Generators of SO(8) group have been described by using direct product of the Gamma matrices and the Pauli Sigma matrices. We have obtained these generators in terms of generalized split octonion also. These generators have been used to describe the rotational transformation of vectors for SO(7) symmetry group.
The 8 $times$ 8 matrix representation of SO(8) Symmetry has been defined by using the direct product of Pauli matrices and Gamma matrices. These 8 $times$ 8 matrices are being used to describe the rotations in SO(8) symmetry. The comparison of 8$times$8 matrices with octonions has also been shown. The transformations of SO(8) symmetry are represented with the help of Octonions and split Octonions spinors.
An attempt has been made to investigate the global SU(2) and SU(3) unitary flavor symmetries systematically in terms of quaternion and octonion respectively. It is shown that these symmetries are suitably handled with quaternions and octonions in order to obtain their generators, commutation rules and symmetry properties. Accordingly, Casimir operators for SU(2)and SU(3) flavor symmetries are also constructed for the proper testing of these symmetries in terms of quaternions and octonions.
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 lambda 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}times SU(2)_{L}times 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 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.
Starting with the quaternionic formulation of isospin SU(2) group, we have derived the relations for different components of isospin with quark states. Extending this formalism to the case of SU(3) group we have considered the theory of octonion variables. Accordingly, the octonion splitting of SU(3) group have been reconsidered and various commutation relations for SU(3) group and its shift operators are also derived and verified for different iso-spin multiplets i.e. I, U and V- spins. Keywords: SU(3), Quaternions, Octonions and Gell Mann matrices PACS NO: 11.30.Hv: Flavor symmetries; 12.10-Dm: Unified field theories and models of strong and electroweak interactions
We consider the questions of whether the damped Lyman-alpha (DLA) and sub-DLA absorbers in quasar spectra differ intrinsically in metallicity, and whether they could arise in galaxies of different masses. Using the recent measurements of the robust metallicity indicators Zn and S in DLAs and sub-DLAs, we confirm that sub-DLAs have higher mean metallicities than DLAs, especially at $z lesssim 2$. We find that the intercept of the metallicity-redshift relation derived from Zn and S is higher than that derived from Fe by 0.5-0.6 dex. We also show that, while there is a correlation between the metallicity and the rest equivalent width of Mg II $lambda 2796$ or Fe II $lambda 2599$ for DLAs, no correlation is seen for sub-DLAs. Given this, and the similar Mg II or Fe II selection criteria employed in the discovery of both types of systems at lower redshifts, the difference between metallicities of DLAs and sub-DLAs appears to be real and not an artefact of selection. This conclusion is supported by our simulations of Mg II $lambda 2796$ and Fe II $lambda 2599$ lines for a wide range of physical conditions. On examining the velocity spreads of the absorbers, we find that sub-DLAs show somewhat higher mean and median velocity spreads ($Delta v$), and an excess of systems with $Delta v > 150$ km s$^{-1}$, than DLAs. Compared to DLAs, the [Mn/Fe] vs. [Zn/H] trend for sub-DLAs appears to be steeper and closer to the trend for Galactic bulge and thick disk stars, possibly suggesting different stellar populations. The absorber data appear to be consistent with galaxy down-sizing. The data are also consistent with the relative number densities of low-mass and high-mass galaxies. It is thus plausible that sub-DLAs arise in more massive galaxies on average than DLAs.
We have studied the adsorption of NO on small Rh clusters, containing one to five atoms, using density functional theory in both spin-polarized and non-spin-polarized forms. We find that NO bonds more strongly to Rh clusters than it does to Rh(100) or Rh(111); however, it also quenches the magnetism of the clusters. This (local) effect results in reducing the magnitude of the adsorption energy, and also washes out the clear size-dependent trend observed in the non-magnetic case. Our results illustrate the competition present between the tendencies to bond and to magnetize, in small clusters.
