Book Description

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.