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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 36 (8 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.
A qdeformation of the Gauss distribution
 J. Math. Phys
, 2000
"... The qdeformed commutation relation aa #  qa # a = 11 for the harmonic oscillator is considered with q # [1, 1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a + a # ..."
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Cited by 14 (2 self)
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The qdeformed commutation relation aa #  qa # a = 11 for the harmonic oscillator is considered with q # [1, 1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a + a # in the vacuum state is explicitly calculated. This distribution is to be regarded as the natural q deformation of the Gaussian. 1995 PACS numbers: 02.50.Cw, 05.40.+j, 03.65.Db, 42.50.Lc 1991 MSC numbers: 81S25, 33D90, 81Q10 1 1
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...
Multiresolution wavelets analysis of integer scale Bessel functions
 J. Math. Phys
"... theory, special functions, recurrence relations, Hilbert space. AIP classification: mathematical methods in physics 02.30.f, 02.30.Mv, 02.30.Tb, 02.30.Uu, 02.50.Cw We identify multiresolution subspaces giving rise via Hankel transforms to Bessel functions. They emerge as orthogonal systems derived ..."
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Cited by 4 (4 self)
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theory, special functions, recurrence relations, Hilbert space. AIP classification: mathematical methods in physics 02.30.f, 02.30.Mv, 02.30.Tb, 02.30.Uu, 02.50.Cw We identify multiresolution subspaces giving rise via Hankel transforms to Bessel functions. They emerge as orthogonal systems derived from geometric Hilbertspace considerations, the same way the wavelet functions from a multiresolution scaling wavelet construction arise from a scale of Hilbert spaces. We study the theory of representations of the C ∗algebra Oν+1 arising from this multiresolution analysis. A connection with Markov chains and representations of Oν+1 is found. Projection valued measures arising from the multiresolution analysis give rise to a Markov trace for quantum groups SOq. 1 1
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