Geometry, spinors and applications.

*(English)*Zbl 0933.53001
London: Springer. Chichester: Praxis Publishing, xvii, 369 p. (2000).

The present monograph is an exploration and exposition of the general ideas required for an understanding of spinors. As such it includes a variety of topics ranging from linear and multilinear algebra, fiber bundles, Lie groups, Clifford algebras, representation theory, to differential forms.

Contents include the following chapters: Preliminaries and algebraic aspects of spinors (70 pages); Preliminaries and geometric aspects of spinors (122 pages); General spinorial differentiation (72 pages); Illustrations and applications (82 pages); and two Appendices on the Infeld-van der Waerden symbols, and Maxwell’s equations and complements (12 pages).

While the exposition is clear and often insightful, the approach is primarily mathematical with emphasis on the fundamental ideas, rather than the development of a concrete methodology or an explanation of the formalism currently employed in the literature. Thus the present monograph will be useful mostly to someone interested in the mathematical preliminaries, but not as a means of learning how spinors are actually used in general relativity; see, e.g., the text of R. Penrose and W. Rindler [“Spinors and space-time”, Vol. I (1986; Zbl 0538.53024)]. However, with this caveat, the book is nicely done, and a welcome addition to the growing spinor literature.

Contents include the following chapters: Preliminaries and algebraic aspects of spinors (70 pages); Preliminaries and geometric aspects of spinors (122 pages); General spinorial differentiation (72 pages); Illustrations and applications (82 pages); and two Appendices on the Infeld-van der Waerden symbols, and Maxwell’s equations and complements (12 pages).

While the exposition is clear and often insightful, the approach is primarily mathematical with emphasis on the fundamental ideas, rather than the development of a concrete methodology or an explanation of the formalism currently employed in the literature. Thus the present monograph will be useful mostly to someone interested in the mathematical preliminaries, but not as a means of learning how spinors are actually used in general relativity; see, e.g., the text of R. Penrose and W. Rindler [“Spinors and space-time”, Vol. I (1986; Zbl 0538.53024)]. However, with this caveat, the book is nicely done, and a welcome addition to the growing spinor literature.

Reviewer: J.D.Zund (Las Cruces)

##### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

15A66 | Clifford algebras, spinors |

83C60 | Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism |

53C27 | Spin and Spin\({}^c\) geometry |

55R10 | Fiber bundles in algebraic topology |

58A10 | Differential forms in global analysis |