Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting proper...Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.展开更多
On the basis of a unified definition of the dual operation and the (anti )self dual operation, the connections of the su(2,2｜1) main cluster was used as the fundamental field variables to construct the self dual L...On the basis of a unified definition of the dual operation and the (anti )self dual operation, the connections of the su(2,2｜1) main cluster was used as the fundamental field variables to construct the self dual Lagrangian of conformal supergravity. A Yang Mill like Lagrangian is obtained and a new gauge theory of supergravity is put forward. The spatial projects of its spin connection are Ashtekar variables.展开更多
In the SU（3） simple group model, the new neutral gauge boson Z＇ couples to pairs of SM fermions with couplings fixed in terms of the SM gauge couplings and depending only on the choice of the fermion embedding. In ...In the SU（3） simple group model, the new neutral gauge boson Z＇ couples to pairs of SM fermions with couplings fixed in terms of the SM gauge couplings and depending only on the choice of the fermion embedding. In this paper, we calculate the contributions of this new particle to the processes e^＋e^-→l^＋l^-, bb^-, and cc^- and study the possibility of detecting this new particle via these processes in the future high-energy linear e^＋e^- collider（LC） experiments with √s= 500 GeV and ￡int= 340 fb^-1. We find that the new gauge boson Z＇ is most sensitive to the process e^＋e^-→b^＋b^-. As long as Mz,≤2 TeV , the absolute values of the relative correction parameter are larger than 5%. We calculate the forward-backward asymmetries and left-right asymmetries for the process e^＋e^-→c^＋c^-, with both the universal and anomaly-free fermion embeddings. Bounds on Z＇ masses are also estimated within 95% confidence level.展开更多
The Higgs-like boson discovered at CERN in 2012 is tentatively assigned to a newly found bound state of two charged gauge bosons W<sup>+</sup>W<sup>-</sup> with a mass of E<sub>B</sub&...The Higgs-like boson discovered at CERN in 2012 is tentatively assigned to a newly found bound state of two charged gauge bosons W<sup>+</sup>W<sup>-</sup> with a mass of E<sub>B</sub> ≈ 117 GeV, much closer to the measured 125 GeV than 110 GeV predicted in a paper with the same title earlier this year. The improvement is due to a shift from the earlier SU(2) representation assignment for the gauge bosons to the more realistic SU(3) one and that the computations are carried out with much greater accuracy.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
文摘Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.
文摘On the basis of a unified definition of the dual operation and the (anti )self dual operation, the connections of the su(2,2｜1) main cluster was used as the fundamental field variables to construct the self dual Lagrangian of conformal supergravity. A Yang Mill like Lagrangian is obtained and a new gauge theory of supergravity is put forward. The spatial projects of its spin connection are Ashtekar variables.
基金supported in part by a grant from Henan Institute of Science and Technology under Grant No.06040
文摘In the SU（3） simple group model, the new neutral gauge boson Z＇ couples to pairs of SM fermions with couplings fixed in terms of the SM gauge couplings and depending only on the choice of the fermion embedding. In this paper, we calculate the contributions of this new particle to the processes e^＋e^-→l^＋l^-, bb^-, and cc^- and study the possibility of detecting this new particle via these processes in the future high-energy linear e^＋e^- collider（LC） experiments with √s= 500 GeV and ￡int= 340 fb^-1. We find that the new gauge boson Z＇ is most sensitive to the process e^＋e^-→b^＋b^-. As long as Mz,≤2 TeV , the absolute values of the relative correction parameter are larger than 5%. We calculate the forward-backward asymmetries and left-right asymmetries for the process e^＋e^-→c^＋c^-, with both the universal and anomaly-free fermion embeddings. Bounds on Z＇ masses are also estimated within 95% confidence level.
文摘The Higgs-like boson discovered at CERN in 2012 is tentatively assigned to a newly found bound state of two charged gauge bosons W<sup>+</sup>W<sup>-</sup> with a mass of E<sub>B</sub> ≈ 117 GeV, much closer to the measured 125 GeV than 110 GeV predicted in a paper with the same title earlier this year. The improvement is due to a shift from the earlier SU(2) representation assignment for the gauge bosons to the more realistic SU(3) one and that the computations are carried out with much greater accuracy.
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.
