Gauge fields, knots, and gravity
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.
Stats for nerds
6recommend
9mentions
9people
2009–2025years
67% recommend33% neutral
Mentions per year
What Hacker News says
- I think the idea of knots as a basis for everything has come and gone several times. One of those were in the 90s, which is when I became aware thanks to the excellent "Gauge Fields, Knots and Gravity" by John Baez and Javier P Muniain, that was part of the "Series on Knots and Everything" [1]. Those are really intriguing ideas.
- If you do want to learn the geometric formulation, part 1 of Gauge Fields, Knots and Gravity is a good resource (and has exercises!).
- A lot of recommendations depend on what you're trying to learn.
But I've enjoyed the following texts to a larger extent than others:
- Algebra: Chapter 0 (Aluffi)
- Real Mathematical Analysis (Pugh)
- Mathematics and its History (Stillwell)
- An Introduction to Manifolds (Tu)
- Gauge Fields, Knots and Gravity (Baez)
- A First Look at Rigorous Probability Theory (Rosenthal)
- All of Statistics (Wasserman)
There are some authors I trust and am happy to buy so long as the topic vaguely interests me: VI Arnold, Tristan Needham and John Stillwell.
I really like the list put out by @enriquto in a separate comment, but I've avoided duplicating those recommendations in the list above.
- I think the standard reference is probably Spivak's 'Calculus on Manifolds' but this never really did it for me.
If you have a background in physics then some combination of Nakahara's 'Geometry, Topology and Physics' and Baez and Muniain's 'Gauge Fields, Knots and Gravity' might be good (I haven't included relativity textbooks as I assume it you have a background in GR then you have enough differential geometry).
An unusual recommendation that I think is really nice is 'Stochastic Models, Information Theory and Lie Groups' by Chirikjian. It covers a few other topics mentioned in this thread and is really nice. It's _extremely_ concrete and spells out a lot of calculations in great detail. Plus, the connection to engineering applications is much more obvious.
- I haven't fully worked through it yet, but I've been really enjoying reading John Baez's Gauge Fields, Knots, and Gravity [1].
[1]: http://www.worldscientific.com/worldscibooks/10.1142/2324
- One of the authors, Baez, wrote "Gauge Fields, Knots, and Gravity" a lovely, cheap, and fairly accessible intro to applications of topology in mathematical physics. Worth a glance if you find yourself wondering about all this
Related
Geometry, topology, and physics
2 recommend · 2
A Course in Modern Mathematical Physics
4 recommend · 5
2019–2025
Modern Classical Physics
4 recommend · 7
New foundations for classical mechanics
5 recommend · 16
Geometric Algebra for Physicists
12 recommend · 20
Spacetime and geometry
8 recommend · 12
