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Title: Clifford (Geometric) Algebras With Applications to Physics, Mathematics, and Engineering by William E. Baylis, Summer School on Theoretical Physics of the Canadian Association of Ph ISBN: 0-8176-3868-7 Publisher: Birkhauser Boston Pub. Date: 01 July, 1996 Format: Hardcover Volumes: 1 List Price(USD): $76.95 |
Average Customer Rating: 4 (1 review)
Rating: 4
Summary: Good compilation
Comment: 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.
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Title: Geometric Algebra for Physicists by Chris Doran, Anthony Lasenby ISBN: 0521480221 Publisher: Cambridge University Press Pub. Date: 01 May, 2003 List Price(USD): $95.00 |
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Title: On Quaternions and Octonions by John Horton Conway, Derek Alan Smith ISBN: 1568811349 Publisher: AK Peters, Ltd. Pub. Date: 01 January, 2003 List Price(USD): $29.00 |
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Title: Clifford Algebras and Spinors (London Mathematical Society Lecture Note Series) by Pertti Lounesto, J. W. S. Cassels ISBN: 0521005515 Publisher: Cambridge University Press Pub. Date: 03 May, 2001 List Price(USD): $55.00 |
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