its Hilbert space representations. In usual (linear) quantization, this corresponds to the construction of the Weyl $C^*$-algebra. In this paper, we produce such a construction for the above mentioned polynomial extension of the Weyl $C^*$-algebra. The result of this construction is a factorizable system of local alge- bras localized on bounded Borel subsets of $mathbb{R}$ and obtained as induc- tive limit of tensor products of finite sets of copies of the one mode $C^*$-algebra. The $C^*$-embeddings of the inductive system require some non{trivial re{scaling of the generators of the algebras involved. These re{scalings are responsible of a $C^*$-analogue of the no-go theorems, first met at the level of Fock second quantization, namely the proof that the family of Fock states defined on the inductive family of $C^*$-algebras is projective only in the linear case (i.e. the case of the usual Weyl algebra). Thus the solution of the representa- tion problem at $C^*$-level does not automatically imply its solution at Hilbert space level.

Repeated quantum interactions Quantum Langevin equation and the low density limit

We consider a repeated quantum interaction model describing a small system $Hh_S$ in interaction with each one of the identical copies of the chain $bigotimes_{N^*}C^{n+1}$, modeling a heat bath, one after another during the same short time intervals $[0,h]$. We suppose that the repeated quantum interaction Hamiltonian is split in two parts: a free part and an interaction part with time scale of order $h$. After giving the GNS representation, we establish the relation between the time scale $h$ and the classical low density limit. We introduce a chemical potential $mu$ related to the time $h$ as follows: $h^2=e^{betamu}$. We further prove that the solution of the associated discrete evolution equation converges strongly, when $h$ tends to 0, to the unitary solution of a quantum Langevin equation directed by Poisson processes.