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Book review

Author(s) Micali, A. (ed.)
Boudet, R. (ed.)
Helmstetter, J. (ed.)
Title Clifford Algebras and their Applications in Mathematical Physics
Publisher Kluwer Academic Publishers
Year of publication 1992
Reviewed by M. Marinov

Clifford algebras, generated by elements subject to anti-commutation relations, has been one of the most important mathematical concepts used in theoretical physics of this century, first of all, because of the role they play in quantum theory, in particular, in description of spinning particles and fermion fields. Actually, Pauli and Dirac matrices are representations of the corresponding finite Clifford algebras, while the electron field operators in quantum electrodynamics can be considered as generators of an infinite-dimensional Clifford algebra. The interest to Clifford algebras was excited substantially during the past two decades because of two reasons. First, after the discovery of the pattern of Bose-Fermi supersymmetry with its numerous applications to string models of elementary particles and the theory of gravity (the supergravity), the algebra and analysis with anticommuting elements (the Grassmann algebra, which can be embedded into the Clifford algebra) became one of the standard tools of modern theoretical physics. Second, the superstring models, which can be quantized consistently in 10-dimensional space-time only, prompted an investigation of geometry of (pseudo) Euclidean spaces of higher dimensionalities and of the representation theory of the corresponding transformation groups. The importance of the Clifford algebras for the group theory is due to the fact that they provide with spinor representations to orthogonal groups.

The book contains a selection of more than 40 papers contributed to the Second Workshop on Clifford Algebras and their Applications, held at Montpellier, France, in September of 1989. The opening paper is a general survey by D. Hestenes, which concerns some aspects of modern geometry where the Clifford algebras have been applied. Other contributions cover an ample range of problems in algebra, geometry, analysis and theoretical physics. For some cases, explicit constructions are given, in particular, with a use of computer programs. Various extensions have been considered, including the algebras over finite fields, Clifford groups on arbitrary quadratic forms (including non-positive definite), and on higher forms. Problems of topology of the group representations have been also considered. A number of contributions have been dealing with analysis on Clifford algebras and the theory of hyper-complex functions. Physical applications include various problems in the theory of gauge fields, supersymmetry, supergravity, and relativistic field equations. The Workshop was dedicated to two scientists who did outstanding contributions to the subject: Marcel Riesz and Mário Schenberg. A short biography of M. Schenberg (1914-1990) is included, together with his paper "Algebraic Structures of Finite Point Sets", which has not been published previously.

The volume will be useful for physicists working in particle theory, and for mathematicians interested in modern geometry, representation theory and the theory of functions of many complex variables. The presentation of the material is not homogeneous, but an expert reader would certainly benefit from the abundance of results elaborated explicitly.