多模双参数变形相干态

借助一个满足量子Heisenberg-Weyl代数(H-W_q,s代数)的多模算符,给出了量子代数SU(2)_q,s和SU(1,1)_q,s的k(k≥2)模实现,并构造了相应的相干态.     

利用量子代数SU(1,1)q,s的 Holstein-Primakoff实现讨论双参数变形Jaynes-Cummings模型

本文利用量子代数 SU(1,1)_q,s的 Holstein-Primakoff 实现讨论了双参数变形Jaynes-Cummings 模型,并给出了布居数反转的时间演化表达式.     

Nonclassical properties and algebraic characteristics of negative binomial states in quantized radiation fields

We study the nonclassical properties and algebraic characteristics of the negative binomial states introduced by Barnett recently. The ladder operator formalism and displacement operator formalism of the negative binomial states are found and the algebra involved turns out to be the SU(1,1) Lie algebra via the generalized Holstein-Primarkoff realization. These states are essentially Perelomov's SU(1,1) coherent states. We reveal their connection with the geometric states and find that they are excited geometric states. As intermediate states, they interpolate between the number states and geometric states. We also point out that they can be recognized as the nonlinear coherent states. Their nonclassical properties, such as sub-Poissonian distribution and squeezing effect are discussed. The quasiprobability distributions in phase space, namely the Q and Wigner functions, are studied in detail. We also propose two methods of generation of the negative binomial states. d 32.80.Pj Optical cooling of atoms; trapping Received 8 May 1999 and Received in final form 8 November 1999     

SU(2) andSU(1,1) algebra eigenstates: A unified analytic approach to coherent and intelligent states

We introduce the concept of algebra eigenstates which are defined for an arbitrary Lie group as eigenstates of elements of the corresponding complex Lie algebra. We show that this concept unifies different definitions of coherent states associated with a dynamical symmetry group. On the one hand, algebra eigenstates include different sets of Perelomov's generalized coherent states. On the other hand, intelligent states (which are squeezed states for a system of general symmetry) also form a subset of algebra eigenstates. We develop the general formalism and apply it to theSU(2) andSU(1,1) simple Lie groups. Complete solutions to the general eigenvalue problem are found in both cases by a method that employs analytic representations of the algebra eigenstates. This analytic method also enables us to obtain exact closed expressions for quantum statistical properties of an arbitrary algebra eigenstate. Important special cases such as standard coherent states and intelligent states are examined and relations between them are studied by using their analytic representations.     

Fidelity susceptibility and geometric phase in critical phenomenon

Motivated by recent developments in quantum fidelity and fidelity susceptibility,we study relations among Lie algebra,fidelity susceptibility and quantum phase transition for a two-state system and the Lipkin-Meshkov-Glick model. We obtain the fidelity susceptibilities for SU(2) and SU(1,1) algebraic structure models. From this relation,the validity of the fidelity susceptibility to signal for the quantum phase transition is also verified in these two systems. At the same time,we obtain the geometric phases in these two systems by calculating the fidelity susceptibility. In addition,the new method of calculating fidelity susceptibility is used to explore the two-dimensional XXZ model and the Bose-Einstein condensate (BEC).     

Quaternion Octonion Reformulation of Grand Unified Theories

In this paper, Grand Unified theories are discussed in terms of quaternions and octonions by using the relation between quaternion basis elements with Pauli matrices and Octonions with Gell Mann λ matrices. Connection between the unitary groups of GUTs and the normed division algebra has been established to re-describe the SU(5) gauge group. We have thus described the SU(5) gauge group and its subgroup SU(3) _C ×SU(2) _L ×U(1) by using quaternion and octonion basis elements. As such the connection between U(1) gauge group and complex number, SU(2) gauge group and quaternions and SU(3) and octonions is established. It is concluded that the division algebra approach to the theory of unification of fundamental interactions as the case of GUTs leads to the consequences towards the new understanding of these theories which incorporate the existence of magnetic monopole and dyon.     

Contracted Hamiltonian on Symmetric Space <Emphasis Type="Italic">SU</Emphasis>(3)/<Emphasis Type="Italic">SU</Emphasis>(2) and Conserved Quantities

We study the quantum model on symmetric space SU(3)/SU(2). By using the Inonu-Wigner contraction to Lie algebra su(3), we arrive at a special case of three-body Sutherland model. It has shown that by calculating conservative quantities of this model, it has Poincare Lie algebra, too.     

A spectrum generating algebra for meson resonances

The algebraSO(6,1) is considered as a unification ofSO(6), which is isomorphic toSU(4) SU(3), and the de Sitter algebraSO(4,1). The latter replaces the Poincaré algebra as the algebra of the group of motions of physical space-time. A representation ofSO(6,1) is constructed, which, on restriction toSU(3), decomposes into the direct sum of allSU(3) representations, each occurring just once in the decomposition. The expectation values of the mass-squared operator, when evaluated in the octet, give accurate mass formulae for the octets of 1^– and 2⁺ meson resonances.     

Octonion Quantum Chromodynamics

Starting with the usual definitions of octonions, an attempt has been made to establish the relations between octonion basis elements and Gell-Mann λ matrices of SU(3) symmetry on comparing the multiplication tables for Gell-Mann λ matrices of SU(3) symmetry and octonion basis elements. Consequently, the quantum chromo dynamics (QCD) has been reformulated and it is shown that the theory of strong interactions could be explained better in terms of non-associative octonion algebra. Further, the octonion automorphism group SU(3) has been suitably handled with split basis of octonion algebra showing that the SU(3) _C gauge theory of colored quarks carries two real gauge fields which are responsible for the existence of two gauge potentials respectively associated with electric charge and magnetic monopole and supports well the idea that the colored quarks are dyons.     

The algebra and geometry ofSU(3) matrices

We give an elementary treatment of the defining representation and Lie algebra of the three-dimensional unitary unimodular groupSU(3). The geometrical properties of the Lie algebra, which is an eight dimensional real linear vector space, are developed in anSU(3) covariant manner. Thef andd symbols ofSU(3) lead to two ways of ‘multiplying’ two vectors to produce a third, and several useful geometric and algebraic identities are derived. The axis-angle parametrization ofSU(3) is developed as a generalization of that forSU(2), and the specifically new features are brought out. Application to the dynamics of three-level systems is outlined.     

The<Emphasis Type="Italic">SU</Emphasis>(1,1) coherent states related to the affine group wavelets

The Perelomov coherent states ofSU(1,1) are labeled by elements of the quotient ofSU(1,1) by its rotation subgroup. Taking advantage of the fact that this quotient is isomorphic to the affine group of the real line, we are able to parameterize the coherent states by elements of that group. Such a formulation permits to find new properties of theSU(1,1) coherent states and to relate them to affine wavelets. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.     

A Locally Compact Quantum Group Analogue of the Normalizer of SU(1,1) in SL(2, ℂ)

 S.L. Woronowicz proved in 1991 that quantum SU(1,1) does not exist as a locally compact quantum group. Results by L.I. Korogodsky in 1994 and more recently by Woronowicz gave strong indications that the normalizer of SU(1,1) in SL(2,ℂ) is a much better quantization candidate than SU(1,1) itself. In this paper we show that this is indeed the case by constructing , a new example of a unimodular locally compact quantum group (depending on a parameter 0<q<1) that is a deformation of . After defining the underlying von Neumann algebra of we use a certain class of q-hypergeometric functions and their orthogonality relations to construct the comultiplication. The coassociativity of this comultiplication is the hardest result to establish. We define the Haar weight and obtain simple formulas for the antipode and its polar decomposition. As a final result we produce the underlying C ^* -algebra of . The proofs of all these results depend on various properties of q-hypergeometric ₁ϕ₁ functions. Received: 28 June 2001 / Accepted: 25 July 2002 Published online: 10 December 2002 RID="*" ID="*" Post-doctoral researcher of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.) Communicated by L. Takhtajan     

Semiclassical quantization of SU(3) skyrmions

Semiclassical quantization of the SU(3)-skyrmion zero modes is performed by means of the collective coordinate method. The quantization condition known for SU(2) solitons quantized with SU(3) collective coordinates is generalized for SU(3) skyrmions with strangeness content different from zero. The quantization of the dipole-type configuration with large strangeness content found recently is considered as an example and the spectrum and the mass splittings of the quantized states are estimated. The energy and baryon number density of SU(3) skyrmions are presented in a form emphasizing their symmetry in different SU(2) subgroups of SU(3), and a lower bound for the static energy of SU(3) skyrmions is derived. Zh. éksp. Teor. Fiz. 112, 1941–1958 (December 1997) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Singlet Deformation of <Emphasis Type="Italic">N</Emphasis> = (1,1) Superspace

We discuss the SO(4) × SU(2) invariant deformation of the Euclidean N = (1,1) supersymmetric theories in the framework of the harmonic superspace.     

Mapping of Shape Invariant Potentials by the Point Canonical Transformation

In this paper by using the method of point canonical transformation we find that the Coulomb and Kratzer potentials can be mapped to the Morse potential. Then we show that the Pöschl-Teller potential type I belongs to the same subclass of shape invariant potentials as Hulthén potential. Also we show that the shape-invariant algebra for Coulomb, Kratzer, and Morse potentials is SU(1,1), while the shape-invariant algebra for Pöschl-Teller type I and Hulthén is SU(2).     

Duality in Interacting Particle Systems and Boson Representation

In the context of Markov processes, we show a new scheme to derive dual processes and a duality function based on a boson representation. This scheme is applicable to a case in which a generator is expressed by boson creation and annihilation operators. For some stochastic processes, duality relations have been known, which connect continuous time Markov processes with discrete state space and those with continuous state space. We clarify that using a generating function approach and the Doi-Peliti method, a birth-death process (or discrete random walk model) is naturally connected to a differential equation with continuous variables, which would be interpreted as a dual Markov process. The key point in the derivation is to use bosonic coherent states as a bra state, instead of a conventional projection state. As examples, we apply the scheme to a simple birth-coagulation process and a Brownian momentum process. The generator of the Brownian momentum process is written by elements of the SU(1,1) algebra, and using a boson realization of SU(1,1) we show that the same scheme is available.     

Random matrix ensembles with random interactions: Results for EGUE(2)-SU(4)

We introduce in this paper embedded Gaussian unitary ensemble of random matrices, for m fermions in Ω number of single particle orbits, generated by random twobody interactions that are SU(4) scalar, called EGUE(2)-SU(4). Here the SU(4) algebra corresponds to Wigner’s supermultiplet SU(4) symmetry in nuclei. Formulation based on Wigner-Racah algebra of the embedding algebra U(4Ω) ? U(Ω) ? SU(4) allows for analytical treatment of this ensemble and using this analytical formulas are derived for the covariances in energy centroids and spectral variances. It is found that these covariances increase in magnitude as we go from EGUE(2) to EGUE(2)-s to EGUE(2)-SU(4) implying that symmetries may be responsible for chaos in finite interacting quantum systems.     

SU(1, 1) and SU(2) Perelomov number coherent states: algebraic approach for general systems

We study some properties of the SU(1, 1) Perelomov number coherent states. The Schrödinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized for the standard coherent states. We obtain the time evolution of the number coherent states by supposing that the Hamiltonian is proportional to the third generator K₀ of the su(1, 1) Lie algebra. Analogous results for the SU(2) Perelomov number coherent states are found. As examples, we compute the Perelomov coherent states for the pseudoharmonic oscillator and the two-dimensional isotropic harmonic oscillator.     

Quantum groupSU(1, 1) ⋊ Z2 and “super-tensor” products

A quantum analogue of the groupSU(1,1)Z ₂—the normalizer ofSU(1, 1) inSL ₂(C)—is introduced and studied. Although there isno correctly defined tensor product in the category of *-representations of the quantum algebraC[SU(1, 1)] _q of regular functions, some categories of *-representations ofC[SU(1, 1)Z ₂] _q turn out to be endowed with a certainZ ₂-graded structure which can be considered as a super-generalization of the monoidal category structure. This quantum effect may be considered as a step to understanding the concept of quantum topological locally compact group.In fact, there seems to be afamily of quantum groupsSU(1, 1)Z ₂ parameterized by unitary characters T ¹ of the fundamental group of the two-dimensional symplectic leaf ofSU(1, 1)/T, whereT is the subgroup of diagonal matrices.It is shown that thequasi-classical analogues of the results of the paper are connected with the decomposition of Schubert cells of the flag manifoldSL ₂(C)_R/B (whereB is the Borel subgroup of upper-triangular matrices) into symplectic leaves.Supported by the Rosenbaum Fellowship.     

On the exchange matrix for WZNW model

The connection between the exchange algebra in theSU(2) Wess-Zumino Novikov-Witten model and the quantum groupSU(2) is discussed. It is shown that on the quasiclassical level this connection has the simple interpretation in terms of the Lie-Poisson action ofSU(2) on the chiral components of the fields in the WZNW model.Dedicated to Res Jost and Arthur Wightman