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21
qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation ..."
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Cited by 64 (2 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian 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 qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Positive representations of general commutation relations allowing wick ordering
 FUNCT ANAL
, 1995
"... We consider the problem of representing in Hilbert space commutation relations of the form aia ∗ j = δij1 + ∑ kℓ T kℓ ij a ∗ ℓ ak, where the T kℓ ij are essentially arbitrary scalar coefficients. Examples comprise the qcanonical commutation relations introduced by Greenberg, Bozejko, and Speicher, ..."
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Cited by 29 (7 self)
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We consider the problem of representing in Hilbert space commutation relations of the form aia ∗ j = δij1 + ∑ kℓ T kℓ ij a ∗ ℓ ak, where the T kℓ ij are essentially arbitrary scalar coefficients. Examples comprise the qcanonical commutation relations introduced by Greenberg, Bozejko, and Speicher, and the twisted canonical (anti)commutation relations studied by Pusz and Woronowicz, as well as the quantum group SνU(2). Using these relations, any polynomial in the generators ai and their adjoints can uniquely be written in “Wick ordered form ” in which all starred generators are to the left of all unstarred ones. In this general framework we define the Fock representation, as well as coherent representations. We develop criteria for the natural scalar product in the associated representation spaces to be positive definite, and for the relations to have representations by bounded operators in a Hilbert space. We characterize the relations between the generators ai (not involving a ∗ i) which are compatible with the basic relations. The relations may also be interpreted as defining a noncommutative differential calculus. For generic coefficients T kℓ ij, however, all differential forms of degree 2 and higher vanish. We exhibit conditions for this not to be the case, and relate them to the ideal structure of the Wick algebra, and conditions of positivity. We show that the differential calculus is compatible with the involution iff the coefficients T define a representation of the braid group. This condition is also shown to imply improved bounds for the positivity of the Fock representation. Finally, we study the KMS states of the group of gauge transformations defined by aj ↦ → exp(it)aj.
Noncommutative Symmetric Functions III: Deformations Of Cauchy And Convolution Algebras
"... This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple qanalogue of the shuffle product, which has unexpec ..."
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Cited by 22 (8 self)
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This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple qanalogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in theoretical physics (the quon algebra).
qCanonical Commutation Relations and Stability of the Cuntz Algebra
, 1994
"... . We consider the qdeformed canonical commutation relations a i a j \Gamma q a j a i = ffi ij 1I, i; j = 1; : : : ; d, where d is an integer, and \Gamma1 ! q ! 1. We show the existence of a universal solution of these relations, realized in a C*algebra E q with the property that every ot ..."
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Cited by 21 (9 self)
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. We consider the qdeformed canonical commutation relations a i a j \Gamma q a j a i = ffi ij 1I, i; j = 1; : : : ; d, where d is an integer, and \Gamma1 ! q ! 1. We show the existence of a universal solution of these relations, realized in a C*algebra E q with the property that every other realization of the relations by bounded operators is a homomorphic image of the universal one. For q = 0 this algebra is the Cuntz algebra extended by an ideal isomorphic to the compact operators, also known as the CuntzToeplitz algebra. We show that for a general class of commutation relations of the form a i a j = \Gamma ij (a 1 ; : : : ; a d ) with \Gamma an invertible matrix the algebra of the universal solution exists and is equal to the CuntzToeplitz algebra. For the particular case of the qcanonical commutation relations this result applies for jqj ! p 2 \Gamma 1. Hence for these values E q is isomorphic to E 0 . The example a i a j \Gamma q a i a j = ffi ij 1I ...
An Obstruction for QDeformation of the Convolution Product
"... We consider two independent qGaussian random variables X 0 and X 1 and a function fl chosen in such a way that fl(X 0 ) and X 0 have the same distribution. For q 2 (0; 1) we find that at least the fourth moments of X 0 +X 1 and fl(X 0 )+X 1 are different. We conclude that no qdeformed convolutio ..."
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Cited by 9 (1 self)
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We consider two independent qGaussian random variables X 0 and X 1 and a function fl chosen in such a way that fl(X 0 ) and X 0 have the same distribution. For q 2 (0; 1) we find that at least the fourth moments of X 0 +X 1 and fl(X 0 )+X 1 are different. We conclude that no qdeformed convolution product can exist for functions of independent qGaussian random variables. 1995 PACS numbers: 02.50.Cw, 05.40.+j, 03.65.Db, 42.50.Lc 1991 MSC numbers: 81S25, 33D90, 81Q10 1 Introduction and Notation In 1991 Bozejko and Speicher introduced a deformation of Brownian motion by a parameter q 2 [\Gamma1; 1] (cf. [1, 2]). Their construction is based on a qdeformation, F q (H), of the full Fock space over a separable Hilbert space H. Their random variables are given by selfadjoint operators of the form X(f) := a(f) + a(f) ; f 2 H; where a(f) and a(f) are the annihilation and creation operators associated to f satisfying the qdeformed commutation relation, a(f)a(g) \Gamma qa(g...
Symmetric Hilbert spaces arising from species of structures
"... Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some `one particle space' K are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of symmetrization by building on Joyal's notion of a combinatorial species. An ..."
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Cited by 5 (2 self)
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Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some `one particle space' K are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of symmetrization by building on Joyal's notion of a combinatorial species. Any such species F gives rise to an endofunctor F of the category of Hilbert spaces with contractions mapping a Hilbert space K to a symmetric Hilbert space F (K) with the same symmetry as the species F . A general framework for annihilation and creation operators on these spaces is developed, and compared to the generalised Brownian motions of R. Speicher and M. Bo _ zejko. As a corollary we nd that the commutation relation a i a j a j a i = f(N) ij with Na i a i N = a i admits a realization on a symmetric Hilbert space whenever f has a power series with infinite radius of convergence and positive coefficients.
Extended SUSY quantum mechanics, intertwining operators and coherent states, Phys
 Lett. A, DOI
, 2008
"... We propose an extension of supersymmetric quantum mechanics which produces a family of isospectral hamiltonians. Our procedure slightly extends the idea of intertwining operators. Several examples of the construction are given. Further, we show how to build up vector coherent states of the GazeauKl ..."
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Cited by 4 (4 self)
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We propose an extension of supersymmetric quantum mechanics which produces a family of isospectral hamiltonians. Our procedure slightly extends the idea of intertwining operators. Several examples of the construction are given. Further, we show how to build up vector coherent states of the GazeauKlauder type associated to our hamiltonians. I Introduction and the method In some old papers the concept of supersymmetric quantum mechanics (SUSY qm) has been introduced and analyzed in many details, see [1, 2] and references therein, and [3] for a more recent paper with a rather extended bibliography. The original motivation was to get a deeper insight on SUSY in the elementary particles context. In our opinion, however, the most relevant
The energy operator for a model with a multiparametric infinite statistics, in preparation
"... secondquantized approach, for the multiparameter quon algebras: aia † † j − qija jai = δij, i, j ∈ I with (qij)i,j∈I any hermitian matrix of deformation parameters. We obtain an elegant formula for normally ordered (sometimes called Wickordered) series expansions of number operators (which determi ..."
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Cited by 4 (4 self)
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secondquantized approach, for the multiparameter quon algebras: aia † † j − qija jai = δij, i, j ∈ I with (qij)i,j∈I any hermitian matrix of deformation parameters. We obtain an elegant formula for normally ordered (sometimes called Wickordered) series expansions of number operators (which determine a free Hamiltonian). As a main result (see Theorem 1) we prove that the number operators are given, with respect to a basis formed by ”generalized Lie elements”, by certain normally ordered quadratic expressions with coefficients given precisely by the entries of the inverses of Gram matrices of multiparticle weight spaces. (This settles a conjecture of two of the authors (S.M and A.P), stated in [8]). These Gram matrices are hermitian generalizations of the Varchenko’s matrices, associated to a quantum (symmetric) bilinear form of diagonal arrangements of hyperplanes (see [12]). The solution of the inversion problem of such matrices in [9] (Theorem 2.2.17), leads to an effective formula for the number operators studied in this paper. The one parameter case, in the monomial basis, was studied by Zagier [15], Stanciu
Fourth Moment Theorem and qBrownian Chaos
 Comm. Math. Phys
, 2012
"... Abstract: In 2005, Nualart and Peccati [13] proved the socalled Fourth Moment Theorem asserting that, for a sequence of normalized multiple WienerItô integrals to converge to the standard Gaussian law, it is necessary and sufficient that its fourth moment tends to 3. A few years later, Kemp et al. ..."
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Cited by 4 (3 self)
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Abstract: In 2005, Nualart and Peccati [13] proved the socalled Fourth Moment Theorem asserting that, for a sequence of normalized multiple WienerItô integrals to converge to the standard Gaussian law, it is necessary and sufficient that its fourth moment tends to 3. A few years later, Kemp et al. [9] extended this theorem to a sequence of normalized multiple Wigner integrals, in the context of the free Brownian motion. The qBrownian motion, q ∈ (−1, 1], introduced by the physicists Frisch and Bourret [6] in 1970 and mathematically studied by Bo˙zejko and Speicher [2] in 1991, interpolates between the classical Brownian motion (q = 1) and the free Brownian motion (q = 0), and is one of the nicest examples of noncommutative processes. The question we shall solve in this paper is the following: what does the Fourth Moment Theorem become when dealing with a qBrownian motion?
THE KERNEL OF FOCK REPRESENTATIONS OF WICK ALGEBRAS WITH BRAIDED OPERATOR OF COEFFICIENTS
, 2001
"... It is shown that the kernel of the Fock representation of a certain Wick algebra with braided operator of coefficients T, T   ≤ 1, coincides with the largest quadratic Wick ideal. Improved conditions on the operator T for the Fock inner product to be strictly positive are given. 1. Introduction ..."
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Cited by 3 (1 self)
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It is shown that the kernel of the Fock representation of a certain Wick algebra with braided operator of coefficients T, T   ≤ 1, coincides with the largest quadratic Wick ideal. Improved conditions on the operator T for the Fock inner product to be strictly positive are given. 1. Introduction. The problem of positivity of the Fock space inner product is central in the study of the Fock representation of Wick algebras (see [2], [3], [5], [6]). The paper [6] presents several conditions on the coefficients of the Wick algebra for the Fock inner product to be positive. If the operator of coefficients of