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Geometric algebra for physicists. (English) Zbl 1078.53001
Cambridge: Cambridge University Press (ISBN 0-521-48022-1/hbk). xiv, 578 p. (2003).
This book grew out of an undergraduate lecture course taught at physics department of Cambridge University (UK). Although the title “Geometric Algebra” is somewhat misleading the book is exceptionally well written. To fix the problem with the title, the authors provide an explanation at the very begining as to what they actually had in mind by choosing “Geometric Algebra” for the title. Under the umbrella of this title they included actually various applications of the Clifford, Grassmann and quaternion algebras to various physical problems. Although the book presents material in historical perspective, I was surprised not to find names of Cartan, PoincarĂ©, Chern and many other less famous mathematicians whose contribution to the subject matters of this book can hardly be underestimated.
Given that the book is aimed at physicists this is not completely surprising. Nevertheless, in its present form it serves to preserve and maintain the boundaries between physical and mathematical ways of looking at nature. Other than this, the subject matters are covered in the book with exceptional clarity which is especially valuable for those who are just entering science.
The book consists of 14 chapters. Selection of material for these chapters is a bit subjective but this is true for any book of such kind. The authors are willing to convince readers that Clifford, Grassmann and quaternion algebras provide the language of modern physics. Their claims are illustrated by examples from classical mechanics, Chapter 3, classical electrodynamics, Chapter 7, special relativity, Chapter 5, gravity, Chapter 14. In addition, the authors provide a rather sketchy (as compared with classical mechanics) description of single particle quantum mechanics, Chapter 8, and, even more sketchy, the multiparticle quantum mechanics, including quantum entanglement (Chapter 9). The authors condense all geometrical notions in a single Chapter 10 supplemented by even sketchier facts from group theory in Chapter 11. They use the content of these chapters later in Chapter 13 on symmetry and gauge theory and in Chapter 14 on gravitation.
In my opinion, the students reading such a book can learn a lot in a rather short time. Hopefully, in future editions the authors may want to provide some important guide for further reading supplemented with similar historical remarks, e.g. in the style accepted in the series of monographs by Bourbaki. This might help younger people to keep a balanced view on various aspects of mathematical physics.

53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
15A66 Clifford algebras, spinors
53C27 Spin and Spin\({}^c\) geometry
53C80 Applications of global differential geometry to the sciences
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis