Feza Gürsey

A Universal Gauge Theory Model Based on E6

Abstract A universal gauge theory based on E6 is suggested as a model for the unification of strong, electromagnetic and weak interactions. Left and right components of six quarks (3 light, 3 charmed) and nine leptons (including heavy charged and neutral leptons) are fifted into two 27-dimensional representations of E6. Two different assignments are proposed. The Weinberg angle and the R ratio are derived and the stability of the proton is discussed.

Quark structure and octonions

The octonion (Cayley) algebra is studied in a split basis by means of a formalism that brings outs its quarkstructure. The groups SO(8), SO(7), and G 2 are represented by octonions as well as by 8 × 8 matrices and the principle of triality is studied in this formalism. Reduction is made through the physically important subgroups SU(3) and SU(2) ⊗ SU(2) of G 2, the automorphism group of octonions.

Group theory approach to scattering

Abstract We show that both bound and scattering states of a certain class of potentials are related to the unitary representations of certain groups. In this class, several potentials of practical interest, such as the Morse and Poschl-Teller potentials, are included. The fact that not only bound states but also scattering states are connected with group representations suggests that an algebraic treatment of scattering problems similar to that of bound state problems may be possible.

Lorentz covariant treatment of the Kerr–Schild geometry

It is shown that a Lorentz covariant coordinate system can be chosen in the case of the Kerr–Schild geometry which leads to the vanishing of the pseudo energy–momentum tensor and hence to the linearity of the Einstein equations. The retarded time and the retarded distance are introduced and the Lienard–Wiechert potentials are generalized to gravitation in the case of world‐line singularities to derive solutions of the type of Bonnor and Vaidya. An accelerated version of the de Sitter metric is also obtained. Because of the linearity, complex translations can be performed on these solutions, resulting in a special relativistic version of the Trautman–Newman technique and Lorentz covariant solutions for spinning systems can be derived, including a new anisotropic interior metric that matches to the Kerr metric on an oblate spheroid.

Generalized vector products, duality, and octonionic identities in D = 8 geometry

In an explicit, unified, and covariant formulation, we study and generalize the exceptional vector products in R8 of Zvengrowski, Gray, and Kleinfeld. We derive the associated general quadratic, cubic, and quartic G2 invariant algebraic identities and uncover an octonionic counterpart to the d=4 quaternionic duality. When restricted to seven dimensions the latter is an algebraic statement of absolute parallelism on S7. We further link up with the Ogievetski–Tzeitlin vector product and obtain explicit tensor forms of the SO(7) and G2 structure constants. SO(8) transformations related by Triality are characterized by means of invariant tensors. Possible physical applications are discussed.

Octonionic torsion on S7 and englert's compactification of d=11 supergravity

Abstract In an octonionic realization of the Cartan-Schouten parallelism on S 7 we exhibit the algebraic nature and Hopf topology of Englerts solutions for spontaneously compactified d =11 supergravity. The S 7 torsion and its dual, the index 4 gauge field are given locally by the Cayley structure constants and the associator of the imaginary octonion units, respectively.

Group theory of the Morse oscillator

Abstract A group theoretical approach to the one-dimensional Morse oscillator, including both bound and scattering states, is presented. It is shown that the group describing the scattering states, U(1,1), can be obtained from that describing the bound states. U(2), by analytic continuation. The inclusion of the continuum allows one to treat algebraically dissociation and scattering in a Morse potential.

Group theory approach to scattering. II. The Euclidean connection

We show that the calculation of the S-matrix associated with a class of solvable potentials with SU(1, 1) dynamic symmetry can be formulated in purely algebraic terms. This formulation makes use of the Euclidean group in two dimensions, E(2), and is amenable to generalization to a larger number of dimensions.

Group theory approach to scattering. IV. Solvable potentials associated with SO(2,2)

We use the potential group SO(2, 2) to realize a class of solvable potentials. The scattering matrices can be obtained by purely algebraic techniques.

Noncommutative geometry and Higgs mechanism in the standard model

Abstract We study a model that is based on the idea of noncommutative geometry. In this approach, space-time is extended to include new coordinates that are elements of a noncommutative algebra. Taking this algebra to be that Pauli matrices, we construct a generalized gauge potential that incorporates both the Yang-Mills and the Higgs sector of the standard model. The lagrangian constructed from the generalized field strength contains the Higgs potential that leads to spontaneous symmetry breaking. The model also predicts sin 2 θ w to be about 0.26 and the Higgs mass to be about 80 GeV after including renormalization group corrections.