arXiv subjects

Garret Sobczyk

Publications and source records attributed to Garret Sobczyk.

At least 19 recordsLinked to original sources

arXiv2023

Global and Local Nilpotent Bases of Matrices

In studying the unusual properties of a special Witt basis of a Clifford geometric algebra with a Lorentz metric, a new concept of local duality makes it possible to define any real geometric algebra by complexifying this structure. Whereas a global basis of Witt null vectors is defined in terms of a pair of correlated Grassmann algebras in a geometric algebra of neutral signature, the special Witt basis of a Lorentz geometric algebra is defined in terms of a single Grassmann algebra. The relationship between these different concepts of duality, and their matrix representations, is studied in terms of simple examples. A surprising connection is exhibited between discrete Fourier and Wavelet transforms and the concept of local duality.

math-ph↗

arXiv2023

Itinerant Quantum Integers: The Language of Quantum Computers

The concept of positively and negatively compatible null vectors arises in the study of Clifford geometric algebras with a Lorentz-Minkowski metric. In previous works, the basic properties of such algebras have been set down in terms of a new principle of quantum duality. In the present work, the same structure is studied in terms of real and complex quantum integers, which generalize the real and complex number systems. It seems natural to identify a qubit as a pair of compatible null vectors; the up state of the qubit being their sum, and the down state being their difference. Basic identities are developed to make calculations routine, and two different representations of the symmetric group are given.

physics.gen-ph↗

arXiv2023

Geometric Algebras of Light Cone Projective Graph Geometries

A null vector is an algebraic quantity with square equal to zero. I denote the universal algebra generated by taking all sums and products of null vectors over the real or complex numbers by N. The rules of addition and multiplication in N are taken to be the familiar rules of addition and multiplication of real or complex square matrices. A pair of null vectors is positively or negatively correlated if their inner product is positive or negative, respectively. A large class of (Clifford) geometric algebras are isomorphic to real or complex matrix algebras, or a pair of such algebras. I begin the study of the eigenvector-eigenvalue problem of linear operators in the geometric algebra G(1,n) of R^{n+1}, and by restricting to barycentric coordinates, n-simplices whose n+1 vertices are non-zero null vectors. These ideas provide a foundation for a new Cayley-Grassmann Theory of Linear Algebra, with many possible applications in pure-applied areas of science and engineering.

math.GM↗

arXiv2021

Spheroidal Domains and Geometric Analysis in Euclidean Space

Clifford's geometric algebra has enjoyed phenomenal development over the last 60 years by mathematicians, theoretical physicists, engineers and computer scientists in robotics, artificial intelligence and data analysis, introducing a myriad of different and often confusing notations. The geometric algebra of Euclidean 3-space, the natural generalization of both the well-known Gibbs-Heaviside vector algebra, and Hamilton's quaternions, is used here to study spheroidal domains, spheroidal-graphic projections, the Laplace equation and its Lie algebra of symmetries. The Cauchy-Kovalevska extension and the Cauchy kernel function are treated in a unified way. The concept of a quasi-monogenic family of functions is introduced and studied.

math.GM↗

arXiv2020

Nested Coordinate Systems in Geometric Algebra

A nested coordinate system is a reassigning of independent variables to take advantage of geometric or symmetry properties of a particular application. Polar, cylindrical and spherical coordinate systems are primary examples of such a regrouping that have proved their importance in the separation of variables method for solving partial differential equations. Geometric algebra offers powerful complimentary algebraic tools that are unavailable in other treatments.

math.GM↗

arXiv2020

Periodic Table of Geometric Numbers

Perhaps the most significant, if not the most important, achievements in chemistry and physics are the Periodic Table of the Elements in Chemistry and the Standard Model of Elementary Particles in Physics. A comparable achievement in mathematics is the Periodic Table of Geometric Numbers discussed here. In 1878 William Kingdon Clifford discovered the defining rules for what he called geometric algebras. We show how these algebras, and their coordinate isomorphic geometric matrix algebras, fall into a natural periodic table, sidelining the superfluous definitions based upon tensor algebras and quadratic forms.

math.GM↗

arXiv2019

Scaffolding of Spacetime

The mathematical foundations of relativistic quantum mechanics is largely based upon the discovery of the Pauli and Dirac matrices. An algebra which lies at an even more fundamental level is the geometric Clifford algebra with metric signature (2,3)=(++---). In this geometric algebra both the fundamental Pauli vectors of space and the Dirac vectors of spacetime are factored into complex bivectors.

physics.gen-ph↗

arXiv2019

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.

math.GM↗

arXiv2018

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.

math.GM↗

arXiv2018

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.

math.GM↗

arXiv2018

From Vectors to Geometric Algebra

Geometric algebra is the natural outgrowth of the concept of a vector and the addition of vectors. After reviewing the properties of the addition of vectors, a multiplication of vectors is introduced in such a way that it encodes the famous Pythagorean theorem. Synthetic proofs of theorems in Euclidean geometry can then be replaced by powerful algebraic proofs. Whereas we largely limit our attention to 2 and 3 dimensions, geometric algebra is applicable in any number of dimensions, and in both Euclidean and non-Euclidean geometries.

math.GM↗

arXiv2017

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.

math.GM↗

arXiv2017

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.

math.GM↗

arXiv2015

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.

physics.gen-ph↗

arXiv2015

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.

math-ph↗

arXiv2015

Part II: Spacetime Algebra of Dirac Spinors

In "Part I: Vector Analysis of Spinors", the author studied the geometry of two component spinors as points on the Riemann sphere in the geometric algebra of three dimensional Euclidean space. Here, these ideas are generalized to apply to four component Dirac spinors on the complex Riemann sphere in the complexified geometric algebra of spacetime, which includes Lorentz transformations. The development of generalized Pauli matrices eliminate the need for the traditional Dirac gamma matrices. We give the discrete probability distribution of measuring a spin 1/2 particle in an arbitrary spin state, assuming that it was prepared in a given state immediately prior to the measurement, independent of the inertial system in which measurements are made. The Fierz identities between the physical observables of a Dirac spinor are discussed.

math-ph↗

arXiv2008

Fundamental Theorem of Calculus

A simple but rigorous proof of the Fundamental Theorem of Calculus is given in geometric calculus, after the basis for this theory in geometric algebra has been explained. Various classical examples of this theorem, such as the Green's and Stokes' theorem are discussed, as well as the new theory of monogenic functions, which generalizes the concept of an analytic function of a complex variable to higher dimensions.

math.HO↗

arXiv2008

Principle of Local Conservation of Energy-Momentum

Starting with Einstein's theory of special relativity and the principle that whenever a celestial body or an elementary particle, subjected only to the fundamental forces of nature, undergoes a change in its kinetic energy then the mass-energy equivalent of that kinetic energy must be subtracted from the rest-mass of the body or particle, we derive explicit equations of motion for two falling bodies. In the resulting mathematical theory we find that there are no singularities and consequently no blackholes.

math-ph↗