Mathematical Physics

Abstract

Based on my own experience of teaching from the first edition, and more importantly based on the comments of the adopters and readers, I have made some significant changes to the new edition of the book: Part I is substantially rewritten, Part VIII has been changed to incorporate Clifford algebras, Part IX now includes the representation of Clifford algebras, and the new Part X discusses the important topic of fiber bundles. I felt that a short section on algebra did not do justice to such an important topic. Therefore, I expanded it into a comprehensive chapter dealing with the basic properties of algebras and their classification. This required a rewriting of the chapter on operator algebras, including the introduction of a section on the representation of algebras in general. The chapter on spectral decomposition underwent a complete overhaul, as a result of which the topic is now more cohesive and the proofs more rigorous and illuminating. This entailed separate treatments of the spectral decomposition theorem for real and complex vector spaces. The inner product of relativity is non-Euclidean. Therefore, in the discussion of tensors, I have explicitly expanded on the indefinite inner products and introduced a brief discussion of the subspaces of a non-Euclidean (the so-called semi-Riemannian or pseudo-Riemannian) vector space. This inner product, combined with the notion of algebra, leads naturally to Clifford algebras, the topic of the second chapter of Part VIII. Motivating the subject by introducing the Dirac equation, the chapter discusses the general properties of Clifford algebras in some detail and completely classifies the Clifford algebras Cν μ(R), the generalization of the algebra C13 (R), the Clifford algebra of the Minkowski space. The representation of Clifford algebras, including a treatment of spinors, is taken up in Part IX, after a discussion of the representation of Lie Groups and Lie algebras.