Infinite Dimensional Analysis, Quantum Probability and Related Topics

The $qq$-bit is the $q$-deformation of the $q$-bit. It arises canonically from the quantum decomposition of Bernoulli random variables and the $q$-parameter has a natural probabilistic and physical interpretation as asymmetry index of the given random variable. The connection between a new type of $q$-deformation (generalizing the Hudson–Parthasarathy bosonization technique and different from the usual one) and the Azema martingale was established by Parthasarathy. Inspired by this result, Schürmann first introduced left and right $q$-JW-embeddings of $M_2$ ($2\times 2$ complex matrices) into the infinite tensor product ${\otimes }_{n\in \mathbb{N}}{M}_2$, proved central limit theorems (CLT) based on these embeddings in the context of ∗-bi-algebras and constructed a general theory of $q$-Levy processes on ∗-bi-algebras. For $q=-1$, left $q$-JW-embeddings define the Jordan–Wigner transformation, used to construct a tensor representation of the Fermi anti-commutation relations (bosonization). For $q=1$, they reduce to the usual tensor embeddings that were at the basis of the first quantum CLT due to von Waldenfels. The present paper is the first of a series of four in which we study these theorems in the tensor product context. We prove convergence of the CLT for all $q\in ℂ$. The moments of the limit random variable coincide with those found by Parthasarathy in the case $q\in [-1,+1)$. We prove that the space where the limit random variable is represented is not the Boson Fock space, as in Parthasarathy, but the monotone Fock space in the case $q=0$ and a non-trivial deformation of it for $q\ne -1,0,+1$. The main analytical tool in the proof is a non-trivial extension of a recently proved multi-dimensional, higher order Cesaro-type theorem. The present paper deals with the standard CLT, i.e. the limit is a single random variable. Paper¹ deals with the functional extension of this CLT, leading to a process. In paper² the left $q$-JW–embeddings are replaced by symmetric $q$-embeddings. The radical differences between the results of the present paper and those of² raise the problem to characterize those CLT for which the limit space provides the canonical decomposition of all the underlying classical random variables (see the Introduction, Lemma 4.5 and Sec. 5 of the present paper for the origin of this problem). This problem is solved in the paper³ for CLT associated to states satisfying a generalized Fock property. The states considered in this series have this property.

In this paper, we unify techniques of Pascal white noise analysis and harmonic analysis on configuration spaces establishing relations between the main structures of both ones. Fix a Random measure $\sigma$ on a Riemannian manifold $X$, we construct on the space of finite compound configuration space ${\mathrm{\Omega }}_0$ the so-called Lebesgue–Pascal measure ${\lambda }_{\widehat{\sigma }}$ and as a consequence we obtain the Pascal measure ${\mu }_{\widehat{\sigma }}$ on the compound configuration space $\mathrm{\Omega }$. Next, the natural realization of the symmetric Fock space over $L^2(\widehat{\sigma })$ as the space $L^2({\lambda }_{\widehat{\sigma }})$ leads to the unitary isomorphism ${{𝒥}}_{\lambda \mu }$ between the space $L^2({\lambda }_{\widehat{\sigma }})$ and $L^2({\mu }_{\widehat{\sigma }})$. Finally, in the first application we study some algebraic products, namely, the Borchers product on the Fock space, the Wick product on the Pascal space, and the ⋆-convolution on the Lebesgue–Pascal space and we prove that the Pascal white noise analysis and harmonic analysis are related through an equality of operators involving ${{𝒥}}_{\lambda \mu }$. The second application is devoted to solve the implementation problem.

A parabolic partial differential equation $u_t^{\prime }(t,x)=Lu(t,x)$ is considered, where $L$ is a linear second-order differential operator with time-independent (but dependent on $x$) coefficients. We assume that the spatial coordinate $x$ belongs to a finite- or infinite-dimensional real separable Hilbert space $H$. The aim of the paper is to prove a formula that expresses the solution of the Cauchy problem for this equation in terms of initial condition and coefficients of the operator $L$. Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup using a Feynman formula (i.e. we write it in the form of a limit of a multiple integral over $H$ as the multiplicity of the integral tends to infinity), which gives us a unique solution to the Cauchy problem in the uniform closure of the set of smooth cylindrical functions on $H$. This solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in $L$ is zero, we prove that the strongly continuous resolving semigroup indeed exists (which implies the existence of a unique solution to the Cauchy problem in the class mentioned above), and that the solution to the Cauchy problem depends continuously on the coefficients of the equation.

We consider a quantum system consisting of two subsystems (open system and environment). We have proved that the Wigner function of an open system is the integral of the Wigner function of a larger system with respect to coordinates of the phase space of the classical subsystem corresponding to the entourage.

Let $\gamma \equiv {\{{\gamma }_{ij}\}}_{(i,j)\in I}$, with $I\subseteq {\mathbb{Z}}_{+}\times {\mathbb{Z}}_{+}$ and $\overline{{\gamma }_{ij}}={\gamma }_{ji}$, be a given complex-valued sequence. The complex moment problem (respectively, the general complex moment problem) associated with $\gamma$ consists in determining necessary and sufficient conditions for the existence of a positive Borel measure (respectively, a charge) $\mu$ on $ℂ$ such that

In this paper, we investigate the notion of recursiveness in the two variable case. We obtain several useful results that we use to deduce new necessary and sufficient conditions for the truncated complex moment problem to admit a solution. In particular, we show that the general complex moment problem always has a solution. A concrete construction of the solution and an illustrating example are also given.

We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial commutation relations. We also discuss some geometric aspects of the quantum orthogonal groups associated with the mixture of classical and free independence discovered by Speicher and Weber. We show that these quantum groups are quantum symmetry groups on some quantum spaces of spherical vectors with partial commutation relations.

Set partitions closed under certain operations form a tensor category. They give rise to certain subgroups of the free orthogonal quantum group $O_n^{+}$, the so-called easy quantum groups, introduced by Banica and Speicher in 2009. This correspondence was generalized to two-colored set partitions, which, in addition, assign a black or white color to each point of a set. Globally colorized categories of partitions are those categories that are invariant with respect to arbitrary permutations of colors. This paper presents a classification of globally colorized categories. In addition, we show that the corresponding unitary quantum groups can be constructed from the orthogonal ones using tensor complexification.