An Introduction to Clifford Algebras and Spinors . By Jayme Vaz Jr and Roldão da Rocha Jr. Oxford University Press, 2019. Paperback, pp. 256. Price GBP 34.99. ISBN 9780198836285.

An Introduction to Clifford Algebras and Spinors by two Brazilian physicists, Jayme Vaz Jr and Roldão da Rocha Jr of IMECC, Universidade Estudual de Campinas and CMCC – Universidade Federal do ABC, was first published by Oxford University Press in hardcover in 2016 and republished in 2019 as a paperback. The 256-page book begins with a sweeping preface that relates the historic works on unifying geometric and algebraic operations by celebrities like G. Leibnitz, J.-R. Argand, C. F. Gauss, R. W. Hamilton, H. G. Grassmann, W. K. Clifford, E. Cartan, W. Pauli, P. Dirac etc. For crystallographers it may be of interest to know that J. G. Grassmann (Justus G. was the father of Hermann G. Grassmann) originally introduced the characterization of crystal planes by orthogonal vectors, now commonly denoted with Miller indices [see Erhard Scholz in Schubring (1996), pp. 37–46]. J. G. Grassmann’s work, including his mathematical school textbooks, provided H. G. Grassmann with fertile ideas for his new concepts of algebra, solely defined by the relations of its elements, from which G. Peano distilled the modern concept of vectors. Grassmann’s pioneering approach was so far ahead of its time that only a few bright minds (like R. W. Hamilton, F. Klein and S. Lie) recognized its genius during his lifetime, late in Grassmann’s life. But the young Cambridge-educated genius W. K. Clifford was truly exceptional, and published in 1878 (one year after Grassmann’s death) his seminal paper ‘Applications of Grassmann’s Extensive Algebra’ in Am. J. Math. It elegantly unified the earlier works of Hamilton on quaternions and Grassmann’s metric-free algebra of extension to geometric algebras (now known as Clifford algebras), by simply adding in the Clifford (or geometric) product the inner product of vectors (necessary for measurements) and the outer product of Grassmann. The authors began to develop their subject 20 years ago in the form of lecture notes in several university courses and have plenty of their own publications on the subject. The text is divided into seven chapters, each completed with a final subsection full of exercises and recommendations for further reading. Readers will have to work their way through the subject-specific notation, for which the well-ordered seven-page Appendix B ‘List of Symbols’ provides essential support. References (7 pp.) and an index (4 pp.) complete the backmatter. The book may therefore in parts be used for lecture courses to graduate students, or alternatively for self-study and as a launch pad for further research. It is of interest to anybody who wants to understand a modern description of spinors in Clifford algebra language. The level is clearly graduate student or above. It will be easiest to read for physicists and mathematicians, but crystallographers with sufficient mathematical background in linear algebra, tensor algebra and group theory should find it accessible. For readers who want an introduction to Clifford algebras, which will make reading this introduction to spinors easier, I would recommend the two college-level textbooks by MacDonald (2011, 2012) and the recent Springer Brief by XambóDescamps (2018). In ‘Preliminaries’ (Chapter 1) vectors are introduced with eight axioms for addition and scalar multiplication (scalars are here taken as elements of a field, e.g. real and complex numbers and quaternions). For quaternions left and right multiplication must be distinguished. Then the dual vector space of linear functionals mapping vectors into scalars equipped with a covector basis is defined. Covariant and contravariant transformations are distinguished and their practical use is demonstrated with two pages of ISSN 2053-2733