An introduction to spinors and geometry with applications in physics

著者

書誌事項

An introduction to spinors and geometry with applications in physics

I.M. Benn, R.W. Tucker

Adam Hilger, 1987

  • : pbk

大学図書館所蔵 件 / 全56件

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注記

Includes bibliographical references and index

内容説明・目次

巻冊次

ISBN 9780852741696

内容説明

This graduate textbook dealing with the modern mathematical techniques of differential geometry and Clifford algebras is written with students of theoretical physics in mind.
巻冊次

: pbk ISBN 9780852742617

内容説明

There is now a greater range of mathematics used in theoretical physics than ever. The aim of this book is to introduce theoretical physicists, of graduate student level upwards, to the methods of differential geometry and Clifford algebras in classical field theory. Recent developments in particle physics have elevated the notion of spinor fields to considerable prominence, so that many new ideas require considerable knowledge of their properties and expertise in their manipulation. It is also widely appreciated now that differential geometry has an important role to play in unification schemes which include gravity. All the important prerequisite results of group theory, linear algebra, real and complex vector spaces are discussed. Spinors are approached from the viewpoint of Clifford algebras. This gives a systematic way of studying their properties in all dimensions and signatures. Importance is also placed on making contact with the traditional component oriented approach. The basic ideas of differential geometry are introduced emphasising tensor, rather than component, methods. Spinor fields are introduced naturally in the context of Clifford bundles. Spinor field equations on manifolds are introduced together with the global implications their solutions have on the underlying geometry. Many mathematical concepts are illustrated using field theoretical descriptions of the Maxwell, Dirac and Rarita-Schwinger equations, their symmetries and couplings to Einsteinian gravity. The core of the book contains material which is applicable to physics. After a discussion of the Newtonian dynamics of particles, the importance of Lorentzian geometry is motivated by Maxwell's theory of electromagnetism. A description of gravitation is motivated by Maxwell's theory of electromagnetism. A description of gravitation in terms of the curvature of a pseudo-Riemannian spacetime is used to incorporate gravitational interactions into the language of classical field theory. This book will be of great interest to postgraduate students in theoretical physics, and to mathematicians interested in applications of differential geometry in physics.

目次

Tensor algebra: The tensor algebra. The exterior algebra of antisymmetric tensors. The exterior algebra as a quotient of the tensor algebra. The Hodge map. The mixed tensor algebra. Bibliography. Clifford algebras and spinors: The Clifford algebra. The structure of the real Clifford algebras. The even subalgebra. The Clifford group. Spinors. Spin-invariant inner products. The complexified Clifford algebras. The confusion of tongues. Bibliography. Pure spinors and triality: Pure spinors. Triality. Bibliography. Manifolds: Topological manifolds. Derivatives of functions R" - R". Differential manifolds. Parameterised curves. Tangent vectors. Vector fields. The tangent bundle. Differential 1-forms. Tensor fields. Exterior derivatives. One-parameter diffeomorphisms and integral curves. Lie derivatives. Integration on manifolds. Metric tensor fields. Bibliography. Applications in physics: Galilean spacetimes. Maxwell's equations and Minkowski spacetime. Observer curves. Electromagnetism. Bibliography. Connections: Linear connections. Examples and Newtonian force. Covariant differentiation of tensors. Curvature and torsion tensors of ... Bianchi identities. Metric-compatible connections. The coveriant exterior derivative. The curvature sealer and Einstein tensor. The pseudo-Riemannian connection. Sectional curvature. The conformal tensor. Some curvature relations in low dimensions. Killing's equations. Bibliography. Gravitation: Lorentzian connections. Fermi-Walker transport. The Einstein field equations. Conservation laws. Some matter fields. The Reissner-Nordstrom solution. Gravitation with torsion. Bibliography. Clifford calculus on manifolds: Covariant differentiation of Clifford products. The operator d. The Kahler equation. The Duffin-Kemmer-Petiau equations. Bibliography. Spinor fields: Spinor bundles. Inner products on spinor fields. Covariant differentiation of spinor fields. Lie derivatives of spinor fields. Representing spinor fields with differential forms. Bibliography. Spinor field equations: The Dirac operator. Covariances of the Dirac equation and conserved currents. The Dirac equation in spacetime. The stress tensor. Tensor spinors. The Lichnerowicz theorem. Killin spinors. Parallel spinors. Appendix. Bibliography. References. Index.

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