The Experts below are selected from a list of 1731 Experts worldwide ranked by ideXlab platform

J Lukierski - One of the best experts on this subject based on the ideXlab platform.

  • internal quark symmetries and colour su 3 entangled with z3 graded lorentz algebra
    Nuclear Physics, 2021
    Co-Authors: Richard Kerner, J Lukierski
    Abstract:

    Abstract In the current version of QCD the quarks are described by ordinary Dirac fields, organized in the following internal symmetry multiplets: the S U ( 3 ) colour, the S U ( 2 ) flavour, and broken S U ( 3 ) providing the family triplets. In this paper we argue that internal and external (i.e. space-time) symmetries are entangled at least in the colour sector in order to introduce the spinorial quark fields in a way providing all the internal quark's degrees of freedom which do appear in the Standard Model. Because the S U ( 3 ) colour algebra is endowed with natural Z 3 -graded discrete automorphisms, in order to introduce entanglement the Z 3 -graded version of Lorentz algebra with its vectorial and spinorial realizations are considered. The colour multiplets of quarks are described by 12-component colour Dirac equations, with a Z 3 -graded triplet of masses (one real and a Lee-Wick Complex Conjugate Pair). We show that all quarks in the Standard Model can be described by the 72-component master quark sextet of 12-component coloured Dirac fields, which is required in order to implement the faithful spinorial representation of the Z 3 -graded Lorentz transformations.

Richard Kerner - One of the best experts on this subject based on the ideXlab platform.

  • internal quark symmetries and colour su 3 entangled with z3 graded lorentz algebra
    Nuclear Physics, 2021
    Co-Authors: Richard Kerner, J Lukierski
    Abstract:

    Abstract In the current version of QCD the quarks are described by ordinary Dirac fields, organized in the following internal symmetry multiplets: the S U ( 3 ) colour, the S U ( 2 ) flavour, and broken S U ( 3 ) providing the family triplets. In this paper we argue that internal and external (i.e. space-time) symmetries are entangled at least in the colour sector in order to introduce the spinorial quark fields in a way providing all the internal quark's degrees of freedom which do appear in the Standard Model. Because the S U ( 3 ) colour algebra is endowed with natural Z 3 -graded discrete automorphisms, in order to introduce entanglement the Z 3 -graded version of Lorentz algebra with its vectorial and spinorial realizations are considered. The colour multiplets of quarks are described by 12-component colour Dirac equations, with a Z 3 -graded triplet of masses (one real and a Lee-Wick Complex Conjugate Pair). We show that all quarks in the Standard Model can be described by the 72-component master quark sextet of 12-component coloured Dirac fields, which is required in order to implement the faithful spinorial representation of the Z 3 -graded Lorentz transformations.

Luca Benvenuti - One of the best experts on this subject based on the ideXlab platform.

Peter Poromaa - One of the best experts on this subject based on the ideXlab platform.

  • Computing eigenspaces with specified eigenvalues of a regular matrix Pair (A, B) and condition estimation: theory, algorithms and software
    Numerical Algorithms, 1996
    Co-Authors: B Kagstrom, Peter Poromaa
    Abstract:

    Theory, algorithms and LAPACK-style software for computing a Pair of deflating subspaces with specified eigenvalues of a regular matrix Pair ( A, B ) and error bounds for computed quantities (eigenvalues and eigenspaces) are presented. The reordering of specified eigenvalues is performed with a direct orthogonal transformation method with guaranteed numerical stability. Each swap of two adjacent diagonal blocks in the real generalized Schur form, where at least one of them corresponds to a Complex Conjugate Pair of eigenvalues, involves solving a generalized Sylvester equation and the construction of two orthogonal transformation matrices from certain eigenspaces associated with the diagonal blocks. The swapping of two 1×1 blocks is performed using orthogonal (unitary) Givens rotations. The error bounds are based on estimates of condition numbers for eigenvalues and eigenspaces. The software computes reciprocal values of a condition number for an individual eigenvalue (or a cluster of eigenvalues), a condition number for an eigenvector (or eigenspace), and spectral projectors onto a selected cluster. By computing reciprocal values we avoid overflow. Changes in eigenvectors and eigenspaces are measured by their change in angle. The condition numbers yield both asymptotic and global error bounds. The asymptotic bounds are only accurate for small perturbations ( E, F ) of ( A, B ), while the global bounds work for all ‖( E, F. )‖ up to a certain bound, whose size is determined by the conditioning of the problem. It is also shown how these upper bounds can be estimated. Fortran 77 software that implements our algorithms for reordering eigenvalues, computing (left and right) deflating subspaces with specified eigenvalues and condition number estimation are presented. Computational experiments that illustrate the accuracy, efficiency and reliability of our software are also described.

  • Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software
    1996
    Co-Authors: B Kagstrom, Peter Poromaa
    Abstract:

    Theory, algorithms and LAPACK-style software for computing a Pair of deflating subspaces with specified eigenvalues of a regular matrix Pair (A; B) and error bounds for computed quantities (eigenvalues and eigenspaces) are presented. The reordering of specified eigenvalues is performed with a direct orthogonal transformation method with guaranteed numerical stability. Each swap of two adjacent diagonal blocks in the real generalized Schur form, where at least one of them corresponds to a Complex Conjugate Pair of eigenvalues, involves solving a generalized Sylvester equation and the construction of two orthogonal transformation matrices from certain eigenspaces associated with the diagonal blocks. The swapping of two 1 x 1 blocks is performed using orthogonal (unitary) Givens rotations. The error bounds are based on estimates of condition numbers for eigenvalues and eigenspaces. The software computes reciprocal values of a condition number for an individual eigenvalue (or a clus..

  • Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A,B) and Condition Estimation: Theory, Algorithms and Software
    1994
    Co-Authors: B Kagstrom, Peter Poromaa
    Abstract:

    Theory, algorithms and LAPACK--style software for computing a Pair of deflating subspaces with specified eigenvalues of a regular matrix Pair (A; B) and error bounds for computed quantities (eigenvalues and eigenspaces) are presented. The reordering of specified eigenvalues is performed with a direct orthogonal transformation method with guaranteed numerical stability. Each swap of two adjacent diagonal blocks in the real generalized Schur form, where at least one of them corresponds to a Complex Conjugate Pair of eigenvalues, involves solving a generalized Sylvester equation and the construction of two orthogonal transformation matrices from certain eigenspaces associated with the diagonal blocks. The swapping of two 1 \Theta 1 blocks is performed using orthogonal (unitary) Givens rotations. The error bounds are based on estimates of condition numbers for eigenvalues and eigenspaces. The software computes reciprocal values of a condition number for an individual eigenvalue (or a clus..

M. Shiroishi - One of the best experts on this subject based on the ideXlab platform.

  • THE SPIN-1/2 XXZ CHAIN AT FINITE MAGNETIC FIELD: CROSSOVER PHENOMENA DRIVEN BY TEMPERATURE
    2001
    Co-Authors: A. Klümper, J. Reyes R. Martínez, C. Scheeren, M. Shiroishi
    Abstract:

    We investigate the asymptotic behaviour of spin-spin correlation functions for the integrable Heisenberg chain. To this end we use the Quantum Transfer Matrix (QTM) technique developed in 1 which results in a set of non-linear integral equations (NLIE). In the case of the largest eigenvalue the solution to these equations yields the free energy and by modifications of the paths of integration the next-leading eigenvalues and hence the correlation lengths are obtained. At finite field h> 0 and sufficiently high temperature T the next-leading eigenvalue is unique and given by a 1-string solution to the QTM taking real and negative values thus resulting into exponentially decaying correlations with antiferromagnetic oscillations. At sufficiently low temperatures a different behaviour sets in where the next-leading eigenvalues of the QTM are given by a Complex Conjugate Pair of eigenvalues resulting into incommensurate oscillations. The above scenario is the result of analytical and numerical investigations of the QTM establishing a well defined crossover temperature Tc(h) at which the 1-string eigenvalue to the QTM gets degenerate with the 2-string solution. Among other things we find a simple particle-hole picture for the excitations of the QTM allowing for a description by the dressed charge formulation of CFT

  • THE 1/2-XXZ SPIN-CHAIN AT FINITE MAGNETIC FIELD: CROSSOVER PHENOMENA DRIVEN BY TEMPERATURE
    2000
    Co-Authors: A. Klümper, J. Reyes R. Martínez, C. Scheeren, M. Shiroishi
    Abstract:

    We investigate the asymptotic behaviour of spin-spin correlation functions for the integrable Heisenberg chain. To this end we use the Quantum Transfer Matrix (QTM) technique developed in 1 which results in a set of non-linear integral equations (NLIE). In the case of the largest eigenvalue the solution to these equations yields the free energy and by modifications of the paths of integration the next-leading eigenvalues and hence the correlation lengths are obtained. At finite field h> 0 and sufficiently high temperature T the next-leading eigenvalue is unique and given by a 1-string solution to the QTM taking real and negative values thus resulting into exponentially decaying correlations with antiferromagnetic oscillations. At sufficiently low temperatures a different behaviour sets in where the next-leading eigenvalues of QTM are given by a Complex Conjugate Pair of eigenvalues resulting into incommensurate oscillations. The above scenario is the result of analytical and numerical investigations of the QTM establishing a well defined crossover temperature Tc(h) at which the 1-string eigenvalue to the QTM gets degenerate with the 2-string solution. Among other things we find a simple particle-hole picture for the excitations of the QTM and we make contact with the dressed charge formulation of CFT.