Su Algebra Research Articles

Two-qutrit entanglement witnesses and Gell-Mann matrices

The Gell-Mann λ matrices for Lie algebra su(3) are the natural basis for the Hilbert space of Hermitian operators acting on the states of a three-level system(qutrit). So the construction of EWs for two-qutrit states by using these matrices may be an interesting problem. In this paper, several two-qutrit EWs are constructed based on the Gell-Mann matrices by using the linear programming (LP) method exactly or approximately. The decomposability and non-decomposability of constructed EWs are also discussed and it is shown that the λ-diagonal EWs presented in this paper are all decomposable but by adding λ-non-diagonal terms, one can obtain various non-decomposable EWs.

Z-Stable ASH Algebras

The Jiang--Su algebra Z has come to prominence in the classification program for nuclear C*-algebras of late, due primarily to the fact that Elliott's classification conjecture predicts that all simple, separable, and nuclear C*-algebras with unperforated K-theory will absorb Z tensorially (i.e., will be Z-stable). There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable and Z-stable C*-algebras. We prove that virtually all classes of nuclear C*-algebras for which the Elliott conjecture has been confirmed so far, consist of Z-stable C*-algebras. This result follows in large part from the following theorem, also proved herein: separable and approximately divisible C*-algebras are Z-stable.

A homological definition of the HOMFLY polynomial

We give a new definition of the knot invariant associated to the Lie algebra su_{N+1}. The knot or link must be presented as the plat closure of a braid. The invariant is then a homological intersection pairing between two submanifolds of a configuration space of points in a disk. This generalizes previous work on the Jones polynomial, which is the case N=1.

SU2 Nonstandard Bases: Case of Mutually Unbiased Bases

This paper deals with bases in a finite-dimensional Hilbert space. Such a space can be realized as a subspace of the representation space of SU(2) corresponding to an irreducible representation of SU(2). The representation theory of SU(2) is reconsidered via the use of two truncated deformed oscillators. This leads to replace the familiar scheme {j^2, j_z} by a scheme {j^2, v(ra)}, where the two-parameter operator v(ra) is defined in the enveloping algebra of the Lie algebra su(2). The eigenvectors of the commuting set of operators {j^2, v(ra)} are adapted to a tower of chains SO(3) > C(2j+1), 2j integer, where C(2j+1) is the cyclic group of order 2j+1. In the case where 2j+1 is prime, the corresponding eigenvectors generate a complete set of mutually unbiased bases. Some useful relations on generalized quadratic Gauss sums are exposed in three appendices.

PHANTOM ENERGY FROM GRADED ALGEBRAS

We construct a model of phantom energy using the graded Lie algebra SU(2/1). The negative kinetic energy of the phantom field emerges naturally from the graded Lie algebra, resulting in an equation of state with w < -1. The model also contains ordinary scalar fields and anticommuting (Grassmann) vector fields which can be taken as two-component dark matter. A potential term is generated for both the phantom fields and the ordinary scalar fields via a postulated condensate of the Grassmann vector fields. Since the phantom energy and dark matter arise from the same Lagrangian, the phantom energy and dark matter of this model are coupled via the Grassmann vector fields. In the model presented here phantom energy and dark matter come from a gauge principle rather than being introduced in an ad hoc manner.

REALIZATION OF DYNAMICAL GROUP FOR A SYMMETRIC WELL POTENTIAL tan2(πx/a) AND ITS CONTROLLABILITY

The exact solutions of quantum system with a symmetric well potential V(x) = D tan 2(πx/a) are obtained. The ladder operators are constructed directly from the normalized eigenfunctions with the factorization method. It is shown that these ladder operators satisfy the commutation relations of the generators for an su(1, 1) algebra. The infinitely deep square well and harmonic limits of this potential are briefly studied. The controllability of this system is also investigated. It is demonstrated that this system with discrete bound states can be strongly completely controlled. This may be realized theoretically by acting the creation operator [Formula: see text] on the ground state.

The two-level pairing model in the Schwinger representation

The two-level pairing model was investigated in the framework of the Schwinger representation. It has been shown that zero-point fluctuations, which are important in the crossover region, are well described by certain coherent states which conserve the constants of motion. These so-called conserving approximations provide excellent descriptions of the ground-state energy and of the first excitation energy over the whole range of coupling constant values. The description of the ground-state energy provided by the Glauber coherent state is also reasonable, but not so good. Although the Glauber coherent state fails to provide the description of the excitation energy around the critical point, its performance in the strong coupling limit is very good both for the description of the ground-state energy and of the first excitation energy. The present results also show that already at the mean-field level, the bosonic description in the independent particle approximation incorporates important correlation corrections to the conventional mean-field description based on the fermionic realization of the su(2) algebra (BCS theory).

Surfaces immersed in Lie algebras obtained from the sigma models

We study some geometrical aspects of two dimensional orientable surfaces arrising from the study of CP^N sigma models. To this aim we employ an identification of R^(N(N+2)) with the Lie algebra su(N+1) by means of which we construct a generalized Weierstrass formula for immersion of such surfaces. The structural elements of the surface like its moving frame, the Gauss-Weingarten and the Gauss-Codazzi-Ricci equations are expressed in terms of the solution of the CP^N model defining it. Further, the first and second fundamental forms, the Gaussian curvature, the mean curvature vector, the Willmore functional and the topological charge of surfaces are expressed in terms of this solution. We present detailed implementation of these results for surfaces immersed in su(2) and su(3) Lie algebras.

A new rational conformal field theory extension of the fully degenerate W1+∞(m)

We found new identities among the Dedekind eta-function, the characters of the W_{m} algebra and those of the level 1 affine Lie algebra su(m)_{1}. They allow to characterize the Z_{m}-orbifold of the m-component free bosons u(1)_{K_{m,p}} (our theory TM) as an extension of the fully degenerate representations of W_{1+infty}^{(m)}. In particular, TM is proven to be a Gamma _{theta}-RCFT extension of the chiral fully degenerate W_{1+infty}^{(m)}.

COHERENT STATES FOR THE FINITE su(2)-OSCILLATOR MODEL

A finite oscillator is a system that has a discrete and finite number of spatial positions and energy levels accessible. The su(2)-oscillator model takes the generators of the angular momentum su(2) algebra and, giving to each one a definite physical interpretation, provides a consistent mathematical description of a finite oscillator. This work searches an example of identification of generalized coherent states, within the collection of states and wavefunctions of the finite su(2) oscillator model.

A note on the generalized neutral orthogonal group in dimension four

We study the main properties of the generalized neutral orthogonal group O(2, 2) and its Lie algebra o(2, 2). We also give an explicit isomorphism between the Lie algebras su(1, 1) ⊕ su(1, 1) and o(2, 2). We use this isomorphism to classify the subalgebras of o(2, 2).

Mean field theory forsu(4) simeso(6)

Algebraic mean field theory constructs a group theoretical model of quantum systems that have a weak dynamical symmetry but may break dynamical symmetry. The strong defining condition for dynamical symmetry is that states belong to one irreducible representation space. Weak dynamical symmetry demands that the densities corresponding to the states have a constant value for each Casimir. Quantum phase transitions and other complex systems exhibit weak dynamical symmetry. Furthermore mean field theory often yields analytic formulae for expectations and energy spectra that are not feasible in representation theory. This paper develops mean field theory on any coadjoint orbit of su(4) densities. The simple Lie algebra su(4) so(6) is a 15-dimensional algebra that contains the subalgebra usp(4) so(5) and the angular momentum algebra su(2). The su(4) dual space consists of density matrices which are defined by the expectations of the su(4) generators. A coadjoint orbit is a common level surface in the dual space of the three su(4) Casimirs. A Lax pair determines the dynamics of these densities on each coadjoint orbit. Analytic solutions are reported for rotating su(4) densities in equilibrium for a particular energy function.

Non-Hermitian interactions between harmonic oscillators, with applications to stable, Lorentz-violating quantum electrodynamics

We examine a new application of the Holstein-Primakoff realization of the simple harmonic oscillator Hamiltonian. This involves the use of infinite-dimensional representations of the Lie algebra su(2). The representations contain nonstandard raising and lowering operators, which are nonlinearly related to the standard a† and a. The new operators also give rise to a natural family of two-oscillator couplings. These nonlinear couplings are not generally self-adjoint, but their low-energy limits are self-adjoint, exactly solvable, and stable. We discuss the structure of a theory involving these couplings. Such a theory might have as its ultra-low-energy limit a Lorentz-violating Abelian gauge theory, and we discuss the extremely strong astrophysical constraints on such a model.

Nonunitary squeezing and biorthogonal scalar products in polarization optics

The polarization vectors of monochromatic electromagnetic fields can be transformed by multiplying them by 2 ? 2 matrices representing, respectively, retardation plates, rotators and partial polarizers. The present description of partial polarizers is related to a two-dimensional Lorentz algebra SU(1,1) leading to nonunitary squeezing as the present SU(1,1) algebra includes non-Hermitian operators in addition to the Hermitian ones. The present analysis exploits the use of similarity transformations which are more general than the unitary transformations used usually in quantum mechanics. By performing a similarity transformation on a set of Hermitian physical operators the transformed operators are Hermitian with respect to new scalar product defined by the use of a new metric. It is shown that such transformations can be implemented experimentally by the use of polarization optics and the metric obtained in such transformations is analysed.

Mean field theory for usp(4) sime so(5)

The simple Lie algebra usp(4) so(5) is a ten-dimensional algebra that contains the angular momentum algebra su(2). In nuclear structure physics, the algebra so(5) describes beta-rigid collective modes in the Bohr–Mottelson and interacting boson models. The so(5) dual space consists of density matrices which are defined by the expectations of so(5) generators. A coadjoint orbit is a common level surface in the dual space of the two so(5) Casimirs. This paper develops mean field theory on any coadjoint orbit of so(5) densities. When the densities of a set of quantum states lie on one orbit, the system is said to have a weak dynamical symmetry. A Lax pair determines the dynamics of so(5) densities on each coadjoint orbit. Analytic solutions are reported for rotating so(5) densities in equilibrium for a particular energy function.

N = 4 Supersymmetric Morse Oscillator andIts Spectrum-Generating Algebra**The projectsupported by National Natural Science Foundation of Chinaunder Grant No. 10275038, and partly by the State Key BasicResearch Development Program under Grant No. G2000077400,and Natural Science Foundation of Tsinghua University

In this paper N = 4 supersymmetry of generalized Morse oscillators inone dimension is studied. Both bound states and scattering states of its foursuperpartner Hamiltonians are analyzed by using unitary irreducible representationsof the noncompact Lie algebra su(1,1).The spectrum-generating algebra governing the Hamiltonian of the N = 4supersymmetric Morse oscillator is shown to be connected with the realization of Liesuperalgebra osp(1,2) or B(0,1) in terms of the variables ofa supersymmetric two-dimensional harmonic oscillator.

Infinite statistics and theSU(1, 1) phase operator

A few years ago, Agarwal (1991 Phys. Rev. A 44 8398) showed that the Susskind–Glogower phase operators, expressible in terms of Bose operators, provide a realization of the algebra for particles obeying infinite statistics. In this paper we show that the SU(1, 1) phase operators, constructed in terms of the elements of the su(1, 1) Lie algebra, also provide a realization of the algebra for infinite statistics. There are many realizations of the su(1, 1) algebra in terms of single or multimode bose operators, three of which are discussed along with their corresponding phase states. The Susskind–Glogower phase operator is a special case of the SU(1, 1) phase operator associated with the Holstein–Primakoff realization of su(1, 1).

Simultaneous Quantization of Three-Dimensional Bialgebras

We perform a simultaneous multiparameter quantization of some three-dimensional Lie algebras, such as the Heisenberg algebra, motion algebras of the Euclidean and pseudoeuclidean planes, and the algebra su(2). Such a quantization is possible since any of the mentioned algebras is dual to the same solvable Lie algebra. We present an explicit form of the numerical R-matrix; this allows us to represent some of the commutation relations in the form of the PTT-equation. Bibliography: 14 titles.

Representation Theory and Wigner-Racah Algebra of the SU(2) Group in a Noncanonical Basis

The Lie algebra su(2) of the classical group SU(2) is built from two commuting quon algebras for which the deformation parameter is a common root of unity. This construction leads to (i) a not very well-known polar decomposition of the generators J- and J+ of the SU(2) group, with J+ = J-† = HUr where H is Hermitean and Ur unitary, and (ii) an alternative to the {J2,Jz} quantization scheme, viz., the {J2,Ur} quantization scheme. The representation theory of the SU(2) group can be developed in this nonstandard scheme. The key ideas for developing the Wigner-Racah algebra of the SU(2) group in the {J2,Ur} scheme are given. In particular, some properties of the coupling and recoupling coefficients as well as the Wigner-Eckart theorem in the {J2,Ur} scheme are examined in great detail.

O(N)symmetries, sum rules for generalized Hermite polynomials and squeezed states

Quantum optics has been dealing with coherent states, squeezed states and many other non-classical states. The associated mathematical framework makes use of special functions as Hermite polynomials, Laguerre polynomials and others. In this connection we here present some formal results that follow directly from the group O(N) of complex transformations. Motivated by the squeezed states structure, we introduce the generalized Hermite polynomials , which include as particular cases, the Hermite polynomials as well as the heat polynomials. Using generalized raising operators, we derive new sum rules for the , which are covariant under O(N) transformations. The and the associated sum rules become useful for evaluating Wigner functions in a straightforward manner. As a byproduct, we use one of these sum rules, on the operator level, to obtain raising and lowering operators for the Laguerre polynomials and show that they generate an sl(2, R) su(1, 1) algebra.