## Geometric Algebra book requests for January 2012

I sent our department’s library coordinator the following list of book requests to be sent to the Ateneo de Manila University Library.  I culled the list from my Geometric Algebra Facebook page:

1. “Geometric Algebra Computing: In Engineering and Computer Science” (2010) by Bayro-Corrochano, Gerik Scheuermann
2. “Classical Groups and Geometric Algebra” (2001) by Larry C. Grove
3. “Geometric Algebra and Applications to Physics (2006) by Bidyut Kumar Datta and Venzo de Sabbata
4. “A New Approach to Differential Geometry using Clifford’s Geometric Algebra” (2012) by John Snygg
5. “Quaternions, Clifford Algebras, and Relativistic Physics” (2007) by Patrick R. Girard
6. “Clifford Algebras and Their Application in Mathematical Physics” (1993) by Gerhard Jank, Klaus Habetha, and Volker Dietrich
7. “Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry” (2007) by Leo Dorst, Daniel Fontijne, and Stephen Mann

## JOSA A 2012: Polarization ellipse and Stokes parameters in geometric algebra

Adler G. Santos, Quirino M. Sugon, Jr., and Daniel J. McNamara, “Polarization ellipse and Stokes parameters in geometric algebra,” J. Opt. Soc. Am. A 29, 89-98 (2012)

Abstract

In this paper, we use geometric algebra to describe the polarization ellipse and Stokes parameters. We show that a solution to Maxwell’s equation is a product of a complex basis vector in Jackson and a linear combination of plane wave functions. We convert both the amplitudes and the wave function arguments from complex scalars to complex vectors. This conversion allows us to separate the electric field vector and the imaginary magnetic field vector, because exponentials of imaginary scalars convert vectors to imaginary vectors and vice versa, while exponentials of imaginary vectors only rotate the vector or imaginary vector they are multiplied to. We convert this expression for polarized light into two other representations: the Cartesian representation and the rotated ellipse representation. We compute the conversion relations among the representation parameters and their corresponding Stokes parameters. And finally, we propose a set of geometric relations between the electric and magnetic fields that satisfy an equation similar to the Poincaré sphere equation.

© 2012 Optical Society of America

Original Manuscript: July 7, 2011
Revised Manuscript: October 24, 2011
Manuscript Accepted: October 27, 2011
Published: December 12, 2011

## Geometric algebra now has a Facebook page

William Kingdon Clifford

I created a Facebook page for geometric algebra: http://www.facebook.com/geometricalgebra.    It is difficult to post the latest updates on geometric algebra and Clifford algebra through this blog.  Facebook makes it easier for me to just simply copy the webpage’s url and first paragraph, and voila! its a blog post, with a preview on the actual view of the original webpage.  So if you need to get the latest updates on geometric algebra, subscribe to the Facebook page.

The mission and vision of the geometric algebra Facebook page is to collect in one repository the web pages I cull from Google Alerts for CA and GA.  If I need to lengthy comment on a particular entry, I shall do it in this blog.  I will try to make this blog more personal–my thoughts on the subject–rather than duplicate information from somewhere else.  Google does not love copycats, as I learned from books on Search Engine Optimization.  What this blog will do is to organize the references in the Facebook page, particularly the books and journal articles, which are handy when writing research papers.  The books I shall really keep track and ask the Ateneo de Manila University Library to purchase them.  My aim is to make the Ateneo Library host the best geometric algebra  collection in this part of the world.

## Modern Geometric Algebra: A (Very Incomplete!) Survey by Jeff Suzuki

from the Mathematics Teacher

August 2009, volume 103, issue 1, page 26

Abstract

Representing products as rectangles can be used to introduce, connect, and reinforce concepts across mathematics.

Keywords:
Article

# John W. Bales

(Submitted on 3 Aug 2011)

Gives an elementary exposition of the twisted group algebra rep- resentation of simple Clifford algebras

 Subjects: Rings and Algebras (math.RA) Cite as: arXiv:1108.0953v1 [math.RA]

## Geometric Algebra and Applications to Physics: a book by Venzo de Sabbata and Bidyut Kumar Datta

from Amazon:

### Product Description

Bringing geometric algebra to the mainstream of physics pedagogy, Geometric Algebra and Applications to Physics not only presents geometric algebra as a discipline within mathematical physics, but the book also shows how geometric algebra can be applied to numerous fundamental problems in physics, especially in experimental situations.

This reference begins with several chapters that present the mathematical fundamentals of geometric algebra. It introduces the essential features of postulates and their underlying framework; bivectors, multivectors, and their operators; spinor and Lorentz rotations; and Clifford algebra. The book also extends some of these topics into three dimensions. Subsequent chapters apply these fundamentals to various common physical scenarios. The authors show how Maxwell’s equations can be expressed and manipulated via space-time algebra and how geometric algebra reveals electromagnetic waves’ states of polarization. In addition, they connect geometric algebra and quantum theory, discussing the Dirac equation, wave functions, and fiber bundles. The final chapter focuses on the application of geometric algebra to problems of the quantization of gravity.

By covering the powerful methodology of applying geometric algebra to all branches of physics, this book provides a pioneering text for undergraduate and graduate students as well as a useful reference for researchers in the field.

Harcover, 184 pages, Taylor and Francis, 1st Edition, 2 December 2006.

## Conformal geometric algebra in Wikipedia

In mathematics, more specifically in the area of computational geometryconformal geometric algebra (CGA) is the geometric algebra(Clifford algebra) that results if an n-dimensional Euclidean (or pseudo-Euclidean) space ℝp,q is projectively mapped into ℝp+1,q+1 in a particular way, and a geometric algebra is then constructed over that n+2 dimensional space. This allows translations of the n-dimensional space, as well as rotations, to be represented using elements of a geometric algebra; and it is found that spheres and circles as well as lines and planes gain particularly natural, as well as computationally amenable, representations.

The projective mapping from ℝp,q to ℝp+1,q+1 is constructed so that the higher dimensional space contains the full projective space of the underlying original space; so that generalized spheres in the underlying space map onto (hyper-)planes orthogonal to a particular vector inn+2 dimensional space (which can thus be used to represent them); and so that the effect of a translation (or in fact any conformal mapping) of the underlying space corresponds to the effect of an n+2 dimensional rotation in the higher dimensional space.

Introduction of the geometric algebra of this space, based on the Clifford product of vectors, allows such transformations to be represented using the algebra’s characteristic sandwich operations, similar to the use of quaternions for spatial rotation in 3D, which combine very efficiently using the Clifford product rule. A consequence of the sandwich transformation structure is that the representations of spheres, planes, circles and other geometrical objects, and equations connecting them, all transform covariantly. Some intersection operations also acquire a very tidy algebraic form: for example, applying the wedge product to the vectors corresponding to two spheres produces abivector corresponding to their circle of intersection; alternatively the tetra-vector of the hypersurface which is the dual of such a vector can be decomposed as the successive wedge product of vectors representing four points on the sphere.