arXiv subjects
Khaled Abdel-Khalek
Publications and source records attributed to Khaled Abdel-Khalek.
Ring Division Algebras, Self-Duality and Supersymmetry
We argue that once octonions are formulated as soft Lie algebras, they may be safely used and the non-associativity can be overcame. The necessary points are: (a) Fixing the direction of action by introducing the δoperator. (b) Closing the δalgebra by using structure functions f_{ijk} (ϕ). (c) Representation of the δalgebra can be developed. The E or E(ϕ) can be found and their structure functions can be computed easily. There may be different applications of soft seven sphere in physics. We have given two cases where the ring division algebras occupies a special position. Self-duality and Simple supersymmetric Yang-Mills theories are two promising places where soft seven sphere prove to be useful and essential.
Off-Shell Formulation of Simple Supersymmetric Yang-Mills
An off-shell formulation for 6 and 10 dimensions simple supersymmetric Yang-Mills theories is presented. While the fermionic fields couple to left action of S^3 and S^7 respectively, the auxiliary ones couple to right action (and vice versa). To close the algebra off-shell, left and right actions must commute. For 6 dimensions quaternions work fine. The 10 dimensional case needs special care. Pure spinors and soft Lie algebra (algebra with structure functions instead of structure constants) are essential. Some tools useful for constructing the superspace are also derived. We show how to relate our results to the early works of Evans and Berkovits.
Beyond Octonions
We investigate Clifford Algebras structure over non-ring division algebras. We show how projection over the real field produces the standard Attiyah-Bott-Shapiro classification.
Algebraic Investigation of the Soft Seven Sphere
We investigate the seven sphere as a soft Lie algebra i.e. an algebra with structure functions instead of structure constants. We calculate its structure functions explicitly and also discuss some relevant points such as the validity of the Jacobi identities. Furthermore, we emphasis some important features such as the pointwise reduction, closure and some other consistency checks.
The Ring Division Self Duality
We present a simple construction of the instantonic type equation over octonions where its similarities and differences with the quaternionic case are very clear. We use the unified language of Clifford Algebra. We argue that our approach is the pure algebraic formulation of the geometric based soft Lie algebra. The topological criteria for the stability of our solution is given explicitly to establish its solitonic property. Many beautiful features of the parallelizable ring division spheres and Absolute Parallelism (AP) reveal their presence in our formulation.
The Compatibility between the Higher Dimensions Self Duality and the Yang-Mills Equation of Motion
We study the compatiblity between the higher dimension dualities and the Yang-Mills equation of motion. Taking a 't Hooft solution as a starting point, we come to the conclusion that for only 4 dimensions the self duality implies the equation of motion for generic instanton size. Whereas in higher dimensions, the self duality is compatable with the equation of motion, approximately, for small instanton size i.e. the zero curvature condition. At the mathematical level, the self duality is still useful since it transforms a second order into a first order differential equation.
Unified Octonionic Representation of the 10-13 Dimensional Clifford Algebra
We give a one dimensional octonionic representation of the different Clifford algebra Cliff(5,5)\sim Cliff(9,1), Cliff(6,6)\sim Cliff(10,2) and lastly Cliff(7,6)\sim Cliff(10,3) which can be given by (8x8) real matrices taking into account some suitable manipulation rules.
Octonions and Super Lie algebra
We discuss how to represent the non-associative octonionic structure in terms of the associative matrix algebra using the left and right octonionic operators. As an example we construct explicitly some Lie and Super Lie algebra. Then we discuss the notion of octonionic Grassmann numbers and explain its possible application for giving a superspace formulation of the minimal supersymmetric Yang-Mills models.
Unified Octonionic Representation of the 10-13 Dimensions Clifford Algebra
We give a one dimensional octonionic representation of the different Clifford algebra $Cliff(5,5)\sim Cliff(1,9), Cliff(6,6)\sim Cliff(2,10)$ and lastly $Cliff(7,6)\sim Cliff(3,10)$.
Quaternion Analysis
Quaternion analysis is considered in full details where a new analyticity condition in complete analogy to complex analysis is found. The extension to octonions is also worked out.
Octonionic Quantum Mechanics and Complex Geometry
The use of complex geometry allows us to obtain a consistent formulation of octonionic quantum mechanics (OQM). In our octonionic formulation we solve the hermiticity problem and define an appropriate momentum operator within OQM. The nonextendability of the completeness relation and the norm conservation is also discussed in details.
Octonionic Dirac Equation
In order to obtain a consistent formulation of octonionic quantum mechanics (OQM), we introduce left-right barred operators. Such operators enable us to find the translation rules between octonionic numbers and $8\times 8$ real matrices (a translation is also given for $4\times 4$ complex matrices). We develop an octonionic relativistic free wave equation, linear in the derivatives. Even if the wave functions are only one-component we show that four independent solutions, corresponding to those of the Dirac equation, exist.
The Super P-Brane Scan and S Duality
Taking into account the recent dualities we rederive the super p-brane scan. Our main results are the importance of the metric's signature and the existence of an S self-dual super 5-brane at D=14 with signature (7,7) or (11,3).
Octonionic Representations of GL(8,R) and GL(4,C)
Octonionic algebra being nonassociative is difficult to manipulate. We introduce left-right octonionic barred operators which enable us to reproduce the associative GL(8,R) group. Extracting the basis of GL(4,C), we establish an interesting connection between the structure of left-right octonionic barred operators and generic 4x4 complex matrices. As an application we give an octonionic representation of the 4-dimensional Clifford algebra.