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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].
Study of Gram matrices in Fock representation of multiparametric canonical commutation relations, extended Zagier’s conjecture, hyperplane arrangements and quantum groups
 Math. Commun
, 1996
"... Abstract.In this Colloqium Lecture (by one of the authors (D.S)) a thorough presentation of the authors research on the subjects,stated in the title,is given.By quite laborious mathematics it is explained how one can handle systems in which each Heisenberg commutation relation is deformed separately ..."
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Cited by 5 (1 self)
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Abstract.In this Colloqium Lecture (by one of the authors (D.S)) a thorough presentation of the authors research on the subjects,stated in the title,is given.By quite laborious mathematics it is explained how one can handle systems in which each Heisenberg commutation relation is deformed separately.For Hilbert space realizability a detailed determinant computations (extending Zagier’s one parametric formulas) are carried out.The inversion problem of the associated Gram matrices on Fock weight spaces is completely solved (Extended Zagier’s conjecture) and a counterexample to the original Zagier’s conjecture is presented in detail. Saˇzetak.U ovom Kolokviju (jednog od autora (D.S)) cjelovito su prikazana istraˇzivanja autora o temama formuliranima u naslovu.S poprilično matematike objaˇsnjeno je kako se mogu obradivati sustavi u kojima je svaka Heisenbergova komutacijska relacija deformirana odvojeno.Za realizabilnost na Hilbertovu prostoru provedeno je detaljno računanje determinanata (koje proˇsiruje Zagierove jednoparametarske formule).Problem inverzije pridruˇzenih Gramovih matrica na Fockovim teˇzinskim prostorima je potpuno rijeˇsen (Proˇsirena Zagierova hipoteza) i kontraprimjer za originalnu Zagierovu hipotezu je detaljno prikazan. Key words and phrases.Multiparametric canonical commutation relations,deformed partial derivatives,lattice of subdivisions,deformed regular representation,quantum bilinear form,Zagier’s conjecture. Ključne riječi i pojmovi.Multiparametarske kanonske komutacijske relacije,deformirane parcijalne derivacije,reˇsetka subdivizija,deformirana regularna reprezentacija, kvantna bilinearna forma,Zagierova hipoteza.
Generalised Brownian Motion and Second Quantisation
, 2000
"... A new approach to the generalised Brownian motion introduced by M. ..."
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Cited by 5 (1 self)
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A new approach to the generalised Brownian motion introduced by M.
Functors of White Noise Associated to Characters of the Infinite Symmetric Group
, 2001
"... The characters ; of the in nite symmetric group are extended to multiplicative positive de nite functions t; on pair partitions by using an explicit representation due to Versik and Kerov. The von Neumann algebra ; (K) generated by the elds !; (f) with f in an in nite dimensional real Hilbert space ..."
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Cited by 2 (0 self)
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The characters ; of the in nite symmetric group are extended to multiplicative positive de nite functions t; on pair partitions by using an explicit representation due to Versik and Kerov. The von Neumann algebra ; (K) generated by the elds !; (f) with f in an in nite dimensional real Hilbert space K is in nite and the vacuum vector is not separating. For a family tN depending on an integer N < f j): The algebras N (` 2 R (Z)) are type I1 factors. Functors of white noise N are constructed and proved to be nonequivalent for dierent values of N . 1
Symmetric Hilbert Spaces Arising from Species of Structures. http://arxiv.org/abs/mathph/0007005
"... 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. A ..."
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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 ΓF of the category of Hilbert spaces with contractions mapping a Hilbert space K to a symmetric Hilbert space ΓF(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 find that the commutation relation aia ∗ j −a∗j ai = f(N)δij 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. 1