Tech@Addebook

Clifford Algebra: A Computational Tool for Physicists

Clifford Algebra: A Computational Tool for Physicists
By John Snygg
Publisher: Oxford University Press, USA
Number Of Pages: 356
Publication Date: 1997-02-06
ISBN-10 / ASIN: 0195098242
ISBN-13 / EAN: 9780195098242
Binding: Hardcover

Clifford algebras have become an indispensable tool for physicists at the cutting edge of theoretical investigations. Applications in physics range from special relativity and the rotating top at one end of the spectrum, to general relativity and Dirac’s equation for the electron at the other. Clifford algebras have also become a virtual necessity in some areas of physics, and their usefulness is expanding in other areas, such as algebraic manipulations involving Dirac matrices in quantum thermodynamics; Kaluza-Klein theories and dimensional renormalization theories; and the formation of superstring theories. This book, aimed at beginning graduate students in physics and math, introduces readers to the techniques of Clifford algebras.

Summary: A gem!
Rating: 5

I would rate this book as a gem! To calibrate that let me say that I think Weinreich’s Geometrical Vectors and Foster and Nightingale’s General Relativity are gems. Chapter 1 gives a beautiful, clear and concise introduction to Clifford Algebra in flat 3-space using Dirac’s anti-commuting gamma matrices. If you have ever wondered about off-hand comments that rotations are double reflections and why half angles enter into this business this is the place to get enlightened. In an amusing series of photographs the author illustrates the 4-pi periodicity of certain objects. The object here is a copy of MTW’s Gravitation - one of the more imaginative uses of this tome. As an example of the application of the CA results the chapter ends with a treatment of the spinning top without using Euler’s equations for rigid body motion. If you have ever struggled through Goldstein’s Classical Mechanics treatment of this problem, from Euler angles to infinitessimal rotations to d-Omega which is not a differential of a vector to dyadics to body diferentials and space differentials to Euler’s equations, you will really appreciate Snygg’s direct solution using CA. Sure, I know Goldstein’s has to be a general treatment of solid body motion and thus more complex so he can treat more general problems, but it is good to find a more direct solution that is cristal clear and only a few pages long. This chapter is real little gem. Chapter 2 takes CA to Minkowsky 4-space rotations. Chapters 3 and 4 take you to flat n-dimensional spaces and curved subspaces embedded in them. Again beautiful explanations are presented of the meaning of tangent spaces, parallel transport and how the covariant derivative arises naturally in curved spaces. I had the silly hope that with Clifford numbers and their products all would be well and done. Unfortunately the exterior product wedges its nose under the tent flap and pretty soon the exterior derivative and its side-kick the co-differential operator soon follow it into the tent. All this is explained in Chapter 7. With Chapter 5 the learning curve steepens with the introduction of Fock-Ivanenko 2-vectors and the curvature 2-vector (or 2-form) and finally the curvature tensor. Chapter 6 solves the field equations for the Schwartzchild metric based on the F-I 2-vector approach. Chapter 8 on the Dirac equation is again an approach different than that found in the usual texts. Chapter 9 derives the Kerr metric, something you won’t find in MTW published 8 to 10 years after Kerr’s papers. Unfortunately the starting point is some obscure problem from an earlier chapter and Snygg does not provide the delightful physical insight of earlier examples. However, there is discussion at the of the chapter. While you might be able to solve the field equations for the Schartzchild metric on your own, once you know it can be done, I certainly would not be able to do so for the Kerr metric. Snygg takes you through step by step, none of them particularly difficult, but the sequence is certainly not something I would have found by myself. Chapter 10 I only skimmed, the index notation, with underscored and bracketed indices, becomes overloaded for my level of sophistication. Chapter 11 organizes all the matrix stuff together, again a beautiful, straightforward and clear presentation. Here is shown how to construct a matrix representation for the gammas. As you might expect, the book is a veritable beehive of sub- and superscripts over bars and carets Greek and Latin indices and full of gamma gymnastics. Even Pauli’s less complementary comment on Dirac algebra comes to mind. The text has a few typos but blessedly few in the Clifford number and gamma indices. By the way, if you expect to find out how to do trace computations on gamma expressions you won’t find it here. The explicit form of the gamma matrices is hardly ever mentioned until chapter 11 nor is it needed in the present context.

Summary: Light, clear, and understandable.
Rating: 5

Snygg’s book is a thoroughly delightful introduction to Clifford Algebra and its applications in physics. It is detailed, readable, and at times even humorous… but always clear and educational. Snygg presents Clifford Algebra above all as a practical tool, rather than as the ultimate algebraic representation of spatial geometry. This gives a refreshing alternative to Hestenes’ writings which, although quite good, can at times be philosophically pedantic and difficult to connect with standard theory.

Summary: Incredibly good!
Rating: 5

From a letter to the author. John, I have to write you to tell you what a wonderful book you wrote. I still can’t believe how good it is. Yesterday I was waiting for a television show to begin in ten minutes and I picked your book up while I sat front of the TV set. When I finally looked up 45 minutes later, I had missed the show! In 35 years as an algebraic topologist, I have tried to learn various things about Clifford algebras because of their role in K-theory and in the Atiyah- Singer Index theorem, and more recently because of the Seiberg-Witten equations. With only mediocre intensity, mostly browsing, I have had little success. In the month since I met you and bought your book, I have browsed through it while occupied with several other competing projects. In the process I have internalized the classification of Clifford algebras, learned how physicists use Dirac’s equation, what they are doing when they talk about gauge theory, understood Hodge duality much better and so the codifferential operator. And I still have only browsed through a small portion of the text. I think we mathematicians should study your book to learn how to improve our own levels of exposition. Sincerely, Daniel Henry Gottlieb

Summary: An educational tool for Clifford algebras
Rating: 5

The author writes in the Introduction: “Much of Clifford algebra is quite simple. If this fact were generally recognized, Clifford algebra would be more widely used as a computational tool.” However, a few applications discussed in the book may require some physics usually covered in the first year of graduate school.

The author starts by Clifford algebras of the 3-dimensional Euclidean space and the 4-dimensional Minkowski space-time. He discusses the Maxwell equations in flat space, the Dirac equation for the electron, the Dirac operator, spherical harmonics, and curved space-times with Schwarzschild and Kerr metrics. The book ends with matrix representation and classification of Clifford algebras of real non-degenerate quadratic spaces and an appendix on Lorentz transformations.

Summary: A pedagogical gem.
Rating: 5

The author is my brother. You may doubt the credibility of my commentary, but check it out for yourself. Based on an early draft, I predict it will be a pedagogical gem. Style and content make anything John Snygg writes a pleasure to read. Snygg is a hidden treasure. As of October 1996, “Clifford Algebra: A Computational Tool for Physicists” was not out.

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