Special relativity in complex vector algebra

Many different mathematical languages have been invented to describe the ideas of Einstein's special relativity. One of the most powerful languages is the Minkowski space-time algebra of D. Hestenes. We discuss the ideas of special relativity in a complex 4-dimensional algebra of observables, which is algebraically isomorphic to the even subalgebra of Hestenes' space-time algebra.Comment: 11 pages, 3 figures, PIRT Moscow, July 200

Spinors in Spacetime Algebra and Euclidean 4-Space

This article explores the geometric algebra of Minkowski spacetime, and its relationship to the geometric algebra of Euclidean 4-space. Both of these geometric algebras are algebraically isomorphic to the 2x2 matrix algebra over Hamilton's famous quaternions, and provide a rich geometric framework for various important topics in mathematics and physics, including stereographic projection and spinors, and both spherical and hyperbolic geometry. In addition, by identifying the time-like Minkowski unit vector with the extra dimension of Euclidean 4-space, David Hestenes' Space-Time Algebra of Minkowski spacetime is unified with William Baylis' Algebra of Physical Space.Comment: 16 pages, 3 figure

Geometrization of the Real Number System

Geometric number systems, obtained by extending the real number system to include new anticommuting square roots of +1 and -1, provide a royal road to higher mathematics by largely sidestepping the tedious languages of tensor analysis and category theory. The well known consistency of real and complex matrix algebras, together with Cartan-Bott periodicity, firmly establishes the consistency of these geometric number systems, often referred to as Clifford algebras. The geometrization of the real number system is the culmination of the thousands of years of human effort at developing ever more sophisticated and encompassing number systems underlying scientific progress and advanced technology in the 21st Century. Complex geometric algebras are also considered.Comment: This new version shows how Cartan periodicity of Clifford algebras is related to Bott periodicity of homotopy groups and Hurwitz-Radon numbers. It also corrects a number of typos and adds a new section. 15 pages, 6 table

Classical r-matrices and construction of quantum groups

A problem of constructing quantum groups from classical r-matrices is discussed.Comment: 9 page

Part I: Vector Analysis of Spinors

Part I: The geometric algebra of space is derived by extending the real number system to include three mutually anticommuting square roots of plus one. The resulting geometric algebra is isomorphic to the algebra of complex 2x2 matrices, also known as the Pauli algebra. The so-called spinor algebra of C(2), the language of the quantum mechanics, is formulated in terms of the idempotents and nilpotents of the geometric algebra of space, including its beautiful representation on the Riemann sphere, and a new proof of the Heisenberg uncertainty principle. In "Part II: Spacetime Algebra of Dirac Spinors", the ideas are generalized to apply to 4-component Dirac spinors, and their geometric interpretation in spacetime.Comment: 20 pages, 4 figure

What's in a Pauli Matrix?

Why is it that after so many years matrices continue to play such an important roll in Physics and mathematics? Is there a geometric way of looking at matrices, and linear transformations in general, that lies at the roots of their success? We take an in depth look at the Pauli matrices, 2x2 matrices over the complex numbers, and examine the various possible geometric interpretations of such matrices. The geometric interpretation of the Pauli matrices explored here natualy extends to what the author has dubbed the study of "geometric matrices". A geometric matrix is a matrix of order 2^n x 2^n over the real or complex numbers, and has its geometric roots in its algebraically isomorphic Clifford geometric algebras.Comment: 19 pages, 2 figures, 1 tabl

Generalized Taylor's Theorem

The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to include any number of points with given multiplicities. Taken together with a simple expression for the remainder, this theorem becomes a powerful tool for approximation and interpolation in numerical analysis. We also have a corresponding theorem for rational approximation.Comment: 3 page

Geometric Number Systems and Spinors

The real number system is geometrically extended to include three new anticommuting square roots of plus one, each such root representing the direction of a unit vector along the orthonormal coordinate axes of Euclidean 3-space. The resulting geometric (Clifford) algebra provides a geometric basis for the famous Pauli matrices which, in turn, proves the consistency of the rules of geometric algebra. The flexibility of the concept of geometric numbers opens the door to new understanding of the nature of space-time, and of Pauli and Dirac spinors as points on the Riemann sphere, including Lorentz boosts.Comment: 7 pages, 2 figures, Conference paper: http://www.fcfm.buap.mx/cima/2015/programa

Notes on Plucker's relations in Geometric Algebra

Grassmannians are of fundamental importance in projective geometry, algebraic geometry, and representation theory. A vast literature has grown up utilizing using many different languages of higher mathematics, such as multilinear and tensor algebra, matroid theory, and Lie groups and Lie algebras. Here we explore the basic idea of the Plucker relations in Clifford's geometric algebra. We discover that the Plucker Relations can be fully characterized in terms of the geometric product, without the need for a confusing hodgepodge of many different formalisms and mathematical traditions found in the literature.Comment: 8 pages, 1 figur

Geometric Matrices and the Symmetric Group

We construct real and complex matrices in terms of Kronecker products of a Witt basis of 2n null vectors in the geometric algebra over the real and complex numbers. In this basis, every matrix is represented by a unique sum of products of null vectors. The complex matrices provide a direct matrix representation for geometric algebras with signatures p+q <= 2n+1. Properties of irreducible representations of the symmetric group are presented in this geometric setting.Comment: 15 page