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N. I. Stoilova

Publications and source records attributed to N. I. Stoilova.

At least 37 records · Page 2Linked to original sources
arXiv2016

Gel'fand-Zetlin basis for a class of representations of the Lie superalgebra gl(\infty|\infty)

A new, so called odd Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(n|n). The related Gel'fand-Zetlin patterns are based upon the decomposition according to a particular chain of subalgebras of gl(n|n). This chain contains only genuine Lie superalgebras of type gl(k|l) with k and l nonzero (apart from the final element of the chain which is gl(1|0)=gl(1)). Explicit expressions for a set of generators of the algebra on this Gel'fand-Zetlin basis are determined. The results are extended to an explicit construction of a class of irreducible highest weight modules of the general linear Lie superalgebra gl(\infty|\infty).

math-ph↗
arXiv2015

Generalized Quantum Statistics and Lie (Super)Algebras

Generalized quantum statistics, such as paraboson and parafermion statistics, are characterized by triple relations which are related to Lie (super)algebras of type B. The correspondence of the Fock spaces of parabosons, parafermions as well as the Fock space of a system of parafermions and parabosons to irreducible representations of (super)algebras of type B will be pointed out. Example of generalized quantum statistics connected to the basic classical Lie superalgebra B(1|1)=osp(3|2) with interesting physical properties, such as noncommutative coordinates, will be given. Therefore the article focuses on the question, addressed already in 1950 by Wigner: do the equation of motion determine the quantum mechanical commutation relation?

math-ph↗
arXiv2015

A class of infinite-dimensional representations of the Lie superalgebra osp(2m+1|2n) and the parastatistics Fock space

An orthogonal basis of weight vectors for a class of infinite-dimensional representations of the orthosymplectic Lie superalgebra osp(2m+1|2n) is introduced. These representations are particular lowest weight representations V(p), with a lowest weight of the form [-p/2,...,-p/2|p/2,...,p/2], p being a positive integer. Explicit expressions for the transformation of the basis under the action of algebra generators are found. Since the relations of algebra generators correspond to the defining relations of m pairs of parafermion operators and n pairs of paraboson operators with relative parafermion relations, the parastatistics Fock space of order p is also explicitly constructed. Furthermore, the representations V(p) are shown to have interesting characters in terms of supersymmetric Schur functions, and a simple character formula is also obtained.

math-ph↗
arXiv2014

On a pair of difference equations for the $_4F_3$ type orthogonal polynomials and related exactly-solvable quantum systems

We introduce a pair of novel difference equations, whose solutions are expressed in terms of Racah or Wilson polynomials depending on the nature of the finite-difference step. A number of special cases and limit relations are also examined, which allow to introduce similar difference equations for the orthogonal polynomials of the $ _3 F_2$ and $ _2 F_1$ types. It is shown that the introduced equations allow to construct new models of exactly-solvable quantum dynamical systems, such as spin chains with a nearest-neighbour interaction and fermionic quantum oscillator models.

math-ph↗
arXiv2013

The parastatistics Fock space and explicit Lie superalgebra representations

It is known that the defining triple relations of m pairs of parafermion operators and n pairs of paraboson operators with relative parafermion relations can be considered as defining relations for the Lie superalgebra osp(2m+1|2n) in terms of 2(m+n) generators. With the common Hermiticity conditions, this means that the parastatistics Fock space of order p corresponds to an infinite-dimensional unitary irreducible representation V(p) of osp(2m+1|2n), with lowest weight (-p/2,...,- p/|p/2,...,p/2). These representations (also in the simplest case m=n=1) had never been constructed due to computational difficulties, despite their importance. In the present paper we solve partially the problem in the general case using group theoretical techniques, in which the u(m|n) subalgebra of osp(2m+1|2n) plays a crucial role: a set of Gelfand-Zetlin patterns of u(m|n) can be used to label the basis vectors of V(p). An explicit and elegant construction of these representations V(p) for m=n=1, and the actions or matrix elements of the osp(3|2) generators are given.

math-ph↗
arXiv2011

The su(2)_α Hahn oscillator and a discrete Hahn-Fourier transform

We define the quadratic algebra su(2)_α which is a one-parameter deformation of the Lie algebra su(2) extended by a parity operator. The odd-dimensional representations of su(2) (with representation label j, a positive integer) can be extended to representations of su(2)_α. We investigate a model of the finite one-dimensional harmonic oscillator based upon this algebra su(2)_α. It turns out that in this model the spectrum of the position and momentum operator can be computed explicitly, and that the corresponding (discrete) wavefunctions can be determined in terms of Hahn polynomials. The operation mapping position wavefunctions into momentum wavefunctions is studied, and this so-called discrete Hahn-Fourier transform is computed explicitly. The matrix of this discrete Hahn-Fourier transform has many interesting properties, similar to those of the traditional discrete Fourier transform.

math-ph↗
arXiv2011

Finite oscillator models: the Hahn oscillator

A new model for the finite one-dimensional harmonic oscillator is proposed based upon the algebra u(2)_α. This algebra is a deformation of the Lie algebra u(2) extended by a parity operator, with deformation parameter α. A class of irreducible unitary representations of u(2)_α is constructed. In the finite oscillator model, the (discrete) spectrum of the position operator is determined, and the position wave functions are shown to be dual Hahn polynomials. Plots of these discrete wave functions display interesting properties, similar to those of the parabose oscillator. We show indeed that in the limit, when the dimension of the representations goes to infinity, the discrete wave functions tend to the continuous wave functions of the parabose oscillator.

math-ph↗
arXiv2010

Gel'fand-Zetlin Basis and Clebsch-Gordan Coefficients for Covariant Representations of the Lie superalgebra gl(m|n)

A Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(m|n). Explicit expressions for the generators of the Lie superalgebra acting on this basis are determined. Furthermore, Clebsch-Gordan coefficients corresponding to the tensor product of any covariant tensor representation of gl(m|n) with the natural representation V ([1,0,...,0]) of gl(m|n) with highest weight (1,0,. . . ,0) are computed. Both results are steps for the explicit construction of the parastatistics Fock space.

math-ph↗
arXiv2008

Representations of the orthosymplectic Lie superalgebra osp(1|4) and paraboson coherent states

We introduce and obtain multimode paraboson coherent states. In appropriate subspaces these coherent states provide a decomposition of unity where the measure, when expressed using the cat-type states, is positive definite. Bicoherent states where the mutually commuting lowering operators are diagonalized are also obtained. Matrix elements in the coherent state basis are calculated.

math-ph↗
arXiv2008

Parafermions, parabosons and representations of so(\infty) and osp(1|\infty)

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so(\infty) and of the Lie superalgebra osp(1|\infty). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labelled by certain infinite but stable Gelfand-Zetlin patterns, and the transformation of the basis is given explicitly. We also present expressions for the character of the Fock space representations.

hep-th↗
arXiv2007

The parafermion Fock space and explicit so(2n+1) representations

The defining relations (triple relations) of n pairs of parafermion operators f_j^\pm (j=1,...,n) are known to coincide with a set of defining relations for the Lie algebra so(2n+1) in terms of 2n generators. With the common Hermiticity conditions, this means that the ``parafermions of order p'' correspond to a finite-dimensional unitary irreducible representation W(p) of so(2n+1), with highest weight (p/2, p/2,..., p/2). Although the dimension and character of W(p) is known by classical formulas, there is no explicit basis of W(p) available in which the parafermion operators have a natural action. In this paper we construct an orthogonal basis for W(p), and we present the explicit actions of the parafermion generators on these basis vectors. We use group theoretical techniques, in which the u(n) subalgebra of so(2n+1) plays a crucial role: a set of Gelfand-Zetlin patterns of u(n) will be used to label the basis vectors of W(p), and also in the explicit action (matrix elements) certain u(n) Clebsch-Gordan coefficients are essential.

hep-th↗
arXiv2007

The paraboson Fock space and unitary irreducible representations of the Lie superalgebra osp(1|2n)

It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators $b^\pm_i$. In particular, with the usual star conditions, this implies that the ``parabosons of order p'' correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of osp(1|2n). Apart from the simple cases p=1 or n=1, these representations had never been constructed due to computational difficulties, despite their importance. In the present paper we give an explicit and elegant construction of these representations V(p), and we present explicit actions or matrix elements of the osp(1|2n) generators. The orthogonal basis vectors of V(p) are written in terms of Gelfand-Zetlin patterns, where the subalgebra u(n) of osp(1|2n) plays a crucial role. Our results also lead to character formulas for these infinite-dimensional osp(1|2n) representations. Furthermore, by considering the branching $ osp(1|2n) \supset sp(2n) \supset u(n)$, we find explicit infinite-dimensional unitary irreducible lowest weight representations of sp(2n) and their characters.

hep-th↗
arXiv2007

Harmonic oscillator chains as Wigner Quantum Systems: periodic and fixed wall boundary conditions in gl(1|n) solutions

We describe a quantum system consisting of a one-dimensional linear chain of n identical harmonic oscillators coupled by a nearest neighbor interaction. Two boundary conditions are taken into account: periodic boundary conditions (where the nth oscillator is coupled back to the first oscillator) and fixed wall boundary conditions (where the first oscillator and the $n$th oscillator are coupled to a fixed wall). The two systems are characterized by their Hamiltonian. For their quantization, we treat these systems as Wigner Quantum Systems (WQS), allowing more solutions than just the canonical quantization solution. In this WQS approach, one is led to certain algebraic relations for operators (which are linear combinations of position and momentum operators) that should satisfy triple relations involving commutators and anti-commutators. These triple relations have a solution in terms of the Lie superalgebra gl(1|n). We study a particular class of gl(1|n) representations V(p), the so-called ladder representations. For these representations, we determine the spectrum of the Hamiltonian and of the position operators (for both types of boundary conditions). Furthermore, we compute the eigenvectors of the position operators in terms of stationary states. This leads to explicit expressions for position probabilities of the n oscillators in the chain. An analysis of the plots of such position probability distributions gives rise to some interesting observations. In particular, the physical behavior of the system as a WQS is very much in agreement with what one would expect from the classical case, except that all physical quantities (energy, position and momentum of each oscillator) have a finite spectrum.

hep-th↗
arXiv2007

On the eigenvalue problem for arbitrary odd elements of the Lie superalgebra gl(1|n) and applications

In a Wigner quantum mechanical model, with a solution in terms of the Lie superalgebra gl(1|n), one is faced with determining the eigenvalues and eigenvectors for an arbitrary self-adjoint odd element of gl(1|n) in any unitary irreducible representation W. We show that the eigenvalue problem can be solved by the decomposition of W with respect to the branching gl(1|n) --> gl(1|1) + gl(n-1). The eigenvector problem is much harder, since the Gel'fand-Zetlin basis of W is involved, and the explicit actions of gl(1|n) generators on this basis are fairly complicated. Using properties of the Gel'fand-Zetlin basis, we manage to present a solution for this problem as well. Our solution is illustrated for two special classes of unitary gl(1|n) representations: the so-called Fock representations and the ladder representations.

math-ph↗
arXiv2006

A classification of generalized quantum statistics associated with the exceptional Lie (super)algebras

Generalized quantum statistics (GQS) associated to a Lie algebra or Lie superalgebra extends the notion of para-Bose or para-Fermi statistics. Such GQS have been classified for all classical simple Lie algebras and basic classical Lie superalgebras. In the current paper we finalize this classification for all exceptional Lie algebras and superalgebras. Since the definition of GQS is closely related to a certain Z-grading of the Lie (super)algebra G, our classification reproduces some known Z-gradings of exceptional Lie algebras. For exceptional Lie superalgebras such a classification of Z-gradings has not been given before.

math-ph↗
arXiv2006

Harmonic oscillators coupled by springs: discrete solutions as a Wigner Quantum System

We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency $ω$, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner Quantum System, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton's equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1|M). Then we study the properties and spectra of the physical operators in a class of unitary representations of gl(1|M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain.

hep-th↗
arXiv2006

Representations of the Lie Superalgebra gl(1|n) in a Gel'fand-Zetlin Basis and Wigner Quantum Oscillators

An explicit construction of all finite-dimensional irreducible representations of the Lie superalgebra gl(1|n) in a Gel'fand-Zetlin basis is given. Particular attention is paid to the so-called star type I representations (``unitary representations''), and to a simple class of representations V(p), with p any positive integer. Then, the notion of Wigner Quantum Oscillators (WQOs) is recalled. In these quantum oscillator models, the unitary representations of gl(1|DN) are physical state spaces of the N-particle D-dimensional oscillator. So far, physical properties of gl(1|DN) WQOs were described only in the so-called Fock spaces W(p), leading to interesting concepts such as non-commutative coordinates and a discrete spatial structure. Here, we describe physical properties of WQOs for other unitary representations, including certain representations V(p) of gl(1|DN). These new solutions again have remarkable properties following from the spectrum of the Hamiltonian and of the position, momentum, and angular momentum operators. Formulae are obtained that give the angular momentum content of all the representations V(p) of gl(1|3N), associated with the N-particle 3-dimensional WQO. For these representations V(p) we also consider in more detail the spectrum of the position operators and their squares, leading to interesting consequences. In particular, a classical limit of these solutions is obtained, that is in agreement with the correspondence principle.

hep-th↗