A simpler definition for a class of 2-parameter quantum groups associated to semisimple Lie algebras is given in terms of Euler form.Their positive parts turn out to be 2-cocycle deformations of each other under some ...A simpler definition for a class of 2-parameter quantum groups associated to semisimple Lie algebras is given in terms of Euler form.Their positive parts turn out to be 2-cocycle deformations of each other under some conditions.An operator realization of the positive part is given.展开更多
We show that there is a quantum Sl_q(2) group symmetry in Hofstadter problem on square lattice.The cyclic representation of the quantum group is discussed and its application for computing the degeneracy density of th...We show that there is a quantum Sl_q(2) group symmetry in Hofstadter problem on square lattice.The cyclic representation of the quantum group is discussed and its application for computing the degeneracy density of the model is shown.展开更多
The convex PBW-type Lyndon basis for two-parameter quantum group U_(r,s)(F_(4))is given.Assume thatrs^(-1)is a primitive l-th root of unity with l odd,then the restricted quantum group u_(r,s)(F_(4))as a quotient of U...The convex PBW-type Lyndon basis for two-parameter quantum group U_(r,s)(F_(4))is given.Assume thatrs^(-1)is a primitive l-th root of unity with l odd,then the restricted quantum group u_(r,s)(F_(4))as a quotient of U_(r,s)(F_(4))is pointed,and of a Drinfel’d double structure under a certain condition.All of Hopf isomorphisms of u_(r,s)(F_(4))are determined,and the necessary and sufficient condition for u_(r,s)(F_(4))to be a ribbon Hopf algebra is singled out by describing the left and right integrals.展开更多
The present work explores a new phenomenon that not all the transition probability of two photon processes is negligible at low irradiance. The irreducible representation 2B2 of C2v is unexpected, for there is no much...The present work explores a new phenomenon that not all the transition probability of two photon processes is negligible at low irradiance. The irreducible representation 2B2 of C2v is unexpected, for there is no much deviation in oscillator strength for two-photon and single-photon process A1 to 2B2. This new phenomenon is only possible to be explored by the symmetrical consideration: the necessary and sufficient condition is molecular plane coincident with yz plane or the operation σ ’v(yz) for group C2v. It is only possible to be evaluated out by use of the full relativistic quantum mechanical theory.展开更多
Entanglement in quantum theory is a peculiar concept to scientists. With this concept we are forced to re-consider the cluster property which means that one event is irrelevant to another event when they are fully far...Entanglement in quantum theory is a peculiar concept to scientists. With this concept we are forced to re-consider the cluster property which means that one event is irrelevant to another event when they are fully far away. In the recent works we showed that the quasi-degenerate states induce the violation of cluster property in antiferromagnets when the continuous symmetry breaks spontaneously. We expect that the violation of cluster property will be observed in other materials too, because the spontaneous symmetry breaking is found in many systems such as the high temperature superconductors and the superfluidity. In order to examine the cluster property for these materials, we studied a quantum nonlinear sigma model with U(1) symmetry in the previous work. There we showed that the model does have quasi-degenerate states. In this paper we study the quantum nonlinear sigma model with SU(2) symmetry. In our approach we first define the quantum system on the lattice and then adopt the representation where the kinetic term is diagonalized. Since we have no definition on the conjugate variable to the angle variable, we use the angular momentum operators instead for the kinetic term. In this representation we introduce the states with the fixed quantum numbers and carry out numerical calculations using quantum Monte Carlo methods and other methods. Through analytical and numerical studies, we conclude that the energy of the quasi-degenerate state is proportional to the squared total angular momentum as well as to the inverse of the lattice size.展开更多
The q-deformation of W（2, 2） Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the ...The q-deformation of W（2, 2） Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the q-deformation of W（2, 2） Lie algebra are completely determined. Finally, the 1-dimensional central extension of the q-deformed W（2, 2） Lie algebra is studied, which turns out to be coincided with the conventional W（2, 2） Lie algebra in the q → 1 limit.展开更多
Starting from Wigner’s definition of the function named now after him we systematically develop different representation of this quasiprobability with emphasis on symmetric representations concerning the canonical va...Starting from Wigner’s definition of the function named now after him we systematically develop different representation of this quasiprobability with emphasis on symmetric representations concerning the canonical variables (q,p) of phase space and using the known relation to the parity operator. One of the representations is by means of the Laguerre 2D polynomials which is particularly effective in quantum optics. For the coherent states we show that their Fourier transforms are again coherent states. We calculate the Wigner quasiprobability to the eigenstates of a particle in a square well with infinitely high impenetrable walls which is not smooth in the spatial coordinate and vanishes outside the wall boundaries. It is not well suited for the calculation of expectation values. A great place takes on the calculation of the Wigner quasiprobability for coherent phase states in quantum optics which is essentially new. We show that an unorthodox entire function plays there a role in most formulae which makes all calculations difficult. The Wigner quasiprobability for coherent phase states is calculated and graphically represented but due to the involved unorthodox function it may be considered only as illustration and is not suited for the calculation of expectation values. By another approach via the number representation of the states and using the recently developed summation formula by means of Generalized Eulerian numbers it becomes possible to calculate in approximations with good convergence the basic expectation values, in particular, the basic uncertainties which are additionally represented in graphics. Both considered examples, the square well and the coherent phase states, belong to systems with SU (1,1) symmetry with the same index K=1/2 of unitary irreducible representations.展开更多
We quantize the W-algebra W（2,2）, whose Verma modules, Harish-Chandra modules, irreducible weight modules and Lie bialgebra structures have been investigated and determined in a series of papers recently.
基金the National Natural Science Foundation of China(Grant Nos.10431040,10728102)the TRAPOYT,the FUDP and the Priority Academic Discipline from the MOE of China,the SRSTP from the STCSM,the Shanghai Priority Academic Discipline from the SMEC
文摘A simpler definition for a class of 2-parameter quantum groups associated to semisimple Lie algebras is given in terms of Euler form.Their positive parts turn out to be 2-cocycle deformations of each other under some conditions.An operator realization of the positive part is given.
文摘We show that there is a quantum Sl_q(2) group symmetry in Hofstadter problem on square lattice.The cyclic representation of the quantum group is discussed and its application for computing the degeneracy density of the model is shown.
基金Nai Hong Hu is supported by the NNSF of China(Grant Nos.12171155,12071094)in part by Science and Technology Commission of Shanghai Municipality(Grant No.22DZ2229014)。
文摘The convex PBW-type Lyndon basis for two-parameter quantum group U_(r,s)(F_(4))is given.Assume thatrs^(-1)is a primitive l-th root of unity with l odd,then the restricted quantum group u_(r,s)(F_(4))as a quotient of U_(r,s)(F_(4))is pointed,and of a Drinfel’d double structure under a certain condition.All of Hopf isomorphisms of u_(r,s)(F_(4))are determined,and the necessary and sufficient condition for u_(r,s)(F_(4))to be a ribbon Hopf algebra is singled out by describing the left and right integrals.
文摘The present work explores a new phenomenon that not all the transition probability of two photon processes is negligible at low irradiance. The irreducible representation 2B2 of C2v is unexpected, for there is no much deviation in oscillator strength for two-photon and single-photon process A1 to 2B2. This new phenomenon is only possible to be explored by the symmetrical consideration: the necessary and sufficient condition is molecular plane coincident with yz plane or the operation σ ’v(yz) for group C2v. It is only possible to be evaluated out by use of the full relativistic quantum mechanical theory.
文摘Entanglement in quantum theory is a peculiar concept to scientists. With this concept we are forced to re-consider the cluster property which means that one event is irrelevant to another event when they are fully far away. In the recent works we showed that the quasi-degenerate states induce the violation of cluster property in antiferromagnets when the continuous symmetry breaks spontaneously. We expect that the violation of cluster property will be observed in other materials too, because the spontaneous symmetry breaking is found in many systems such as the high temperature superconductors and the superfluidity. In order to examine the cluster property for these materials, we studied a quantum nonlinear sigma model with U(1) symmetry in the previous work. There we showed that the model does have quasi-degenerate states. In this paper we study the quantum nonlinear sigma model with SU(2) symmetry. In our approach we first define the quantum system on the lattice and then adopt the representation where the kinetic term is diagonalized. Since we have no definition on the conjugate variable to the angle variable, we use the angular momentum operators instead for the kinetic term. In this representation we introduce the states with the fixed quantum numbers and carry out numerical calculations using quantum Monte Carlo methods and other methods. Through analytical and numerical studies, we conclude that the energy of the quasi-degenerate state is proportional to the squared total angular momentum as well as to the inverse of the lattice size.
基金Supported by National Natural Science Foundation of China (Grant No. 10825101)
文摘The q-deformation of W（2, 2） Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the q-deformation of W（2, 2） Lie algebra are completely determined. Finally, the 1-dimensional central extension of the q-deformed W（2, 2） Lie algebra is studied, which turns out to be coincided with the conventional W（2, 2） Lie algebra in the q → 1 limit.
文摘Starting from Wigner’s definition of the function named now after him we systematically develop different representation of this quasiprobability with emphasis on symmetric representations concerning the canonical variables (q,p) of phase space and using the known relation to the parity operator. One of the representations is by means of the Laguerre 2D polynomials which is particularly effective in quantum optics. For the coherent states we show that their Fourier transforms are again coherent states. We calculate the Wigner quasiprobability to the eigenstates of a particle in a square well with infinitely high impenetrable walls which is not smooth in the spatial coordinate and vanishes outside the wall boundaries. It is not well suited for the calculation of expectation values. A great place takes on the calculation of the Wigner quasiprobability for coherent phase states in quantum optics which is essentially new. We show that an unorthodox entire function plays there a role in most formulae which makes all calculations difficult. The Wigner quasiprobability for coherent phase states is calculated and graphically represented but due to the involved unorthodox function it may be considered only as illustration and is not suited for the calculation of expectation values. By another approach via the number representation of the states and using the recently developed summation formula by means of Generalized Eulerian numbers it becomes possible to calculate in approximations with good convergence the basic expectation values, in particular, the basic uncertainties which are additionally represented in graphics. Both considered examples, the square well and the coherent phase states, belong to systems with SU (1,1) symmetry with the same index K=1/2 of unitary irreducible representations.
基金Supported by NSF＇of China （Grant Nos. 10825101, 10926166）, Special Grade of the Financial Support from China Postdoctoral Science Foundation （Grant No. 201003326） and the Natural Science Research Project for Higher Institutions of Jiangsu Province （Grant No. 09KJB110001）
文摘We quantize the W-algebra W（2,2）, whose Verma modules, Harish-Chandra modules, irreducible weight modules and Lie bialgebra structures have been investigated and determined in a series of papers recently.
