Timeline for Intuitively what's the relationship between forces and connections?
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| Oct 9, 2019 at 4:08 | answer | added | nothingIsMere | timeline score: 4 | |
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| Jan 4, 2019 at 10:06 | history | edited | Qmechanic♦ |
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| Oct 3, 2018 at 13:22 | answer | added | jak | timeline score: 0 | |
| Aug 10, 2015 at 20:52 | comment | added | Robin Ekman | Even if your fiber bundle is trivial you need a notion of parallel transport to compare stuff over here to stuff over there. If $\sigma : P \to M \times G, p \mapsto (m, g)$ is a trivialization so is $p \mapsto (m, f(m) g)$ for any $f : M \to G$. | |
| Aug 10, 2015 at 19:57 | comment | added | ACuriousMind♦ | Forces do not "require principal bundles". Instead, it is the description of forces in terms of potentials that leads to gauge symmetries (adding total derivatives to the potential, roughly), and gauge symmetries are theories on principal bundles in the Lagrangian formulation. There's no intuition for this because no non-Abelian gauge theory classically appears in the description of a force, and the $\mathrm{U}(1)$ case looks far simpler than this general case. | |
| Aug 10, 2015 at 19:43 | history | edited | Gold | CC BY-SA 3.0 |
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| Apr 16, 2014 at 6:54 | comment | added | Christoph | the idea of parallel transporting particle properties is just how I tend to think about it; if your fiber bundle isn't trivial, you need a notion of parallel transport to compare stuff 'over here' to stuff 'over there'; thus, connections (basically the infinitesimal version of parallel transport) appear naturally; in the Yang-Mills case, the 'stuff' you want to compare depends on some sort of internal symmetry, and we end up with principal connections | |
| Apr 16, 2014 at 6:50 | comment | added | Christoph | @user1620696: in classical Yang-Mills theory, you get equations of motion for particles from force equations (the generalization of the Lorentz-force law); in the Lagrangian formulation, this is achieved via minimal coupling, which requires a generalized charge (in the general case, a coadjoint orbit instead of just a number); this is not the case for gravity; another way in which gravity is different that the gauge symmetries in YM theory are vertical (leaving space-time alone), wheras in case of gravity, they are not | |
| Apr 16, 2014 at 5:18 | history | tweeted | twitter.com/#!/StackPhysics/status/456300462849802240 | ||
| Apr 16, 2014 at 0:32 | comment | added | Gold | Thanks @Christoph for your point. Now, as I've said I'm just starting with connections. What is the difference in general relativity? I know that we stop conceiving gravity as a force and conceive it as a property of spacetime itself. But still, it is described by a connection right? Also, what is this idea of parallel transporting particle properties? Could you tell me where to read more about it? Thanks again. | |
| Apr 15, 2014 at 23:22 | comment | added | Christoph | note that assuming ordinary general relativity, gravity is rather different and shouldn't be lumped together with the other forces (this gets somewhat alleviated in the teleparallel formulation); for a hand-wavy explanation why we care about principal connections: because we need to parallel transport particle properties (which relate to internal symmetries) | |
| Apr 15, 2014 at 22:24 | comment | added | Robin Ekman | Not duplicate, but I think this is related physics.stackexchange.com/q/77368 | |
| Apr 15, 2014 at 22:11 | history | asked | Gold | CC BY-SA 3.0 |