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Clifford (Geometric) Algebras With Applications in Physics, Mathematics, and Engineering, by William Baylis
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This volume is an outgrowth of the 1995 Summer School on Theoretical Physics of the Canadian Association of Physicists (CAP), held in Banff, Alberta, in the Canadian Rockies, from July 30 to August 12,1995. The chapters, based on lectures given at the School, are designed to be tutorial in nature, and many include exercises to assist the learning process. Most lecturers gave three or four fifty-minute lectures aimed at relative novices in the field. More emphasis is therefore placed on pedagogy and establishing comprehension than on erudition and superior scholarship. Of course, new and exciting results are presented in applications of Clifford algebras, but in a coherent and user-friendly way to the nonspecialist. The subject area of the volume is Clifford algebra and its applications. Through the geometric language of the Clifford-algebra approach, many concepts in physics are clarified, united, and extended in new and sometimes surprising directions. In particular, the approach eliminates the formal gaps that traditionally separate clas sical, quantum, and relativistic physics. It thereby makes the study of physics more efficient and the research more penetrating, and it suggests resolutions to a major physics problem of the twentieth century, namely how to unite quantum theory and gravity. The term "geometric algebra" was used by Clifford himself, and David Hestenes has suggested its use in order to emphasize its wide applicability, and b& cause the developments by Clifford were themselves based heavily on previous work by Grassmann, Hamilton, Rodrigues, Gauss, and others.
- Sales Rank: #1823768 in Books
- Published on: 1999-04-23
- Original language: English
- Number of items: 1
- Dimensions: 10.00" h x 1.19" w x 7.01" l, 2.55 pounds
- Binding: Hardcover
- 517 pages
Review
"Of interest due to the many provocative physical interpretations of quantum mechanics and gravitational theory suggested by the Clifford algebra approach to these theories."
―Mathematical Reviews
From the Back Cover
Leading authorities in the emerging field of Clifford (geometric) algebras have contributed to this fundamental and comprehensive text. The subject of Clifford algebras is presented here in efficient geometric language: common concepts in physics are clarified, united and extended in new and sometimes surprising directions. The text may well serve as a pedagogical tool for either self study or in courses at both the undergraduate and graduate level. Bibliographies complete many chapters and an index covers the entire book. Those new to Clifford algebras may start by reading the Introduction, after which practically any set of chapters can be read independently of the others.
Most helpful customer reviews
22 of 22 people found the following review helpful.
Good compilation
By Dr. Lee D. Carlson
This book, a compilation of 33 articles covering many different aspects and applications of Clifford algebras, can be read profitably by anyone desiring an overview of their history, theory, and applications. I did not read every article, and space also prohibits such a comprehensive review, so I will comment only on the ones that I actually studied.
Chapter introduces Clifford algebras as an extension of the real numbers to include vectors and vector products. The familiar representation in Euclidean space is outlined, with emphasis on the exterior product of two vectors, which, the author points out, is associative (unlike the ordinary cross product). The connection with rotations, reflections, and volume elements is pointed out, and the complex numbers and the Pauli algebra are shown to be Clifford algebras.
A short history of Clifford algebras is given in chapter 2. The reader not familiar with Clifford algebras should have no trouble following the ensuing discussion where some elementary geometric constructions are given of the Clifford algebra on the Euclidean plane. In addition, the operator approach to Weyl, Majorana, and Dirac operators is given, illustrating in detail their connection to physics. Recognizing that the Fierz identities do not by themselves give the Weyl and Majorana spinors, the author introduces what he calls the boomerang method for their construction. The boomerang is essentially a linear combination of bilinear covariants for a spinor, and the author details the conditions under which the spinor can be reconstructed. Interestingly, and unknown to me at the time of reading this chapter, the author constructs a new class of spinors, the "flag-dipole" spinors, that are different from the Weyl, Majorana, and Dirac spinors.
The author of chapter 3 considers the construction of Clifford algebras from a more geometric viewpoint, calling them geometric algebras, which he motivates by the consideration of extending the reals by a unipotent ( a number not equal to +1 or -1 but whose square is 1). The resulting unipodal numbers are isomorphic to the diagonal 2 x 2 matrices. The extension of the unipodal numbers so as to make this isomorphism to the full 2 x 2 matrix algebra leads to Clifford algebras.
In Chapter 9, the spacetime algebra is brought in to study electron physics. The "space-time algebra" or STA is used to characterize the observables associated with Pauli and Dirac spinors. The material presented is standard in physics, wherein the Green's function (propagator) for the Dirac equation is given, along with scattering theory. The typical problem of scattering off a potential barrier of finite width is discussed, along with the Klein paradox.
The space-time algebra is also discussed in the context of the interpretation of quantum mechanics in Chapter 11. The authors really do not add anything new here (in terms of what one might consider "strange" behavior in quantum physics). They interpret Dirac currents as measurable quantities, avoiding seemingly any notion of wave packet collapse and difficulties with defining tunneling time(s), but not answering at all how to measure these currents. In addition, the Pauli principle is interepreted in the context of space-time algebra, without any quantum field theory. Howerver, it is not shown that such an approach satisfies cluster decomposition, casting suspicion on its utility.
In Chapters 21, 22, and 23 the author shows how spinors fit into the framework of the Lorentz group, their relationship to the Clifford algebra, and in general relativity. It is shown how the Dirac spinor can be defined in three different ways, namely as an element of the representation space of the Clifford algebra of spacetime, an element of the representation space of the fundamental representation of the Dirac spinor metric-preserving automorphism group of the Clifford algebra, and as an element of the representation space of the fundamental representation of the covering group of the conformal group.
The most interesting discussion in the book is chapter 28 on extending the Grassmann algebra. Dispensing with any scalar product on a vector space, the author shows how to obtain the relative magnitude between two vectors and this leads to the notion of a multivector. The duals to these are called outer forms, and are the familiar differential forms when depending on spatial position. Many helpful diagrams are used to illustrate the properties of multivectors and pseudomultivectors, the linear span of which is called the extended Grassmann algebra of multivectors. Adding a scalar product reduces the number of directed quantities to four, and electrodynamics can be formulated in a way that is independent of the scalar product.
0 of 1 people found the following review helpful.
Five Stars
By JF
Classical book about electromagnetism with geometric algebra approach.
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