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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 expec ..."
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Cited by 65 (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].
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
A Mixed QuantumClassical Central Limit Theorem
"... . A randomized qcentral or qcommutative limit theorem on a family of bialgebras with one complex parameter is shown. 1. Introduction In [Spe92] Speicher proved a noncommutative limit theorem, in which the commutation relations of the quantum random variables are given by classical f\Gamma1; 1g ..."
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. A randomized qcentral or qcommutative limit theorem on a family of bialgebras with one complex parameter is shown. 1. Introduction In [Spe92] Speicher proved a noncommutative limit theorem, in which the commutation relations of the quantum random variables are given by classical f\Gamma1; 1gvalued random variables, i.e. Bernoulli random variables. We will show a similar limit theorem, but placed in a bialgebra setting. Here it is the coproduct that depends on a sequence of i.i.d. complexvalued random variables. This also leads to qcommutation relations for the increments, but with less independence in the choice of the commutation factors, as we will see. We will also see that the limit distribution can be expressed as an exponential with respect to a (nonassociative!) averaged convolution. We begin by briefly recalling the results of Speicher[Spe92] and Sch¨urmann[Sch93]. In Secion 2 we introduce the bialgebras used to formulate the main theorem (Section 3), and the algebr...