3,100 research outputs found
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
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
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
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
The Spectral Basis and Rational Interpolation
The Euclidean Algorithm is the often forgotten key to rational approximation
techniques, including Taylor, Lagrange, Hermite, osculating, cubic spline,
Chebyshev, Pade and other interpolation schemes. A unified view of these
various interpolation techniques is eloquently expressed in terms of the
concept of the spectral basis of a factor ring of polynomials. When these
methods are applied to the minimal polynomial of a matrix, they give a family
of rational forms of functions of that matrix.Comment: 8 pages, 2 figure
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