Topology, Geometry, and Gauge Fields: Interactions (Applied Mathematical Sciences) [Hardcover]

Gregory L. Naber (Author)

Book Description

Publication Date: March 10, 2000 | ISBN-10: 0387989471 | ISBN-13: 978-0387989471 | Edition: 1

A study of topology and geometry, beginning with a comprehensible account of the extraordinary and rather mysterious impact of mathematical physics, and especially gauge theory, on the study of the geometry and topology of manifolds. The focus of the book is the Yang-Mills-Higgs field and some considerable effort is expended to make clear its origin and significance in physics. Much of the mathematics developed here to study these fields is standard, but the treatment always keeps one eye on the physics and sacrifices generality in favor of clarity. The author brings readers up the level of physics and mathematics needed to conclude with a brief discussion of the Seiberg-Witten invariants. A large number of exercises are included to encourage active participation on the part of the reader.

Editorial Reviews

Review

From the reviews:

MATHEMATICAL REVIEWS

"The presentation in the remaining five chapters is enriched by detailed discussions about the physical interpretations of connections, their curves and characteristic classes. I particularly enjoyed Chapter 2 where many fundamental physical examples are discussed at great length in a reader friendly fashion. No detail is left to the reader’s imagination or interpretation. I am not aware of another source where these very important examples and ideas are presented at a level accessible to beginners…The topics covered in this book can be found in many other sources, but the present volume discusses with great care those aspects and notions which are particularly important in gauge theory. Moreover, the presentation is backed by many useful and relevant examples and I am convinced that any beginner in gauge theory will find them very useful."

NZMS NEWSLETTER

"It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics...Naber combines a knowledge of his subject with an excellent informal writing style."

SIAM REVIEW

"Naber writes in a most unpretentious style. His prose is not terse like Rudin’s, but not verbose either. He gives full details to all difficult calculations and shows good judgment in deciding what is difficult versus what is not. This is one way in which a writer demonstrates rapport with his/her readers. Never once has Naber omitted anything out of laziness, under the pretense that it is routine. The book is carefully thought out and lecture-tested account of the subject matter listed earlier. It is rigorous, with an emphasis on the details in the examples. Naber favors examples that deal with concrete spaces and revisits them whenever appropriate…In terms of its ability to teach a subject to the novice, this book ranks right up there with many classics…People who collect classics should consider buying this one, whether or not they plan to study it chapter by chapter. For someone who plans to compute right along with the examples, this book is a must-buy. Naber’s goal is not to teach a sterile course on geometry and topology, but rather to enable us to see the subject in action, through gauge theory. The book is capable of fulfilling this goal because of Naber’s efforts. He has undertaken the arduous task of researching the broad field with its extensive literature, learning the material himself, class testing it in lectures, and agonizing over the best ways to present it. Amazingly, the fruits of his labor can be had for less than $70, thanks to Springer’s consumer-friendly pricing…[the reviewer] hopes that Naber will continue the scholarly program of bringing exciting mathematics and physics to a level of clarity that is within our reach."

 

REVIEWS OF TOPOLOGY, GEOMETRY, AND GAUGE FIELDS:  FOUNDATIONS

"It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics...Naber combines a knowledge of his subject with an excellent informal writing style."
NZMS NEWSLETTER

"...this book should be very interesting for mathematicians and physicists (as well as other scientists) who ae concerned with gauge theories."
ZENTRALBLATT FUER MATHEMATIK

From the Back Cover

This is a book on topology and geometry, and like any book on subjects as vast as these, it has a point of view that guided the selection of topics. The author’s point of view is that the rekindled interest that mathematics and physics have shown in each other of late should be fostered, and that this is best accomplished by allowing them to cohabit. The goal is to weave together rudimentary notions from the classical gauge theories of physics and the topological and geometrical concepts that became the mathematical models of these notions. The reader is assumed to have a minimal understanding of what an electromagnetic field is, a willingness to accept a few of the more elementary pronouncements of quantum mechanics, and a solid background in real analysis and linear algebra with some of the vocabulary of modern algebra. To such a reader we offer an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2)-connections on S4 with instanton number -1. This second edition of the book includes a new chapter on singular homology theory and a new appendix outlining Donaldson’s beautiful application of gauge theory to the topology of compact, simply connected , smooth 4-manifolds with definite intersection form. Reviews of the first edition: “It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics…Naber combines a deep knowledge of his subject with an excellent informal writing style.” (NZMS Newsletter) "...this book should be very interesting for mathematicians and physicists (as well as other scientists) who are concerned with gauge theories." (ZENTRALBLATT FUER MATHEMATIK) “The book is well written and the examples do a great service to the reader. It will be a helpful companion to anyone teaching or studying gauge theory …” (Mathematical Reviews) --This text refers to an alternate Hardcover edition.

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Product Details

• Hardcover: 477 pages
• Publisher: Springer; 1 edition (March 10, 2000)
• Language: English
• ISBN-10: 0387989471
• ISBN-13: 978-0387989471
• Product Dimensions: 9.4 x 6.5 x 1.3 inches
• Shipping Weight: 1.8 pounds (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (7 customer reviews)
• Amazon Best Sellers Rank: #2,782,941 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

13 of 15 people found the following review helpful:

5.0 out of 5 stars required reading for a topologist interested in physics, May 13, 2000

By 

Sean Leckey (Staten Island, NY USA) - See all my reviews
(REAL NAME)   

This review is from: Topology, Geometry and Gauge fields: Foundations (Texts in Applied Mathematics) (Hardcover)

As a mathematician turned physics grad student, it is often difficult to read "Math for Physicists" books simply because of the focus on making "numbers churn out;" which, at least for me personally, more difficult to get a handle on the subject and then, in turn, use it fruitfully.

This book on the other hand, is exemplary of why I got into physics in the first place. The first chapter (Physical motivations) and the last chapter (Gauge Fields and Instantons) can be read by any one with undergraduate topology under their belt and come away with a more powerful understanding of gauge theory than, in my opinion, can be found in other introductory gauge theory texts I've been directed to.

Of course I'll read all those said texts as well, but I'm thankful that I found this one.

8 of 9 people found the following review helpful:

5.0 out of 5 stars Easy reading, complete proofs, plenty of exercises, October 29, 2005

By 

Rehan Dost (Canada) - See all my reviews
(REAL NAME)   

This review is from: Topology, Geometry and Gauge fields: Foundations (Texts in Applied Mathematics) (Hardcover)

This text is by far the best introductory text marrying basic concepts of physics with pure mathematics.

Some background in the basic concepts of vector calculus, linear algebra, complex numbers and group theory is required.

The author begins by motivating the mathematics by the pursuit of finding a vector potential to represent a magnetic monopole. We see that the topology of R3-0 precludes such a vector potential from existing. We see here a simple example of how the topology of a space affects the physics associated with it.

The importance of the vector potential as something other than a convenient computational tool is highlighted by a reference to essential inclusion in quantum mechanics. Thus we NEED such a potential.

The author now asks whether there is a "trick" or device to get around this difficulty. The device are principal bundles and connections. For example the potentials noted above must keep track of the phase of a charged test particle as it moves thru the field of a magnetic monopole. We need a "bundle" of circles ( representing the phase at each point ) over S2 ( the author explains why we need only consider S2 instead of R3-0, briefly we need only keep track of 2 of the 3 spherical co-ordinates ).

Thus a curve in S2 thought of as the particles trajectory will have to be "lifted" to the bundle space by a lifting procedure called a connection.

In a more general setting elementary particles have an internal structure ( spin etc ) which becomes apparent during interactions although may not be apparent in uniform motion thru a vacuum. Since the phase of the particle does not alter the modulus when calculating probabilities these do not change. However, when the particles interact phase differences are important. We need to keep track of such phases as the particles interact.

Thus we need a "bundle" over a 4-manifold ( keeps track of the particles space-time path ) to keep track of such internal states. One sees we also need a group to transform states into one another ( usually incorporated into the bundle ). Connections then model physical phenomena which mediate changes in the internal states.

We see that some connections satisfy the Yang-Mills equations and using the appropriate equivalence relation form Moduli spaces.

Now that may seem like alot to digest with only a spattering of mathematical maturity.

The beauty of the book is that the author starts from FIRST principles.

Chapter 1 introduces topological concepts of topology, continuity, quotient topology, projective spaces, compactness, connectivity, covering spaces and topological groups.

Chapter 2 introduces concepts of path lifting, fundamental groups, contractability, simple connectedness, covering homotopy theorem, higher homotopy groups

Chapter 3 introduces principle bundles, transition functions, bundle maps and principle bundles over spheres.

Chapter 4 introduces manifolds, derivatives on manifolds, tangent/cotangent spaces, submanifolds, vector fields, matrix lie groups, vector valued 1- forms, 2 forms and Riemann metrics

Chapter 5 gets to some physics with gauge fields and connections, curvature, Yang-Mills functional, moduli spaces, Hodge dual , matter fields and covariant derivatives.

At each step the author carefully provides complete proofs and easy exercises to ensure understanding.

It was a pleasure to read the book and complete the exercises. At no point did I feel frustration or boredom.

5 of 5 people found the following review helpful:

5.0 out of 5 stars An Introduction with Mathematical Integrity, December 31, 2008

By 

A Reader - See all my reviews

This review is from: Topology, Geometry and Gauge fields: Foundations (Texts in Applied Mathematics) (Hardcover)

Gregory Naber is to be commended for writing a thorough introduction to gauge field theory in which the mathematics is presented with clarity and rigor. For the professional mathematician who is interested in physics, or for the graduate student who prefers to see the mathematics "done right" in advanced applications to physics, Naber's wonderful two-volume set stands apart from its major competitors, nearly all of which were written by physicists, for physicists.

Despite the attention to mathematical rigor, it is clear that Naber intended his books to be accessible to a dual audience of physicists and mathematicians. For the physicists, he has included gentle introductory chapters on topological spaces, homotopy groups, principal bundles, manifolds and Lie groups, and differential forms. For mathematicians, the chapters on physical motivation, gauge fields and instantons, Yang-Mills-Higgs theory, Spinor structures, etc., provide unusually accessible introductions to some difficult physics materials.

Chapter 0 of the first volume is worth the price of both books, as it leads the reader, in 26 succinct pages, to a compelling appreciation of the natural "fit" of the Hopf Bundle to the task of providing a quantum mechanical analysis of the exterior of a single magnetic monopole. For outsiders who have become incredulous about the increasingly sophisticated uses of topology and geometry in theoretical physics, this example provides some much-needed assurance. As the reader quickly learns, the use of connections on principal fiber bundles is neither gratuitous nor mathematical overkill: indeed, the bundle machinery emerges quite NATURALLY as the simplest and best mathematical tool, perfectly fitted to the special problem at hand.

Any serious reader will want to buy both volumes of this set: Topology, Geometry, and Gauge Fields: Foundations (volume 1), and Topology, Geometry, and Gauge Fields: Interactions (volume 2). These books take their place alongside the work of authors such as Jerrold Marsden, Theodore Frankel, Barrett O'Neill, and Walter Thirring, all of whom write about modern mathematical physics in a way that does not obscure the true role of the mathematics.