Table of Contents

Selected Publications


University of Windsor
Essex Hall
401 Sunset Avenue Windsor, Ontario Canada N9B 3P4
Phone: (519) 253-3000 ext:2673
FAX:(519) 973-7075

William Baylis

Office Hours
MW 14-17 or by appointment

Welcome to the website for my research on Clifford's Geometric Algebra of Physical Space (APS). The intention is to explain the use of paravectors in the algebra to model spacetime. For other information on my research, please see my webpage under Faculty Overview at

Paravectors are sums of scalars and vectors. If the vector space is 3-dimensional physical space with a Euclidean metric, the paravectors form a linear space of four dimensions with the Minkowski spacetime metric. They provide a covariant formulation of relativity that avoids matrix and tensor components but brings powerful tools such as spinors and projectors.

APS allows an introduction to relativity, suitable for beginning physics students, that avoids the mathematical formalisms of matrices and tensors. It lets students concentrate on the fascinating geometry of spacetime without getting lost in the math. With a minimal subset of the full algebra, students can perform quantitative Lorentz transformations (boosts and rotations)RelEasyPs.pdf (also available as

To gain a working knowledge of geometric algebra and an introduction to APS, work through the attached workbook. You will need Adobe Acrobat Reader to view the file: gaworkbook.pdf(421 kB).

Here's a somewhat more detailed lecture on Applications of Clifford's Geometric Algebra in Physics: cainphys.pdf(604 kB)

An application of APS to the motion of charges in electromagnetic waves, with and without axial electric or magnetic fields, is given in this 1999 Phys. Rev aritcle:PRA00785.pdf, and an application to understanding electromagnetic radiation is given in this preprint (new version June 21, 2003):emrad.pdf.

A paper with Garret Sobczyk compares relative and absolute formulations of special relativity in APS and STA. See

Two recent papers that use paravectors in APS: Geometry of Paravector Space, with Applications to Relativistic Physics: baylis1.pdfand Quantum/Classical Interface: an Approach from the Classical Side: baylis2.pdf
An application of multiple APS to quantum computation: QComp.pdf

Bill Baylis

Comments about our web pages? Send e-mail to: Web Administrator, University of Windsor. Last modified on 20/08/2004 by CN=William Baylis/O=University of Windsor. Copyright 2007, University of Windsor. Although care has been taken in preparing the information in this site the University of Windsor cannot guarantee its accuracy.