Gauge Theory and Variational Principles | 
 enlarge | Author: David Bleecker
Publisher: Dover Publications
Category: Book
List Price: $12.95
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Rating:  5 reviews
Sales Rank: 444679
Media: Paperback
Pages: 208
Number Of Items: 1
Shipping Weight (lbs): 0.5
Dimensions (in): 8.4 x 5.3 x 0.6
ISBN: 0486445461
Dewey Decimal Number: 530.1435
EAN: 9780486445465
Publication Date: December 10, 2005
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Product Description
Detailed and self-contained, this text supplements its rigor with intuitive ideas and is geared toward beginning graduate students and advanced undergraduates. Topics include principal fiber bundles and connections; curvature; particle fields, Lagrangians, and gauge invariance; inhomogeneous field equations; free Dirac electron fields; calculus on frame bundle; and unification of gauge fields and gravitation. 1981 edition
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| Customer Reviews:
 Spivak on Steroids February 24, 2007
Rehan Dost (Canada)
12 out of 12 found this review helpful
As the title suggests this "text" serves as an introduction to the QFT and guage theories recast in the "modern" mathematical setting of differential geometry.
This book is only 167 ( 1/2 regular size paper ) pages long. Although self-contained I highly recommend the reader have a working knowledge of QFT and at least an introductory course in GR. The mathematical tools of the reader should include a course in analysis on manifolds at the Spivak level or higher, acquintance with fibre bundles and basic lie groups. For example in the first chapter ( 22 pages ) the author covers differential forms, manifolds, Stokes theorem, lie derivative, deRham cohomology, lie groups and algebras. The next 20 page chapter covers principal bundles and connections ( 3 definitions all shown to be equivalent and these turn out to be the physical equivalent of gauge potentials ) followed by lie algebra valued forms, exterior covariant derivative curvature and the structure equation.
Chapter 3 defines particle fields as mappings from the principal bundle to a vector space which are equivariant or to the space of sections of the associated vector bundle. We now see that guage transformations are nothing more but the automorphisms of the bundle with certain requirements. Langrangians are developed as mappings of the space of 1 jets to the reals. G invariance of langrangians is defined and is shown to be an insufficient criteria for invariance under gauge transformations. However, we see that introduction of a connection and hence covariant exterior derivative on the bundle our Langrangian becomes gauge invariant.
The next chapter introduces action densities and shows that a particular particle field obeys the principle of least action iff it is stationary which is true iff Langrange's equation holds. There is alot of mathematical notation and machinery developed here. At this stage spin zero electrodynamics are treated.
Current are defined on PFB and a conservation of charge for g-invariant Langrangians with stationary particle fields is shown to be true. This chapter introduces the "self interaction" term for the gauge field and shows that a particle field and connection ( gauge potential ) obey the principle of least action iff they satisfy BOTH langrange equation and the inhomgeneous field equation.
The next chapter introduces spinor bundles ( to add spin particles to our repretoire ) requiring modifications of the previous mathematical tools leading to the appropriate Langrangian. Lagrange's equation is shown to reduce to the dirac free field equation.
The next chapter shows how to deal with interactions between the particle fields with spin and guage fields. This requires "spliced bundles" ( one where the particles with spin live and the other where guage fields live ) which requires another straight forward modification of our mathematical tools...redefining particle fields, Langrangians, currents, and action densities in the process. We see that in this general setting the particle fields and gauge potentials ( connection ) are stationary ( satisfy principle of least action ) iff they satisfy two generalized versions of lagranges equation and an inhomegeneous field equation. The author shows how these reduce the the special cases of the dirac electron field and yang mills field to the dirac and yang-mills equations and the inhomegeneous maxwell and yang-mills equations, respectively.
Chapter 8 goes over the mathematics of general relativity in about 10 pages.
Chapter 9 attempts to unify gauge theories and gravitation showing that the Einstein field equations and the Yang-Mills equations follow from a single variational principle dependent upon the scalar curvature of the metric defined on a suitable PFB. Problems of this unification are explained.
The final chapter explores symmetry breaking monopoles and instantons.
As you can see there is alot to absorb with the prerequisites noted above.
The book has no examples or exercise but each theorem is proved. There is also a page with corrections which is refreshing and a page summarizing the notation and page they are first introduced.
Clean and modern exposition June 19, 2006
Jeffrey M. Lee
9 out of 10 found this review helpful
Despite what another reviewer said, this book uses standard differential geometry notation. The notation is of the invariant (no index) style of Kobayashi and Nomizu.
I find it a delightful little book. It should be good for anyone with a background in manifold theory and Lie groups.
 A mathematically tight and well-motivated work July 5, 2006
David A. Anderson
10 out of 10 found this review helpful
Professor Bleecker has succeeded in writing a book for mathematicians and physicists. And, it's all there. I would rate this work 5-star, except I fear some physicists might find the mathematical format a bit tough (definition, theorem, lemma, etc.) As a mathematician studying physics I hope I am wrong. I find this book user-friendly due to its formality and "compactness". I caution those w/o a fair degree of mathematical acumen that this big, little book is a good deal more formal than, say, Gilmore's "Lie Groups, Lie Algebras, and Some of their Applications." But, then we all must bite the bullet. With effort, I think you will find this a chewable bullet.
 A useful reference text for gauge theory. July 28, 2007
A. Van Dyk (Philadelphia)
A useful reference text for gauge theory. While Bleecker includes an introductory chapter to cover prerequisites (tensor analysis, differential forms, Lie groups etc) the text assumes some familiarity with these techniques and gauge theory. The text is in the "definition, theorem, proof" format. Bleecker gives fairly detailed proofs which help the reader to follow most steps. The text is short and the price reasonable, so the absence of second quantization gauge potentials, for example, is perhaps understandable. While there are chapters on gauge invariance and Action density I found the material didn't go into the detail I was looking for on invariance problems (discussion of theorems of Noether, Caratheodory's methods, Lie algebras and groups). I rated this text lower because of the complete absence of exercises and the limited references and bibliography (about three pages in total).
 This book is all that is bad about abstract mathematical physics writing May 15, 2006
R. Bagula (Lakeside, Ca United States)
11 out of 30 found this review helpful
Am I new to gauge theory: no.
Why does he make me feel I am?
Are these new generalized symbols really necessary to his treatment.
Apparently the author believes that rather than use words to explain
theorems...
He can rely almost entirely on this new set of symbols
that he has used to translate gauge theory to fiber bundles.
Why is it that so many times
the reader is the one who is made to do the work
of teaching himself and learning new "languages"
when the author claims he will be teaching you?
I feel that I have to warn the reader, that although he claims to be "mainstream"
differential geometry, he lies.
he uses very non-standard ( at least for the papers I've read)
notation and instead of starting at the simple gives the most abstract examples
of the notation even in chapter "0".
I grade this book as a "F"... way below even the worst
group theory book I have by an English set of professors!
I'm really as sorry as I can be, but he got me for the money,
by false advertising...
It is a very badly written book,
more a "show off"... see what I can do book,
than a I will teach you to understand book.
I had to go back to other books to compare the notation
so many times that I stack them with this book.
I spent half a year plotting solitons in Mathematica,
and I don't recognize anything in his treatment of them.
Monopoles and Instantons are even worse in this book, if possible.
It is not that I I don't believe he is using correct mathematics,
it is just that it is so hard to tell if he is!
Roger L. Bagula
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