Jayme Vaz, Jr and Roldão Da Rocha, Jr
OXFORD UNIVERSITY PRESS 2016, 256 pages
PRICE (HARDBACK) £55.00 ISBN 978-0-19-878292-6
This book wonderfully captures the essence of progress in the study of Clifford algebras and spinors. Throughout the text, from the word go, the reader finds various worked examples to help understand the ideas presented.
In summary, the book is divided into six chapters with a set of exercises provided at the end of each one. To lay the foundations, key ideas and concepts from linear algebra are discussed as subsequent ideas are developed on the basis of these. The notion of the tensor algebra is introduced leading towards the definitions of exterior algebras and Grassmann algebras, which although synonymous, are presented in their own rights respectively. Three different definitions of the geometric algebras, better known as Clifford algebras, are presented in order to highlight the excellent properties of the universal Clifford algebras. Classification and representations of the Clifford algebras is followed by a study of the groups associated with them, Pin and Spin, and consequently the corresponding Lie algebras associated with these groups are presented with a brief look at one principle application: conformal transformations associated with the Minkowski space-time. The final chapter of the book is devoted to spinors. Three different types of definitions, each with its own set of properties, can be found: an algebraic spinor, a classical spinor and a spinor operator. Furthermore, the relation amongst the three definitions is explored and the reader finds that under certain sets of conditions the three otherwise distinct definitions are equivalent. The chapter ends with an elaborate study of Weyl spinors. The appendix bears witness to the standard two-component spinor formalism in the Van der Waerden framework. Concepts such as the Lorentz transformations, the classical counterparts of dotted/undotted Weyl spinors, the Penrose flagpole and supersymmetry algebra are briefly covered.
As a mathematical physicist, I was fascinated by the chapter on spinors. Roughly speaking spinors are mathematical entities that allow a more general treatment of the notion of invariance under rotations and Lorentz boosts. Spinors arise naturally in discussions of the Lorentz group. As previously mentioned, three different types of definitions are given in the book: algebraic, classical, and operational, each emphasising a different perspective. The algebraic and classical definitions are well known, while the third definition is gaining prominence in the literature slowly. In physics, spinors emerged as a result of the study of both relativistic and non-relativistic quantum mechanics.
In 1975, Haag, Lopuszanski and Sohnius presented their proof that by weakening the assumptions of the Coleman-Mandula theorem and allowing both commuting and anti-commuting symmetry generators, there is a non-trivial extension of the Poincaré algebra, namely the supersymmetry (SUSY) algebra. Aspects of the SUSY algebra where spinors arise in the form of dotted/undotted Weyl spinors are briefly discussed in the appendix of the book.
To conclude, this book provides a unique pedagogical approach to the formal developments in the study of Clifford algebras with a focus on spinors, and bridges the gap between mathematics and physics, accompanied with several illustrative examples.
Johar Ashfaque AMIMA
Book review published in Mathematics Today February 2017
