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What group is renormalization group?by ChrisVer
Tags: renormalization 
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#1
May2414, 03:20 PM

P: 914

What type of group is the Renormalization Group?
All I've seen is people giving a (differential) equation for betafunction when they teach for the RG... Also I haven't been able to find an algebra characterizing the RG... Any clues? 


#3
May2414, 05:07 PM

P: 914

even SU(2) or SO(3) (maybe more) can be parametrized by a single parameter though...



#4
May2414, 05:15 PM

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P: 2,850

What group is renormalization group?
Huh? I may not know much about the normalization group, but I find it unlikely that SU(2) or SO(3) can be parametrized by one parameter since they are of dimension 3...can you show such a parametrization?



#5
May2414, 05:37 PM

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P: 16,379

Everyone seems to be thinking this is a gauge group. It's not  it's just a plain old group, with the usual group properties (closure, etc).



#6
May2414, 05:49 PM

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P: 8,657

The renormalization group is either a group or semigroup. In most cases, information is lost (you cannot run the flow backwards), so it is a semigroup.
http://jfi.uchicago.edu/~leop/Physic...malization.pdf http://www.physics.rutgers.edu/~frie...StonyBrook.pdf 


#7
May2414, 08:02 PM

Thanks
P: 1,948

Wait, what? Why can't you run the flow backwards?



#8
May2414, 09:22 PM

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#9
May2414, 10:39 PM

Thanks
P: 1,948




#10
May2614, 09:36 AM

P: 641




#11
May2614, 10:14 AM

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P: 8,657

The other explanation I've heard is that renormalization integrates out degrees of freedom, and so one can't recover these uniquely. However, although integrating out and using new effective degrees of freedom at lower energies is conceptually part of renormalization, I don't know if this is formally the case, since if there's a change to new effective degrees of freedom, I think these are usually guessed at, rather than coming from mechanically integrating out degrees of freedom. 


#12
May2614, 03:48 PM

P: 914

is there any theory giving Landau poles at plausible energy scales?
Well I think QED appears for example at [itex]10^{277} GeV[/itex] or so... that's for sure not a plausible pole, since you have new physics at least at Planck's scale. What about the Higg's field? 


#13
May2614, 04:00 PM

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#14
May2614, 05:47 PM

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#15
May2814, 08:55 PM

P: 366

Usually, though, the thresholds do mess things up I would say. This more standard way of doing the RGE running means you switch between a whole series of effective field theories where indeed you chop out certain fields as they decouple and add threshold corrections to the RGE running to consistently account for switching theories. You can of course put the fields back in by hand just as you chopped them out, but that is only because you know what they are. You of course cannot uniquely find the "correct" UV completion of each effective theory automatically. But yeah this alternate method I allude to never actually gets rid of those degrees of freedom, they just decouple continuously rather than at discrete thresholds, so you can continuously run the RGE's up and down as a single set of differential equations. edit: It could become incredibly sensitive to perturbations though, since the RGE trajectories may converge on fixed points and so on. Some tiny numerical error may then put you on a different RGE trajectories when you run in the opposite direction. I think this can happen in both the IR and UV though, and of course is rather related to finetuning. 


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