《Sadri Hassani Mathematical Physics A Modern I》第二版

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《Sadri Hassani Mathematical Physics A Modern I》

Preface to Second Edition
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.
Fiber bundles have become a significant part of the lore of fundamental
theoretical physics. The natural setting of gauge theories, essential in
describing electroweak and strong interactions, is fiber bundles. Moreover,
differential geometry, indispensable in the treatment of gravity, is most elegantly
treated in terms of fiber bundles. Chapter 34 introduces fiber bundles
and their complementary notion of connection, and the curvature form arising
from the latter. Chapter 35 on gauge theories makes contact with physics
and shows how connection is related to potentials and curvature to fields. It
also constructs the most general gauge-invariant Lagrangian, including its
local expression (the expression involving coordinate charts introduced on
the underlying manifold), which is the form used by physicists. In Chap. 36,
vii
viii Preface to Second Edition
by introducing vector bundles and linear connections, the stage becomes
ready for the introduction of curvature tensor and torsion, two major players
in differential geometry. This approach to differential geometry via fiber
bundles is, in my opinion, the most elegant and intuitive approach, which
avoids the ad hoc introduction of covariant derivative. Continuing with differential
geometry, Chap. 37 incorporates the notion of inner product and
metric into it, coming up with the metric connection, so essential in the general
theory of relativity.
All these changes and additions required certain omissions. I was careful
not to break the continuity and rigor of the book when omitting topics. Since
none of the discussions of numerical analysis was used anywhere else in the
book, these were the first casualties. A few mathematical treatments that
were too dry, technical, and not inspiring were also removed from the new
edition. However, I provided references in which the reader can find these
missing details. The only casualty of this kind of omission was the discussion
leading to the spectral decomposition theorem for compact operators in
Chap. 17.
Aside from the above changes, I have also altered the style of the book
considerably. Now all mathematical statements—theorems, propositions,
corollaries, definitions, remarks, etc.—and examples are numbered consecutively
without regard to their types. This makes finding those statements
or examples considerably easier. I have also placed important mathematical
statements in boxes which are more visible as they have dark backgrounds.
Additionally, I have increased the number of marginal notes, and
added many more entries to the index.
Many readers and adopters provided invaluable feedback, both in spotting
typos and in clarifying vague and even erroneous statements of the
book. I would like to acknowledge the contribution of the following people
to the correction of errors and the clarification of concepts: Sylvio Andrade,
Salar Baher, Rafael Benguria, Jim Bogan, Jorun Bomert, John Chaffer,
Demetris Charalambous, Robert Gooding, Paul Haines, Carl Helrich,
Ray Jensen, Jin-Wook Jung, David Kastor, Fred Keil, Mike Lieber, Art Lind,
Gary Miller, John Morgan, Thomas Schaefer, Hossein Shojaie, Shreenivas
Somayaji, Werner Timmermann, Johan Wild, Bradley Wogsland, and Fang
Wu. As much as I tried to keep a record of individuals who gave me feedback
on the first edition, fourteen years is a long time, and I may have omitted
some names from the list above. To those people, I sincerely apologize.
Needless to say, any remaining errors in this new edition is solely my responsibility,
and as always, I’ll greatly appreciate it if the readers continue
pointing them out to me.
I consulted the following three excellent books to a great extent for the
addition and/or changes in the second edition:
Greub, W., Linear Algebra, 4th ed., Springer-Verlag, Berlin, 1975.
Greub,W., Multilinear Algebra, 2nd ed., Springer-Verlag, Berlin, 1978.
Kobayashi, S., and K. Nomizu, Foundations of Differential Geometry,
vol. 1, Wiley, New York, 1963.
Maury Solomon, my editor at Springer, was immeasurably patient and
cooperative on a project that has been long overdue. Aldo Rampioni has
Preface to Second Edition ix
been extremely helpful and cooperative as he took over the editorship of
the project. My sincere thanks go to both of them. Finally, I would like to
thank my wife Sarah for her unwavering forbearance and encouragement
throughout the long-drawn-out writing of the new edition.
Normal, IL, USA Sadri Hassani
November, 2012