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Product details
File Size: 13091 KB
Print Length: 208 pages
Publisher: Dover Publications (January 18, 2013)
Publication Date: January 18, 2013
Sold by: Amazon Digital Services LLC
Language: English
ASIN: B00A73FDD4
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This is the first gauge theory introduction which has made sense to me. After wading through numerous review articles and bits and pieces of books, all purporting to explain the overlap between gauge theory and fibre bundles, finally this book by David Dudley Bleecker (1948-2016) has clarified the basics. I think this is because Bleecker actually understood the FB/GT overlap himself, unlike most other "introductions".There was a flurry of long GT review articles in journals around about 1980 and thereafter, all supposedly explaining either gauge theory to mathematicians, or fibre bundles to physicists. They mostly failed because the authors did not themselves understand the FB/GT overlap. I make one exception here for the 1980 Daniel/Viallet article "The geometrical setting of gauge theories of the Yang-Mills type" in Reviews of Modern Physics, vol 52, pages 175-197, which is clear up until the point where the physics really gets going. They were physicists who explained the mathematics very well. Bleecker was a mathematician who understood the physics very well. (Unfortunately he died about 12 months ago.)The first 41 pages of this tiny 1981 book by Bleecker are a crash course in the relevant differential geometry, with special emphasis on the way in which physicists represent connections on principal bundles. I'm sure that these 41 pages are not intended to be an introduction to DG for physicists. They are extremely concise and compressed. I think you are expected to know the DG already. These 41 pages merely tell you which choices of definitions and notations are being used here. They also show how to convert the pure mathematician's connection forms into physicists' connection forms via local pull-backs from the PFB total space to Lie algebra valued forms on the base space.Following the 41 page DG crash course are 129 pages of applications to physics. (You can see the table of contents in the Amazon preview.) The application of connection forms to particle physics requires the application of the connection to an associated fibre bundle, which yields a covariant derivative, which is then incorporated into the Lagrangian, which then yields equations of motion via the Euler-Lagrange equations. (I'm not 100% certain I've understood all of this yet. So hopefully I am not misleading you!)I am not aware of any better introduction to gauge theory for mathematicians. The vast majority of books on gauge theory are from the physicist's point of view, which for pure mathematicians seems like some kind of stream of consciousness where nothing is justified or justifiable. The most difficult aspect of the physics GT literature is the lack of mathematical definitions whereby you can decode the meanings of symbols as ZF sets. Bleecker has made GT decodable for me. But I also recommend the long Daniel/Viallet review article.By the way, there is no quantization in this book. There's an excellent review of Bleecker's book in Bulletin Amer Math Soc, Vol 9, 1983 by Mayer, which explains that QM and QFT are not presented in this book.
A very good book that connects field theory to differential geometry.I like this book because it is VERY PRECISE and clean!It explains things vague in QFT and GR textbooks, sometimes they write down a notation without definition, or they just express things in "physical intuition", which means you cannot ask why.I admit the first time I opened it I considered it is an alien's book, full of notations and definitions, wondering how can physics be written this way? But after a while, I gradually attracted by the author, David Bleecker, I think he really has something deep to express.This book addresses physics beautifully without losing rigor or physical interpretation, and indeed helps to understand the idea of gauge theory. A very recommended book!
A little gem. Very detailed exposition, elegant methods, beautiful results. The last two, culminating chapters, show how extra dimensions permit a geometrization of Einstein-Yang-Mills theory. At this price it is a must-have, highly recommended.
Notation is dated, but concepts are adequately explained.
this is denser than neutronium -- it is impossible to imagine thereader who could actually benefit from this (non experts need not evenbother) -- a mistaken conception and a horrible execution -- dover getsdemerits for resurrecting this turkey
It fills in the mathematical gaps in the physics books discussion of gauge theory. It is very helpful to me.
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.
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