Orthogonal and symplectic Clifford algebras. Spinor structures.

*(English)*Zbl 0701.53003
Mathematics and Its Applications, 57. Dordrecht etc.: Kluwer Academic Publishers Group. xiii, 350 p. Dfl. 240.00; £79.00; $ 124.00 (1990).

In his classification of simple representations of simple Lie algebras E. Cartan (1913) discovered a hitherto unknown representation of orthogonal Lie algebras, and with the discovery of spin in quantum theory the objects on which this representation operate were called spinors. Many authors contributed to the explication of spinor theory, and in a seminal paper R. Brauer and H. Weyl (1935) outlined a general theory based on Clifford algebras. A geometric version of this theory was given by E. Cartan in his “Leçons sur la théorie de spineurs” (2 volumes, Hermann et Cie, 1938; Zbl 0019.36301 and Zbl 0022.17101). Subsequently, C. Chevalley in his monograph “The algebraic theory of spinors” (1954; Zbl 0057.259) gave an elegant but rather abstract treatment of this approach. The present monograph is intended to provide a book on Clifford algebras and spinors - in the spirit of Chevalley - which includes an introduction to the recent applications of these topics in differential geometry and mathematical physics. This has long been needed, and the author has admirably succeeded in giving such an account which should be useful to both mathematicians and physicists. The author’s style is concise, but very clear, and assuming the reader has the mathematical/physical prerequisites, this book is a veritable goldmine of information.

The contents include nineteen chapters: Orthogonal and symplectic geometries: Tensor algebras, exterior and symmetric algebras; Orthogonal Clifford algebras; Clifford groups, twisted Clifford groups and their fundamental subgroups; Spinors and spin representations; Fundamental Lie algebras and groups in Clifford algebras, matrix approach to spinors in three and four dimensional spaces; Spinors in maximal index (even dimension); Spinors in maximal index (odd dimension); Hermitian structure on the space of complex spinors (conjugation, etc.); Spinoriality groups; Coverings of the complete conformal group (twistors); triality principle, interaction principle and orthosymplectic graded Lie algebras; Clifford algebra and bundle of a pseudo-Riemannian manifold (existence of spinor structures); Spin derivations; Dirac equation; Symplectic Clifford algebras and associated groups; Symplectic spinor bundles and the Maslov index; algebra of deformations on symplectic manifolds; and three appendices: Primitive idempotents and amorphous spinor fibre bundles; self-dual Yang-Mills fields and the Penrose transform; symplectic and complex structures, symplectic spinors and the Fourier transform. The book concludes with a comprehensive bibliography of items dealing with Clifford algebras and orthogonal spinors, symplectic algebras and symplectic spinors.

Finally one caveat is in order: the author’s viewpoint and treatment make little contact with the 2-component van der Waerden spinor formalism which has proven so fruitful and popular in classical general relativity. Hence, the present monograph must be regarded not as a replacement, but as a companion to the two volume treatise of R. Penrose and W. Rindler “Spinors and space-time” [Vol. 1 (1984; Zbl 0538.53024) paperback reprint (1986; Zbl 0602.53001); Vol. 2 (1986; Zbl 0591.53002)]. Indeed the author’s book begins, just about where the discussion of Penrose and Rindler ends!

The contents include nineteen chapters: Orthogonal and symplectic geometries: Tensor algebras, exterior and symmetric algebras; Orthogonal Clifford algebras; Clifford groups, twisted Clifford groups and their fundamental subgroups; Spinors and spin representations; Fundamental Lie algebras and groups in Clifford algebras, matrix approach to spinors in three and four dimensional spaces; Spinors in maximal index (even dimension); Spinors in maximal index (odd dimension); Hermitian structure on the space of complex spinors (conjugation, etc.); Spinoriality groups; Coverings of the complete conformal group (twistors); triality principle, interaction principle and orthosymplectic graded Lie algebras; Clifford algebra and bundle of a pseudo-Riemannian manifold (existence of spinor structures); Spin derivations; Dirac equation; Symplectic Clifford algebras and associated groups; Symplectic spinor bundles and the Maslov index; algebra of deformations on symplectic manifolds; and three appendices: Primitive idempotents and amorphous spinor fibre bundles; self-dual Yang-Mills fields and the Penrose transform; symplectic and complex structures, symplectic spinors and the Fourier transform. The book concludes with a comprehensive bibliography of items dealing with Clifford algebras and orthogonal spinors, symplectic algebras and symplectic spinors.

Finally one caveat is in order: the author’s viewpoint and treatment make little contact with the 2-component van der Waerden spinor formalism which has proven so fruitful and popular in classical general relativity. Hence, the present monograph must be regarded not as a replacement, but as a companion to the two volume treatise of R. Penrose and W. Rindler “Spinors and space-time” [Vol. 1 (1984; Zbl 0538.53024) paperback reprint (1986; Zbl 0602.53001); Vol. 2 (1986; Zbl 0591.53002)]. Indeed the author’s book begins, just about where the discussion of Penrose and Rindler ends!

Reviewer: J.D.Zund

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

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

15A90 | Applications of matrix theory to physics (MSC2000) |

53B50 | Applications of local differential geometry to the sciences |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

15A66 | Clifford algebras, spinors |

15-02 | Research exposition (monographs, survey articles) pertaining to linear algebra |