scholarly journals From quantum Bernoulli process to creation and annihilation operators on interacting q-Fock space

1998 ◽  
Vol 24 (1) ◽  
pp. 149-167 ◽  
Author(s):  
M. De GIOSA ◽  
Y. G. Lu
Keyword(s):  
Fock Space ◽  
Bernoulli Process ◽  
Creation And Annihilation Operators

scholarly journals κ-DEFORMED STATISTICS AND CLASSICAL FOUR-MOMENTUM ADDITION LAW

2008 ◽  
Vol 23 (09) ◽  
pp. 653-665 ◽  
Author(s):  
MARCIN DASZKIEWICZ ◽  
JERZY LUKIERSKI ◽  
MARIUSZ WORONOWICZ
Keyword(s):  
Minkowski Space ◽  
Fock Space ◽  
Scalar Fields ◽  
Cross Product ◽  
Creation And Annihilation Operators ◽  
Multiplication Rule ◽  
Symmetry Properties ◽  
Deformed Algebra ◽  
Free Scalar

We consider κ-deformed relativistic symmetries described algebraically by modified Majid–Ruegg bi-cross-product basis and investigate the quantization of field oscillators for the κ-deformed free scalar fields on κ-Minkowski space. By modification of standard multiplication rule, we postulate the κ-deformed algebra of bosonic creation and annihilation operators. Our algebra permits one to define the n-particle states with classical addition law for the four-momentum in a way which is not in contradiction with the nonsymmetric quantum four-momentum co-product. We introduce κ-deformed Fock space generated by our κ-deformed oscillators which satisfy the standard algebraic relations with modified κ-multiplication rule. We show that such a κ-deformed bosonic Fock space is endowed with the conventional bosonic symmetry properties. Finally we discuss the role of κ-deformed algebra of oscillators in field-theoretic noncommutative framework.

scholarly journals Completely positive maps for imprimitive complex reflection groups

2021 ◽  
Vol 13 (2) ◽  
pp. 452-459
Author(s):  
H. Randriamaro
Keyword(s):  
Hilbert Space ◽  
Coxeter Groups ◽  
Fock Space ◽  
Inner Product ◽  
Reflection Groups ◽  
Bounded Operators ◽  
Completely Positive ◽  
Creation And Annihilation Operators ◽  
Complex Reflection ◽  
Complex Reflection Groups

In 1994, M. Bożejko and R. Speicher proved the existence of completely positive quasimultiplicative maps from the group algebra of Coxeter groups to the set of bounded operators. They used some of them to define an inner product associated to creation and annihilation operators on a direct sum of Hilbert space tensor powers called full Fock space. Afterwards, A. Mathas and R. Orellana defined in 2008 a length function on imprimitive complex reflection groups that allowed them to introduce an analogue to the descent algebra of Coxeter groups. In this article, we use the length function defined by A. Mathas and R. Orellana to extend the result of M. Bożejko and R. Speicher to imprimitive complex reflection groups, in other words to prove the existence of completely positive quasimultiplicative maps from the group algebra of imprimitive complex reflection groups to the set of bounded operators. Some of those maps are then used to define a more general inner product associated to creation and annihilation operators on the full Fock space. Recall that in quantum mechanics, the state of a physical system is represented by a vector in a Hilbert space, and the creation and annihilation operators act on a Fock state by respectively adding and removing a particle in the ascribed quantum state.

scholarly journals Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

2019 ◽  
Vol 32 (05) ◽  
pp. 2030004
Author(s):  
Alexei Daletskii ◽  
Alexander Kalyuzhny ◽  
Eugene Lytvynov ◽  
Daniil Proskurin
Keyword(s):  
Linear Operator ◽  
Fock Space ◽  
Multicomponent System ◽  
Baxter Equation ◽  
Special Choice ◽  
Bounded Linear ◽  
Creation And Annihilation Operators ◽  
Commutation Relations ◽  
System A ◽  
Fock Representations

Let [Formula: see text] be a separable Hilbert space and [Formula: see text] be a self-adjoint bounded linear operator on [Formula: see text] with norm [Formula: see text], satisfying the Yang–Baxter equation. Bożejko and Speicher ([10]) proved that the operator [Formula: see text] determines a [Formula: see text]-deformed Fock space [Formula: see text]. We start with reviewing and extending the known results about the structure of the [Formula: see text]-particle spaces [Formula: see text] and the commutation relations satisfied by the corresponding creation and annihilation operators acting on [Formula: see text]. We then choose [Formula: see text], the [Formula: see text]-space of [Formula: see text]-valued functions on [Formula: see text]. Here [Formula: see text] and [Formula: see text] with [Formula: see text]. Furthermore, we assume that the operator [Formula: see text] acting on [Formula: see text] is given by [Formula: see text]. Here, for a.a. [Formula: see text], [Formula: see text] is a linear operator on [Formula: see text] with norm [Formula: see text] that satisfies [Formula: see text] and the spectral quantum Yang–Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function [Formula: see text] in the case [Formula: see text] determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its [Formula: see text]-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.

The Hodge Operator in Fermionic Fock Space

2005 ◽  
Vol 70 (7) ◽  
pp. 979-1016 ◽  
Author(s):  
Leszek Z. Stolarczyk
Keyword(s):  
Fock Space ◽  
Differential Forms ◽  
Grassmann Algebra ◽  
Basis Set ◽  
Electron Systems ◽  
Linear Transformations ◽  
Spin Orbital ◽  
Creation And Annihilation Operators ◽  
Finite Dimensional ◽  
Hodge Operator

The Hodge operator ("star" operator) plays an important role in the theory of differential forms, where it serves as a tool for the switching between the exterior derivative and co-derivative. In the theory of many-electron systems involving a finite-dimensional fermionic Fock space, one can define the Hodge operator as a unique (i.e., invariant with respect to linear transformations of the spin-orbital basis set) antilinear operator. The similarity transformation based on the Hodge operator results in the switching between the fermion creation and annihilation operators. The present paper gives a self-contained account on the algebraic structures which are necessary for the construction of the Hodge operator: the fermionic Fock space, the corresponding Grassmann algebra, and the generalized creation and annihilation operators. The Hodge operator is then defined, and its properties are reviewed. It is shown how the notion of the Hodge operator can be employed in a construction of the electronic time-reversal operator.

scholarly journals GENERALIZED DEFORMED PARA-BOSE OSCILLATOR AND NONLINEAR ALGEBRAS

1995 ◽  
Vol 10 (36) ◽  
pp. 2739-2748
Author(s):  
HA HUY BANG
Keyword(s):  
Fock Space ◽  
Single Mode ◽  
Creation And Annihilation Operators ◽  
Commutation Relations ◽  
Special Cases ◽  
Deformed Commutation Relations

Generalized deformed commutation relations for a single mode para-Bose oscillator and for a system of two para-Bose oscillators are constructed. It turns out that generalized deformed para-Bose oscillators are not, in general, exactly independent. Furthermore, we also discuss about the Fock space corresponding to generalized deformed para-Bose oscillators. Finally, we show how SU(2) and SU(1, 1) generators can be constructed in terms of generalized deformed para-Bose creation and annihilation operators. The algebras SU(2) and SU(1, 1) of generalized deformed oscillators14,18 are the special cases of generalized deformed para-Bose oscillators algebras but, interestingly, they have the same form.

scholarly journals ON PARASTATISTICS DEFINED AS TRIPLE OPERATOR ALGEBRAS

1998 ◽  
Vol 13 (13) ◽  
pp. 995-1005 ◽  
Author(s):  
STJEPAN MELJANAC ◽  
MARKO STOJIĆ ◽  
MARIJAN MILEKOVIĆ
Keyword(s):  
Operator Algebras ◽  
Fock Space ◽  
Quantum Systems ◽  
Many Body ◽  
Creation And Annihilation Operators

Parastatistics, defined as triple operator algebras represented on Fock space, are unified in a simple way using the transition number operators. They are expressed as a normal ordered expansion of creation and annihilation operators. We discuss several examples of parastatistics, particularly Okubo's and Palev's parastatistics connected to many-body Wigner quantum systems and relate them to the notion of extended Haldane statistics.

scholarly journals TOWARD SECOND-QUANTIZATION OF D5-BRANE

1998 ◽  
Vol 13 (06) ◽  
pp. 923-963 ◽  
Author(s):  
TOSHIO NAKATSU
Keyword(s):  
Moduli Space ◽  
Cohomology Group ◽  
Fock Space ◽  
Low Energy ◽  
Second Quantization ◽  
Creation And Annihilation Operators ◽  
World Volume

A framework of second-quantization of D5-branes is proposed. It is based on the study of topology of the moduli space of their low energy effective world volume theory. Among the topological cycles whose resolve singularities caused by overlapping D5-branes, there are introduced those cycles which duals, constituting a subspace of cohomology group of the moduli space, turn out to define the Fock space of the second-quantized D5-branes. The second-quantized operators are given by creation and annihilation operators of those cycles or their duals.

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