Help me understand this stuff (re: Aharonov-Bohm Effect, etc.)

[Geezer]

Okay, so I was reviewing the Aharonov-Bohm effect online, and found some related discussion on Dirac Monopoles. Let me quote a bit:

In order to combine this local system into a -principal bundle, on the -coordinate over must be related to the -coordinate over by , with integer . This explains the appearance of Dirac's string singularity when the are extended to , and the fact that it can be removed by a gauge transformation which requires Dirac's quantization condition. Thus, the trivial bundle admits no monopole (charge -monopole). The existence of a monopole indicates non-triviality of a corresponding principal bundle. The monopole of charge is the connection in the Hopf fibration , while the monopole of charge with corresponds to the -bundle over with the lens space as a total space ( is viewed inside as a subgroup of th roots of the unit matrix)

I don't understand a lick of it. What kind of math do I need to take to understand this stuff?

-Geez

 

[jacobrhcp]

Diracs quatization condition is usually treated in quantum field theory classes.

The bundle talked about is treated in manifold classes, sometimes differential geometry (although that is strictly more than you need), but often bundles are just covered in QFT/string theory classes on a physics level of rigor. Moreover, this seems to be about string theory, which you would need to take.

So you would first need a lot of calculus, linear algebra, complex analysis, analysis in many dimensions, advanced QM, E&M, classical field theory, advanced thermal physics, group theory, and maybe even some topology.

In short, you need the basic education of any theoretical physicist

Have fun ;)