What type of group is the Renormalization Group? All I've seen is people giving a (differential) equation for beta-function when they teach for the RG... Also I haven't been able to find an algebra characterizing the RG... Any clues?
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?
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).
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/Physics 352/Chicago course lectures/Part 9 Renormalization.pdf http://www.physics.rutgers.edu/~friedan/talks/flows/Friedan_2009.01.22_StonyBrook.pdf
In the block spin picture, averaging maps different fine configurations to the same coarse configuration, so from a coarse configuration you can't recover a unique fine configuration, the coarse graining can't be run backwards. For example, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.129.3194&rep=rep1&type=pdf defines the coarse graining as a semigroup in Eq 35 and footnote 112, but then gives a continuous version in footnote 121 that "can typically be extended to an Abelian group".
That makes sense. That's a different kind of renormalization than the one I was thinking about. Coming from a particle physics back ground I was thinking about renormalization of quantum field theories which has nothing to do with block spins and is reversible.
I don't think that RG-flows are in general reversible in QFT, or that it is unrelated to block spins. Think of the Wilson method of renormalisation by integrating out large wavenumber field configurations to get an effective theory with a lower energy cutoff. That's not an invertible operation.
I've come across a different explanation for why one can't run the flow backwards towards higher energies in these notes by Polonyi http://www.bolyai.elte.hu/download/eloadas/szakmai/mestermuhely/takacs/ln.pdf. He says that if there is a Landau pole at high energies, then the flow can't be run up all the way. Presumably then only asymptotically free or safe theories would form a true group, if this reasoning is correct? 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.
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?
I think that if the Higgs mass were more a few hundred GeV, the Higgs coupling would run into a Landau pole below the Planck scale (implying that there must be new physics at or below the scale of the Landau pole). You can Google things like "higgs triviality bound" for more info on this.
One example that is quite often given is Fermi's effective theory of weak interactions http://isites.harvard.edu/fs/docs/icb.topic1146665.files/III-8-Non-RenormalizableTheories.pdf (section 3).
I think if you start from the high energy theory and write down all the beta functions, then it is possible to do it in a way that you can run the theory all the way down and back up again just fine, even going over all the particle thresholds. I unfortunately can't find the paper I am thinking of that describes this though... 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 fine-tuning.