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Posted: Tue Apr 15, 2008 10:38 pm
I pretty frequently have half-baked ideas, but occasionally I get excited over them. Here's one I kinda like.
Disclaimer: I've thought through the following more extensively than I shall present it here. The brevity of this post is due to 1) I'm still procrastinating and 2) I'm tired as all sin and barely have any undamaged neural synapses left. Nonetheless, here's the gist of my latest:
Endowing the affine connection on space-time with nontrivial torsion would naturally force local inertial frames to rotate relative to distant frames. If we developed a 'co-Einstein' field theory ( >.< toss me a better name, s'il vous plait) that did for the affine connection what the presently establish field equations do for metrics and then quantized the sucker, do you think the state-space of particles interacting with the field could correspond to spin state space given the right torsion field action? Of course we would have to figure out to what and how the field would couple.
I think it'd be interesting. Let's discuss!
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Posted: Sat Jun 07, 2008 12:59 am
I've had a bit too much alcohol to discuss such issues in depth at this time, but I suggest you look in to Einstein-Cartan theory, which, sans actual quantization (work is pending), is exactly what you're trying to do.
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Posted: Sat Jun 07, 2008 12:27 pm
VorpalNeko I've had a bit too much alcohol to discuss such issues in depth at this time, but I suggest you look in to Einstein-Cartan theory, which, sans actual quantization (work is pending), is exactly what you're trying to do. Yeah, I've read up a bit since then. Einstein-Cartan theories introduce torsion fields, but I haven't come across a paper on it being used to introduce spin classically yet. There's also Einstein-Weyl-Cartan theories that I'm not too familiar with, but apparently it introduces another geometric tensor field.
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Posted: Thu Jun 12, 2008 5:35 pm
Okay, so I read some more recently. Apparently Cartan introduced the idea that space-time could have a nontrivial torsion and noted that it would endow things with an intrinsic torsion (seemed to be heading in the same direction I was going). Then the quantum mechanical description of spin sprang up and Einstein-Cartan theories went by the wayside (weird, I would think it would strengthen his argument). Now the models do the opposite of what I'd like to do: instead of torsion generating spin, spin generates torsion. stare I'm not sure where the reasoning got derailed.
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Posted: Fri Jun 13, 2008 12:47 am
Well, it shouldn't be surprising at all. The bare theory is classical, and so it would apply to classical spin only, and thus would need to be quantized in some way to account for quantum spin (I thought you had realized this, since in your first post you've talked of quantizing it). If physicists could not easily do so, it going by the wayside is only expected--they had more fundamental issues to worry about at the time. As an example of classical spin, the spin density of a localized electromagnetic field in vacuum is proportional to E×A, where A is the electromagnetic potential, which can be found by decomposing the angular momentum contribution as determined by the Poynting vector and extracting an origin-independent term (cf. Jackson exercise 7.19 = 7.27). Classical spin cannot apply to point-particles.
Disclaimer: I've not studied Einstein-Cartan theory, so I've no idea how much finagling if necessary to make it respect particle (rather than field) spin, or if it's even possible. I'd be very surprised if it was done, and much more so if Cartan's original proposal could handle it.
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Posted: Thu Jun 19, 2008 6:50 am
No, it's still surprising. One could quantize the theory to make the torsion fields impart quantized spin values to particles. My issue deals with which generates which. I've always thought spin was kind of ad hoc, albeit a very real ad hoc quantity. It'd be nice to have it arise from first principles. (Granted I suppose postulating a non-zero torsion field isn't much less ad hoc...).
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