Gauge Fields, Knots & Gravity (V4)

by Baez J

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.

[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.