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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].
Wavelets in mathematical physics: qoscillators
 J. Phys. A: Math. Gen
"... We construct representations of a qoscillator algebra by operators on Fock space on positive matrices. They emerge from a multiresolution scaling construction used in wavelet analysis. The representations of the Cuntz Algebra arising from this multiresolution analysis are contained as a special cas ..."
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Cited by 3 (3 self)
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We construct representations of a qoscillator algebra by operators on Fock space on positive matrices. They emerge from a multiresolution scaling construction used in wavelet analysis. The representations of the Cuntz Algebra arising from this multiresolution analysis are contained as a special case in the Fock Space construction. In this paper we establish a connection between multiresolution wavelet analysis on one hand and representation theory for operator on Hilbert spaces depending on a real parameter on the other. These operators arise from a multiresolution wavelet analysis based on Bessel functions. We wish to develop a framework for the study of creation operators on Hilbert space, satisfying simple identities, and allowing a Hopf algebra structure. Examples will include oscillator algebras coming from physical models. In the first section of the paper, we review the background and the motivation for the study of the qrelations, both as it relates to problems in 1 mathematics and in physics. On the mathematical side, the problems concern wavelet analysis and transform theory, especially the Mellin transform, and on the physics side, they relate to the quon gas of statistical mechanics. For the construction of the representations, we then turn to the twisted Fock space and the qoscillator algebra. Our approach is motivated by wavelet analysis, and it uses a certain loop group. Our main result is Theorem 6. 1
A FAMILY OF ∗ALGEBRAS ALLOWING WICK ORDERING: FOCK REPRESENTATIONS AND UNIVERSAL ENVELOPING C*algebras
, 2000
"... We consider an abstract Wick ordering as a family of relations on elements ai and define ∗algebras by these relations. The relations are given by a fixed operator T: h ⊗h → h ⊗h, where h is oneparticle space, and they naturally define both a ∗algebra and an innerproduct space HT, 〈 ·, · 〉T. If ..."
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We consider an abstract Wick ordering as a family of relations on elements ai and define ∗algebras by these relations. The relations are given by a fixed operator T: h ⊗h → h ⊗h, where h is oneparticle space, and they naturally define both a ∗algebra and an innerproduct space HT, 〈 ·, · 〉T. If a ∗ i denotes the adjoint, i.e., 〈aiϕ,ψ〉T = 〈ϕ,a ∗ i ψ〉T, then we identify when 〈 ·, · 〉T is positive semidefinite (the positivity question!). In the case of deformations of the CCRrelations (the qijCCR and the twisted CCR’s), we work out the universal C∗algebras A, and we prove that, in these cases, the Fock representations of the A’s are faithful.