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SU(2) a double cover for Lorentz group? 
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#19
Jan1913, 10:16 PM

P: 982

Now as I understand you, this means that given the p_{μ} and M_{μν}, these commutator relations restrict the kind of internal symmetries that can exist. But I wonder if it can also be read, given the internal symmetries these commutators restrict the spacetime symmetries to the Lorentz group? In other words, once these commutators have been proven to be TRUE, does it matter how they were proven, that they were proven with the assumption of the Lorentz group? Can we not just take these commutators to be inherently true and use them as I've suggested? 


#20
Jan2013, 09:42 PM

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#21
Jan2113, 01:08 PM

P: 982

But perhaps something more is needed for these commutator relations to be meaningful. Is it that they must be a commutation relation between Lorentz group and internal symmetries? In other words, it's not a commutation relation between the Lorentz group and arbitrary groups; it must be with internal symmetrys. And it's not a commutation relation between internal symmetries and arbitrary spacetime symmetries; it must be with Lorentz group, right? Is it the QFT formulism that connects the two symmetries in the commutator, that requires the symmetries to be of the Lorentz group and the internal symmetries? In other words, it's only valid within the framework of a relativistic QFT that has internal symmetries? Or can it involve a QFT that has internal symmetries but is not necessarily Lorentz invariant, and visa versa, if such can exist? 


#22
Jan2113, 05:27 PM

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P: 1,940

Just try being a student and study Greiner's books properly. I have already answered the other things in your last post, and I have other things to do. 


#23
Jan2213, 10:37 AM

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Thanks
P: 2,550

I haven't read through the whole thread (due to lack of time, sorry). I just want to mention that the universal covering group of [itex]\mathrm{O}(1,3)^{\uparrow}[/itex] is [itex]\mathrm{SL}(2,\mathbb{C})[/itex].



#24
Jan2213, 10:53 AM

P: 982

I got a little sidetracked on the ColemanMandula theorem. But I never really got a good answer to the following:



#25
Jan2213, 11:44 AM

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PF Gold
P: 2,602

The appearance of 2 copies of the algebra su(2) here is not because of the signature, but is a property of any so(p,q) algebra for which p+q=4. This is the sense in which the relationship is special to four dimensions. 


#26
Jan2213, 01:00 PM

P: 982




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