The quantum group SU<sub>q</sub>(2) and a q-analogue of the boson operators

Biedenharn, L. C.

doi:10.1088/0305-4470/22/18/004

Cited by 1,500 publications

(1,032 citation statements)

References 8 publications

Supporting

Mentioning

1,017

Contrasting

Unclassified

Order By: Relevance

“…A q-boson algebra [9,10] (deformed harmonic oscillator) is a set of elements called q-boson operators: a (annihilation), a † (creation), N (number), which satisfy the following commutation relations:…”

Section: A Q-deformed Objectsmentioning

confidence: 99%

See 1 more Smart Citation

Some unusual algebraic structures and their applications in many-body problems

Avancini¹

2003

Braz. J. Phys.

View full text Add to dashboard Cite

In this paper, we introduce the q-deformed and quon algebras formalism. Some applications of these algebraic structures are considered. A possible connection of the quon algebra with composite particle systems is discussed and perspectives on using such mathematical objects in the Bose-Einstein condensation is presented. I IntroductionThe identification of algebraic structures in quantum physical systems has been an important tool for their understanding. Concepts of symmetry, invariance and group theory have shown to be of great utility from the beginning of quantum mechanics [1]. In a more recent context some unusual algebraic structures related to quantum inverse scattering methods and statistical mechanics models have appeared [2]. These algebraic structures are usually called q-algebras, qdeformed algebras or quantum groups. In current theoretical investigations there have been extensive applications of qalgebras to many branches of physics [3,4,5]. In a sense, many successful standard applications of group theory in the past may be extended to a quantum group symmetry. The aim of this paper is to discuss the use of q-algebras in the framework of many-body problems. We introduce the mathematical fundamentals as simply as possible and consider applications to q-deformed pseudo-spin models. These models can be considered as convenient theoretical laboratories where one can test the properties of these algebraic structures. We also show that the q-algebra formalism may be useful in the context of boson mappings,i. e., when the substitution of fermion pairs by bosons is convenient. A q-deformed algebra has a closely related algebraic structure which is called the quon algebra [6,7]. Such algebra was proposed to describe particles that violate statistics by a small amount. We consider the applications of quon algebras in boson mappings and nuclear models. We also show that they may be relevant in the study of composite bosonic particles (particles composed by fermion pairs), since deviations of a true bosonic particle could be accommodated in a natural way through the quon algebra. This formalism can also be applied to the study of Bose-Einstein condensation in trapped atoms [8]. This paper is organized as follows: In Sec.II quantum groups or q-algebras are defined and some of their properties and applications are analysed, in Sec.III the quon algebra formalism is considered and ,finally, in Sec.IV we present our conclusions. II q-deformed algebras or quantum groupsThe q-algebras or quantum groups are generalizations of classical Lie groups and Lie algebras and involve two fundamental ideas: Deformation and Non-commutative comultiplication [5]. Next, these two concepts will be discussed in detail. A. q-deformed objectsA q-boson algebra [9,10] (deformed harmonic oscillator) is a set of elements called q-boson operators: a (annihilation), a † (creation), N (number), which satisfy the following commutation relations:Note that the first and second commutation relations are equal to the common harmonic oscillator, ho...

show abstract

Section: A Q-deformed Objectsmentioning

confidence: 99%

“…We can construct the irreducible representations of su q (2) using a method very similar to the commonly used for su(2) [3,10,9]. Denoting the basis states by {|jm }, where m runs from j up to -j, the matrix elements of the generators are given by following expressions:…”

Section: • Shift In Frequencymentioning

confidence: 99%

Some unusual algebraic structures and their applications in many-body problems

Avancini¹

2003

Braz. J. Phys.

View full text Add to dashboard Cite

show abstract

“…In [12] Biedenharn proposed a new realization of quantum group SU q (2) and in order to realize generators of a q-deformation U q (su(2)) of the universal enveloping algebra of the Lie algebra su(2) he defined a pair of mutual commuting q-harmonic oscillator systems (alà Jordan-Schwinger approach to su(2) Lie algebra).…”

Section: Polar Decomposition Of Su Q (2; C) and Su(2; C) Groups' Algementioning

confidence: 99%

“…Generalized Pauli algebras appear naturally also in Z n -Quantum Mechanics [9], in q-deformed Heisenberg algebras [14] , [9] as well as in q = ω -deformed quantum oscillator [15], [9] as suggested by L. C. Biedenharn in [12].…”

Section: Introductionmentioning

confidence: 99%

On generalized Clifford algebraC 4 (n) andGL q (2;C) quantum group

Kwaśniewski

1999

AACA

View full text Add to dashboard Cite

The non commuting matrix elements of matrices from quantum group GLq(2; C) with q ≡ ω being the n-th root of unity are given a representation as operators in Hilbert space with help of C (n) 4 generalized Clifford algebra generators appropriately tensored with unit 2 × 2 matrix infinitely many times. Specific properties of such a representation are presented . Relevance of generalized Pauli algebra to azimuthal quantization of angular momentum alà Lévy -Leblond [10] and to polar decomposition of SUq(2; C) quantum algebra ala Chaichian and Ellinas [6] is also commented.The case of q ∈ C, |q| = 1 may be treated parallely.

show abstract

“…При этом мы определяем соответствующий ос-циллятор и его гамильтониан по заданному набору ортогональных многочленов, которые оказываются собственными функциями гамильтониана, в то время как при стандартном подходе сам осциллятор и его оператор энергии известны, а следует искать систему собственных функций. Заметим, что понятие "обобщенного" осцил-лятора возникает как естественное обобщение широко используемого в современной литературе понятия "деформированного" осциллятора [7]- [9] и сохраняет основные черты алгебры "деформированного" осциллятора. Однако в отличие от "деформи-рованных" осцилляторов, где при предельном переходе параметров (к 0 или к 1) "деформированная" алгебра переходит в алгебру Гейзенберга, в нашем случае от-сутствует связь с алгеброй обычного гармонического осциллятора.…”

Section: Introductionunclassified

Обобщенный Осциллятор И Его Когерентные Состояния

Borzov¹

2007

ТМФ

View full text Add to dashboard Cite

The quantum group SU_q(2) and a q-analogue of the boson operators

Cited by 1,500 publications

References 8 publications

Some unusual algebraic structures and their applications in many-body problems

Some unusual algebraic structures and their applications in many-body problems

On generalized Clifford algebraC 4 (n) andGL q (2;C) quantum group

Обобщенный Осциллятор И Его Когерентные Состояния

Contact Info

Product

Resources

About

The quantum group SUq(2) and a q-analogue of the boson operators

Cited by 1,500 publications

References 8 publications

Some unusual algebraic structures and their applications in many-body problems

Some unusual algebraic structures and their applications in many-body problems

On generalized Clifford algebraC 4 (n) andGL q (2;C) quantum group

Обобщенный Осциллятор И Его Когерентные Состояния

Contact Info

Product

Resources

About

The quantum group SU_q(2) and a q-analogue of the boson operators