| dc.description.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. | en_US |
