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18
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 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].
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.
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 ...
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.
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.
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.
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
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
On C ∗algebras generated by some deformations of CAR relations
, 2008
"... We study the representation theory and enveloping C ∗algebras for Wick analogues of CAR and twisted CAR algebras. The realization of the C ∗algebras under consideration as algebras of continuous matrixfunctions satisfying certain boundary conditions is given. ..."
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We study the representation theory and enveloping C ∗algebras for Wick analogues of CAR and twisted CAR algebras. The realization of the C ∗algebras under consideration as algebras of continuous matrixfunctions satisfying certain boundary conditions is given.