Spinors in physics. Transl. from the French by J. Michael Cole.

*(English)*Zbl 0923.53001
Graduate Texts in Contemporary Physics. New York, NY: Springer. xi, 226 p. (1999).

Over the last thirty years, a number of excellent monographs and basic textbooks on the theory of spinors have appeared. For the purposes of the present review, these may be roughly divided into two categories: those devoted to the basic theory of spinors and their geometric interpretation (without linking them to specific physical applications such as general relativity); and those primarily concerned with the explicit development of a spinor apparatus for a physical theory, e.g., general relativity.

The first category includes the English translation of É. Cartan’s 1937-38 master work [‘The theory of spinors’ (Hermann, Paris) (1966; Zbl 0147.40101)] and the book of M. Morand [‘Géométrie spinorielle’ (Masson, Paris) (1973; Zbl 0257.53019)], while the second category is prominently represented by the book of R. Penrose and W. Rindler [‘Spinors and space-time, Vol. 1: Two-spinor calculus and relativistic fields’ (Cambridge University Press) (1984; Zbl 0538.53024)].

The present monograph clearly belongs in the first category, and is a most welcome addition to the literature. The book by Cartan was very elegant and profound, but being written in the characteristic style of its illustrious author, it naively assumed that the prospective reader possessed his geometric acumen and knowledge. The book by Morand was essentially an attempt to explain Cartan’s view of spinors to physicists, and while it was neatly done, it largely neglected the formal development of the algebraic tools in favor of the geometric picture. It is at this juncture that the book by the present author, who was apparently an associate of Morand, fills in the missing algebraic background.

Assuming a rudimentary acquaintance with group theory and quantum mechanics, the author provides a detailed and essentially self-contained exposition of basic spinor theory. In particular, he develops the representation theory of the rotation group in three dimensions, the Lorentz group in four dimensions, and concludes with a brief introduction to Clifford and Lie algebras. In addition, there is a twenty page appendix devoted to group theory and representation theory.

The text is essentially aimed at advanced undergraduates and first year graduate students in physics, and includes applications to quantum theory as originally explored by Pauli (1927) and Dirac (1928). However, it stops short of the work of van der Waerden (1929) which essentially amalgamated the existing notions of spinor theory into a well-defined spinor analysis analogous to tensor analysis.

As such, the exposition is very clear and detailed, and should be readily accessible to its intended audience, as well as to mathematicians who wish to obtain an introduction to the basic theory of spinors. Granted, much of this material is also available in the more encyclopedic text of Penrose and Rindler, but it is nice to see it collected in a tidy manner, together with exercises and solutions, in a single book.

Contents include chapters on: two-component spinors; spinors and SU(2) group representations; spinor representations of SO(3); Pauli spinors; the Lorentz group; representations of the Lorentz group; Dirac spinors; Clifford and Lie algebras; and an appendix: groups and their representations.

While the book is certainly not a replacement for the technical relativistic analysis as presented by Penrose and Rindler, it can be warmly recommended as a pedagogically sound starting point for anyone wishing to understand what spinors are about, and why they are of importance to physicists.

The first category includes the English translation of É. Cartan’s 1937-38 master work [‘The theory of spinors’ (Hermann, Paris) (1966; Zbl 0147.40101)] and the book of M. Morand [‘Géométrie spinorielle’ (Masson, Paris) (1973; Zbl 0257.53019)], while the second category is prominently represented by the book of R. Penrose and W. Rindler [‘Spinors and space-time, Vol. 1: Two-spinor calculus and relativistic fields’ (Cambridge University Press) (1984; Zbl 0538.53024)].

The present monograph clearly belongs in the first category, and is a most welcome addition to the literature. The book by Cartan was very elegant and profound, but being written in the characteristic style of its illustrious author, it naively assumed that the prospective reader possessed his geometric acumen and knowledge. The book by Morand was essentially an attempt to explain Cartan’s view of spinors to physicists, and while it was neatly done, it largely neglected the formal development of the algebraic tools in favor of the geometric picture. It is at this juncture that the book by the present author, who was apparently an associate of Morand, fills in the missing algebraic background.

Assuming a rudimentary acquaintance with group theory and quantum mechanics, the author provides a detailed and essentially self-contained exposition of basic spinor theory. In particular, he develops the representation theory of the rotation group in three dimensions, the Lorentz group in four dimensions, and concludes with a brief introduction to Clifford and Lie algebras. In addition, there is a twenty page appendix devoted to group theory and representation theory.

The text is essentially aimed at advanced undergraduates and first year graduate students in physics, and includes applications to quantum theory as originally explored by Pauli (1927) and Dirac (1928). However, it stops short of the work of van der Waerden (1929) which essentially amalgamated the existing notions of spinor theory into a well-defined spinor analysis analogous to tensor analysis.

As such, the exposition is very clear and detailed, and should be readily accessible to its intended audience, as well as to mathematicians who wish to obtain an introduction to the basic theory of spinors. Granted, much of this material is also available in the more encyclopedic text of Penrose and Rindler, but it is nice to see it collected in a tidy manner, together with exercises and solutions, in a single book.

Contents include chapters on: two-component spinors; spinors and SU(2) group representations; spinor representations of SO(3); Pauli spinors; the Lorentz group; representations of the Lorentz group; Dirac spinors; Clifford and Lie algebras; and an appendix: groups and their representations.

While the book is certainly not a replacement for the technical relativistic analysis as presented by Penrose and Rindler, it can be warmly recommended as a pedagogically sound starting point for anyone wishing to understand what spinors are about, and why they are of importance to physicists.

Reviewer: J.D.Zund (Las Cruces)

##### MSC:

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

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

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

81R25 | Spinor and twistor methods applied to problems in quantum theory |

53Z05 | Applications of differential geometry to physics |

15A66 | Clifford algebras, spinors |