Purpose of a minimal theory is to achief most with least. Least may be for example the spacetime algebra. But the symmetric unitary groupSU(3) is not a part of any real Clifford algebra of 4-dimensional space, especially not of the algebraCl_1,3 of the Minkowski spacetime, nor of the algebraCl_3,1 in the opposite metric. Therefore we can ask how quantumchromodynamics enters into the theory. A first answer is that the groupSU(3) is an object of both the complexified algebras C ⊕Cl_1,3 and C⊕Cl_3,1. To show this we first define six color spaces which are spanned by conjugate triples of commuting base elements. These contain the six idempotent lattices that can be located inCl_3,1. Their images exist in both C⊕Cl_1,3 and C⊕Cl_3,1. Further in each color space there is defined an octahedral orientation stabilizer group which fixates one lepton and color rotates the states in its quark family. Thus quantum numbers of strong interacting fields such as isospin, charge, hypercharge and color turn out as geometric properties. Next we ask if the artificialty of complexification can be avoided. The answer is yes. Defining the class of Clifford algebras with proper imaginary unit it turns out thatCl_1,3 andCl_3,1 do not belong to this class. ButCl_4,1 andCl_1,6 do. It is shown that in the latter algebra the whole color space Ansatz can be established and the generators ofSU(3) represented most naturally and without complexification. That the proposed theory becomes a physically true statement requires that there exists a non rank preserving freedom of motion within the constituents of primitive idempotents, that is, transpositions among conjugate triples in color space.

Abstract.

A random geometric graph is constructed by randomly choosing a set of points in the unit cube [0, 1]^d and connecting two points by an edge if their Euclidean distance is at most some fixed distance r > 0. Graphs of this type are of particular interest as models of wireless networks. In this paper, the d-dimensional unit cube [0, 1]^d is discretized to create a collection V of vertices used to define geometric graphs. With r fixed, dynamic random walks are defined on the subsets of V, resulting in dynamic random walks on the collection of geometric graphs in the discretized cube. These walks naturally model addition-deletion networks and can be visualized as walks on hypercubes with loops. Adjacency operators are constructed using subalgebras of Clifford algebras and are used to recover information about the cycle structure and connected components of graphs in the sequence.

A variety of universal similarity factorization equalities over real Clifford algebrasR_p,q are established. On the basis of these equalities, real, complex and quaternion matrix representations of elements inR_p,q can easily be explicitly be determined.

Abstract.

We present a simplified derivation of the fact that the set of gauge equivalence classes or moduli space of flat connections (potentials) in the abelian Aharonov-Bohm effect, is isomorphic to the circle. The length of this circle is the absolute value of the electric charge.

Universal geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. This paper treats the fundamentals of the multivector differential calculus part of geometric calculus. The multivector differential is introduced, followed by the multivector derivative and the adjoint of multivector functions. The basic rules of multivector differentiation are derived explicitly, as well as a variety of basic multivector derivatives. Finally factorization, which relates functions of vector variables and multivector variables is discussed, and the concepts of both simplicial variables and derivatives are explained. Everything is proven explicitly in a very elementary level step by step approach. The paper is thus intended to serve as reference material, providing a number of details, which are usually skipped in more advanced discussions of the subject matter. The arrangement of the material closely followschapter 2 of [3].

Abstract.

In this paper we mainly study the so-called isotonic Dirac system over more general types of unbounded domains in Euclidean space of even dimension. In such systems different Dirac operators in the half dimension act from the left and from the right on the functions considered. We obtain the integral representation of isotonic functions satisfying the decay condition over the unbounded domains, and show that the integral representation formula over the unbounded domains for holomorphic functions of several complex variables and for Hermitean monogenic functions may be derived from it.

Abstract.

In this paper we are working with a generalized Kolosov-Muskhelishvili formula. This representation formula expresses the elastic displacements of a solid body in terms of two monogenic functions. Known polynomial systems of monogenic functions are used to construct a basis of polynomial solutions to the Lamé system from linear elasticity.

The similarity and consimilarity of elements in the real quaternion, octonion and sedenion algebras, as well as in the general real Cayley-Dickson algebras are considered by solving the two fundamental equationsax=xb and $$ax = \bar xb$$ in these algebras. Some consequences are also presented.

Abstract.

Recently pure spinor geometry has allowed, after decades of desperate efforts, some definite steps forward for the solution of one of the main unsolved problems of theoretical physics: that of quantum gravity; thus confirming the role of pure spinors in quantum physics. In this paper we try to show that it might be a fundamental role, since pure spinors not only are at the origin of strings, now often adopted in quantum field theory, where they substitute the Euclidean concept of point-event, but also they provide quite simple explanations to several aspects of elementary particle phenomenology, often cumbersome and obscure. Furthermore they naturally define compact manifolds in momentum spaces where a purely geometrical way may be found for the representation and easy solution of some dynamical quantum problems, as first indicated, already a long time ago, by V. Fock.