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

Abstract

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...