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I'm interested in mathematical physics.


Dec
14
comment What is known about the topological structure of spacetime?
Very interesting! Thanks for this answer.
Dec
13
comment Electromagnetic Field as a Connection in a Vector Bundle
Well, if you're working on a vector bundle then the connection forms are really only defined relative to a local frame. If you patch them together to get something global then it doesn't take values in any nice bundle. However, the curvature form is a global two form with values in the adjoint rep. But if you go from a vector bundle to its associated principal bundle, then the connection form is a global 1 form on the principal bundle with values in the Lie algebra (not ad rep) and the curvature form is a 2 form on the principal bundle with values in the Lie algebra.
Dec
12
awarded  Teacher
Dec
12
answered Classical mechanics without coordinates book
Dec
12
comment Electromagnetic Field as a Connection in a Vector Bundle
Just to nitpick: a connection is specified by a one-form with values in the Lie algebra (not the adjoint bundle); but this is a one-form on the associated principal bundle. you only get a form description on the base manifold locally. You might be confused with the curvature which can be seen as a form on the base manifold with values in the adjoint bundle.
Dec
12
asked Where is the Atiyah-Singer index theorem used in physics?
Dec
11
awarded  Nice Question
Dec
10
comment What is known about the topological structure of spacetime?
Oh ok, so that would still be a geometric thing: scaling a cylinder doesn't change any topology.
Dec
10
comment What is known about the topological structure of spacetime?
Thanks for the answer. I do not see why EFEs cannot contain topological data since you need a global solution to them (you can solve it locally but they need to patch together to form a global metric). For example, if the EFEs implied something like positive scalar curvature then that would really limit the topology (being positive at a point is local, being positive everywhere is global). The adding of topological invariants looks very interesting-- I'll have to read more into it.
Dec
10
comment What is known about the topological structure of spacetime?
What do you mean by scale of the topology? But Einstein's equations need to be solved globally so couldn't they put some restrictions on the topology? For example if Einstein's equations implied positive scalar curvature, then that would limit the possible manifolds. Also, With there not being any classification of even simply connected 4-manifolds, it seems likely there are nontrivial ones which wouldn't have the "wrap around" property of light rays.
Dec
10
awarded  Student
Dec
10
asked What is known about the topological structure of spacetime?
Dec
9
awarded  Precognitive
Nov
3
comment Lie theory, Representations and particle physics
you may want to check out this question: math.stackexchange.com/questions/622/…
Nov
2
awarded  Supporter