Gauge Fields, Knots and Gravity

by John C Baez

This is an introduction to the basic tools of mathematics needed to understand the relation between knot theory and quantum gravity. The book begins with a rapid course on manifolds and differential forms, emphasizing how these provide a proper language for formulating Maxwell's equations on arbitrary spacetimes. The authors then introduce vector bundles, connections and curvature in order to generalize Maxwell theory to the Yang-Mills equations. The relation of gauge theory to the newly discovered knot invariants such as the Jones polynomial is sketched. Riemannian geometry is then introduced in order to describe Einstein's equations of general relativity and show how an attempt to quantize gravity leads to interesting applications of knot theory.

Genres: Physics, Mathematics, Science, Textbooks, Nonfiction

480 pages, Paperback

First published January 1, 1994

Reviews

[Michael Nielsen · Rating 5 out of 5]

Superbly written. I spent somewhat over a hundred hours with this back in 2006 and 2007, reading (IIRC) a little over a hundred pages in great detail, and skimming many other parts. It was very well worth the time.

[Jon Paprocki · Rating 4 out of 5]

This text is aimed at undergraduates, but I would readily recommend it to grad students as well. It is a really fast and easy introduction to the topics mentioned in the title. I only skimmed the more basics sections on topics like fiber bundles, connections, etc. since I am already very comfortable with them, but I will say that I continue to be convinced that mathematicians can greatly benefit from the intuition that physicists bring to topics - especially in this case, where physicists invented these ideas independently of mathematicians and had entirely different motivations and intuitions as to why these concepts were useful and true! I also didn't read the section on quantum gravity yet since I'm not very interested in it.

But as far as an introduction into gauge theory, Chern-Simons theory, and applications of quantum field theory to knots and 3-manifold invariants, a grad student (in either physics or math) could do worse than spending a few days going through these sections. The writing is not overly dense, the exercises are well thought out and usually easy, and the authors easily switch back and forth between mathematical and physical language. While I had wished that some topics had more written on them (when do you ever not?) the references and (!) the sections that even explain what the most helpful references will probably be make it easy to keep going without losing any steam.

The lost star is just because I can't see myself ever opening the book again now that I've read it once (unless I decide to read the quantum gravity section). It is very pedagogical and won't be a very useful reference to have around. My ideal textbook is somehow both pedagogical and an excellent reference, an almost impossible feat to achieve.

[Tim Robinson · Rating 4 out of 5]

There is a huge amount of maths here, but not nearly enough physics. Much algebra but far too few numerical examples. An explanation of Yang-Mills theory should lead naturally into an exposition of the weak nuclear force, not least because there are some actual calculations to be made. But the book has no chapter devoted to physical Yang-Mill theories at all.

[David · Rating 4 out of 5]

I read this little book some years ago and I really enjoyed. It shows you all the mathematical bricks you need to know in order to build your understanding some of the most advanced theories in Physics. The explanations are clear and the style is one that physicists will certainly appreciate (also mathematicians, but perhaps they would prefer more theorem and proof style, absent here). It can be read by advanced undergraduates, as it assumes nothing more than advanced calculus and linear algebra, but as it is usually said, some "mathematical maturity" would also help.

The book begins discussing the math tools needed for advanced Physics, basically differential geometry and group theory, and it does it in a very fast way, so don't expect anything about the Gauss-Bonnet theorem or discussion of exotic groups (SU(5) or even more exotic ones). The example of the electromagnetism, rewritting the Maxwell equations in the language of differential forms is very well done. The explanation of the concept of fiber bundle is also brilliant, taking into consideration that it's a difficult concept.

There's a section of knot theory that it wasn't specially interesting for me but that it's very well explained, beginning with the basics. It seems that knots could be involved in the unification of General Relativity and Quantum Mechanics.

The final section of the book is about Einstein's relativity, perhaps the most beautiful theory of Physics, trying to link it to what it has been showed previously: a geometric theory that can be described with Cartan's formalism. Curious point: all the groups involved in the electromagnetic, weak and strong forces are compacts, and when a group is compact is equivalent to a sum of irreducible representations (not proved in the text, and probably an advanced proof, but we physicists have faith). The irreducible representations are the particles. The exception is General Relativity, that it is not compact. A change of a topological property that creates a really big problem and literally divides physics in two.

Overall, a very nice book if you're serious about learning advanced theories in Physics but you're also a beginner.

[Saman · Rating 4 out of 5]

General relativity and quantum mechanics, the main two models to describe the world, are not compatible. In other words, there is no compatible model which includes all the forces in the nature, including gravity. This book is about some of the approaches to find such a model, known as the theory of everything. The book starts from basics and explain all the mathematical prerequisites, such as differential geometry, knot theory, gauge theory, electromagnetism, general relativity, quantum mechanics, etc. The book is (more or less) self-contained. However, it is not very deep and the choices of the topics is pretty biased. To sum up it is a good book to gain a picture about the topic but not a good book to understand the details.

[Thomas Ray]

QC793.3 F5 B33 1994 AMP Library

[Keir Havel · Rating 4 out of 5]

Great book for a high-level overview of gauge theory.

[Ania (安雅) Freymond · Rating 5 out of 5]

Highly recommended.