Another subject I avoided for many years: fiber bundles in physics.
It was highly recommended to me that I should read this paper first: “Fiber Bundles and Quantum Theory” by Herbert J. Bernstein and Anthony V. Phillips published in Scientific American in 1981. It is an old paper but still relevant. I have read it. It was very helpful.
I have also studied the Chapter 15 (“Fibre bundles and gauge connections”) of Roger Penrose’s book “The Road to Reality”. In my opinion, this is the best introduction to fiber bundles in physics. I admit it, the mathematics of Chapter 15 was not easy for me to follow. As they say, struggle is good for the soul. A quote from Chapter 15:
“Instead of regarding these internal dimensions as being part of a higher dimensional spacetime, it will be more appropriate to think of them as providing us with what is called a fibre bundle (or simply a bundle) over spacetime. This is an important notion that is central to the modern gauge theories of particle interactions. We imagine that `above’ each point of spacetime is another space, called a fibre. The fibre consists of all the internal dimensions, according to the physical picture referred to above.” – Roger Penrose
In terms of other books, Richard Healey dedicates a significant part of his book “Gauging What’s Real” to the fiber bundle formulation of gauge field theories in physics. This is one of my favorite books. This book requires a serious effort to study as well.
In terms of web pages, there is this article by Urs Schreiber at the nLab :
https://ncatlab.org/nlab/show/fiber+bundles+in+physics
There are also few helpful comments (answers) at the StackExchange.
“When constructing or defining a mathematical object, you have to strike a certain balance: You want your object to be general enough to have lots of applications but structured enough to let you answer important questions. Fiber bundles are very general (in the sense that many spaces can be viewed as fiber bundles) and very structured (in the sense that you glean a lot of information about maps into and out of a space which decomposes as a fiber bundle).
… Fiber bundles provide another sort of local picture, decomposing your space into subsets possessing a certain symmetry. More importantly, perhaps, they give you some idea about how these local pieces glue up to a global structure. For physicists, vector bundles (like the tangent bundle) are an especially rich source of useful fiber bundles. Physicists care a great deal about “connections” on bundles, which are ways to compare points in nearby fibers.” – Kyle
The answer by David Bar Moshe at the StackExchange is very informative. Selected quotes from his answer:
“In geometric quantization, the physical states are sections of line bundles.”
“Fermions on curved spaces are described by sections of spinor bundles.”
“Berry phases describe holonomies of connections on fiber bundles.”
“Higgs fields are described by sections of vector bundles.”
Other educational material on fiber bundles:
https://mathworld.wolfram.com/FiberBundle.html
https://en.wikipedia.org/wiki/Fiber_bundle
One of the most intriguing fiber bundles is the Hopf bundle (Hopf fibration) which is nicely explained in http://www.dimensions-math.org/Dim_CH7_E.htm . You should watch the chapters 7-8 of their video.
Suresh Emre
Renaissance Universal 