1,721,019 research outputs found
Polynomial extensions of the Weyl -algebra
No full textWe introduce higher order (polynomial) extensions of the unique (up to isomorphisms) non trivial central extension of the Heisenberg algebra. Using the boson representation of the latter, we construct the corresponding polynomial analogue of the Weyl C*-algebra and use this result to deduce the explicit form of the composition law of the associated generalization of the 1-dimensional Heisenberg group. These results are used to calculate the vacuum characteristic func- tions as well as the moments of the observables in the Galilei algebra. The continuous extensions of these objects gives a new type of second quantization which even in the quadratic case is quite different from the quadratic Fock functor.ou
Low Density Limit and the Quantum Langevin equation for the Heat Bath
Full text linkWe consider a repeated quantum interaction model describing a small system \Hh_S in interaction with each one of the identical copies of the chain \bigotimes_{\N^*}\C^{n+1}, modeling a heat bath, one after another during the same short time intervals . We suppose that the repeated quantum interaction Hamiltonian is split in two parts: a free part and an interaction part with time scale of order . After giving the GNS representation, we establish the relation between the time scale and the classical low density limit. We introduce a chemical potential related to the time as follows: . We further prove that the solution of the associated discrete evolution equation converges strongly, when tends to 0, to the unitary solution of a quantum Langevin equation directed by Poisson processes.ou
Open Quantum Random Walks and Quantum Markov Chains
No full textIn the present paper we construct quantum Markov chains associated with open quantum random walks in the sense that the transition operator of a chain is determined by an open quantum random walk and the restriction of the chain to the commutative subalgebra coincides with the distribution Pρ of the walk. This sheds new light on some properties of the measure Pρ. For example, this measure can be considered as the distribution of some functions of a certain Markov process
Open quantum random walks, quantum Markov chains and recurrence
Full text linkIn the present paper, we construct QMCs (Quantum Markov Chains) associated with Open Quantum Random Walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC to the commutative subalgebra coincides with the distribution ρ of OQRW. This sheds new light on some properties of the measure ρ. As an example, we simply mention that the measure can be considered as a distribution of some functions of certain Markov processes. Furthermore, we study several properties of QMC and associated measures. A new notion of φ-recurrence of QMC is studied, and the relations between the concepts of recurrence introduced in this paper and the existing ones are established
Characterization of product probability measures on Rd in terms of their orthogonal polynomials
No full textIn paper [1] the d-dimensional analogue of the Jacobi parameters has been individuated in a pair of sequences ((a.n0),(Ω∼n)), where (a.n0) is a sequence of Hermitean matrices and Ω∼n(n N) a positive definite kernel with values in the linear operators on the n-th space of the orthogonal gradation. In this paper we prove that product measures on Rd are characterized by the property that the (a.n0) are diagonal and the (Ω∼n) quasidiagonal (see Definition 2 below) in the orthogonal polynomial basis
C*-Non-Linear Second Quantization
Full text linkWe construct an inductive system of C*-algebras each of which is isomorphic to a finite tensor product of copies of the one-mode n-th degree polynomial extension of the usual Weyl algebra constructed in our previous paper (Accardi and Dhahri in Open Syst Inf Dyn 22(3):1550001, 2015). We prove that the inductive limit C*-algebra is factorizable and has a natural localization given by a family of C*-sub-algebras each of which is localized on a bounded Borel subset of R. Finally, we prove that the corresponding family of Fock states, defined on the inductive family of C*-algebras, is projective if and only if n = 1. This is a weak form of the no-go theorems which emerge in the study of representations of current algebras over Lie algebras
Polynomial Extensions of the Weyl C-Algebra
No full textWe introduce higher order (polynomial) extensions of the unique (up to isomorphisms) nontrivial central extension of the Heisenberg algebra, which can be concretely realized as sub-Lie algebras of the polynomial algebra generated by the creation and annihilation operators in the Schrödinger representation. The simplest nontrivial of these extensions (the quadratic one) is isomorphic to the Galilei algebra, widely studied in quantum physics. By exponentiation of this representation we construct the corresponding polynomial analogue of the Weyl C-algebra and compute the polynomial Weyl relations. From this we deduce the explicit form of the composition law of the associated nonlinear extensions of the 1-dimensional Heisenberg group. The above results are used to calculate a simple explicit form of the vacuum characteristic functions of the nonlinear field operators of the Galilei algebra, as well as of their moments. The corresponding measures turn out to be an interpolation family between Gaussian and Meixner, in particular Gamma
On the multi-dimensional Favard lemma
No full textAbstract
We prove some properties of the Jacobi sequences and the creator operators, on the d commuting indeterminates polynomial algebra. Moreover, we prove that the matrix representations of the Jacobi sequences associated to product probability measures on
ℝ
d
{\mathbb{R}^{d}}
with finite moments of any order, are diagonal in the basis introduced by the tensor product of the orthogonal polynomials of the factor measures. Finally, we give a characterization of the atomic probability measures on
ℝ
d
{\mathbb{R}^{d}}
with finite number of atoms.</jats:p
Markovian properties of the spin-boson model
Full text linkISBN : 978-3-642-01762-9We systematically compare the Hamiltonian and Markovian approaches of quantum open system theory, in the case of the spin-boson model. We first give a complete proof of the weak coupling limit and we compute the Lindblad generator of this model. We study properties of the associated quantum master equation such as decoherence, detailed quantum balance and return to equilibrium at inverse temperature 0 < β ≤ ∞. We further study the associated quantum Langevin equation, its associated interaction Hamiltonian. We finally give a quantum repeated interaction model describing the spin‐boson system where the associated Markovian properties are satisfied without any assumption
Identification of the theory of orthogonal polynomials in d -indeterminates with the theory of 3 -diagonal symmetric interacting Fock spaces on Cd
Full text linkThe identification mentioned in the title allows a formulation of the multidimensional Favard lemma different from the ones currently used in the literature and which parallels the original 1-dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of positive definite kernels. The above result opens the way to the program of a purely algebraic classification of probability measures on Rd with moments of any order and more generally of states on the polynomial algebra on Rd. The quantum decomposition of classical real-valued random variables with all moments is one of the main ingredients in the proof
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