# Why does renormalization problem still happen in nonperturbative QG?

## Main Question or Discussion Point

I think that renormalization problem happens in perturbative QFT,but not happens in nonperturbation theory.So I do not understand why renormalization problem appears in Quantum Gravity Theory despite of there are many nonperturbation theory to solve the problem?
Thank you very much for your kind helping.

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Demystifier
I think that renormalization problem happens in perturbative QFT,but not happens in nonperturbation theory.
I don't know why do you think that, but that's wrong. Non-perturbative approaches also need renormalization.

"Problem" is an unfortunate word. I'd imagine that the OP has been reading a lot high energy textbooks and the such, probably written a couple of decades ago. I would suggest, as an antidote, some condensed matter theory textbooks, or even just statistical mechanics. A personal favourite (because it is freely available to all) is by Sethna: http://pages.physics.cornell.edu/~sethna/StatMech/

That will provide the OP with some perspective on what renormalisation is, and its use in high energy theory will be less confounding.

tom.stoer
Renormalization in perturbative QFT is constantly teached to be "removing divergences". Unfortunately this is only a dirty trick! Renormalization (non-perturbative renormalization) is something totally different - and it may even appear in systems w/o any divergences.

Have a look at http://en.wikipedia.org/wiki/Renormalization_group

Suppose you have a theory with certain coupling constants ga, gb, ...; suppose you study this theory at some energy scale E. Then you will observe that you can re-express the theory in terms of different coupling constants g*a, g*b,... at a different energy scale E* (and different interactions, i.e. changing E may turn on new couplings). It is interesting that these two 'theories' defined at different energy scales E and E* are essentially the same theory! The way how to find the description at scale E* given the description at E is via studying trajectories ga(E), gb(E) in the space of all coupling constants.

Different points on the same trajectory correspond to different scales E, E*, ... of the same theory; different theories are defined by points not connected by a trajectory. These trajectories define something called renormalization group flow. The renormalization group of a certain theory is defined via a set of coupled differential equation defining this flow of coupling constants.

Perturbative QFT is defined by studying the flow near the point ga = gb = ... = 0; in some case (QCD) it can be shown that this is reasonable in certain regimes; in other cases (QG) there are strong indications that G=0 may never be a valid starting point, neither in the UV (not asymptotically free), nor in the IR (coupling constant of Newtonian gravity G is not zero).

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What does ''w/o'' mean?

tom.stoer
'w/o' = without
'w/' = with
'b/c' = because

Fra
Depending on your perspective RG theory can itself be embedded into yet a larger framwork. RG typically studies how the theory parameterisation changes when you "look at different energy or distance scales".

But IMO the even more general question here; which btw is not well developed (like normal RG is) is to ask how one theory relates to another theory depending on which observer that inferred it (from it's own interactin history).

Here one realizes that looking at different energy or distance scales is merely a subset of the more general observer-observer transformation.

Like Tom said these deeper questions has absolutely nothing in princple do to with perturbation theory, they have to do with the deep question of how changing perspective (observer) FORCES a deformation of the theory itself. RG idea is to track the theories by parametrisation in theory space, but again there may be generalisations to this where the theory space itself also changes.

This hits the absolultey deepest issues in fundamental physics, and is IMO a very underdeveloped area.

/Fredrik

Please tell me what 'IMO'' and ''btw'' mean.

Fra
IMO = in my opinion
btw = by the way

/Fredrik

There is still exist unsolved problem of renormalization in Quantum Gravity Theory,nowaday?

tom.stoer
the approach is called 'asymptotic safety'; there are indications that the theory is non-perturbatively renormalizable, i.e. NOT closed to G=0; in addition there are indications that the theory is defined by a few coupling constants and that higher order curcature terms do not contribute;

the main problem is that in principle the space of coupling constants is infinite- dimensonal and only a small sector can be explored

Fra
the main problem is that in principle the space of coupling constants is infinite- dimensonal and only a small sector can be explored
This is patly why I the whole concept and understanding of "theory space" is needed in the proper context of how a theory is evolving.

By the same token none tries to understand biological systems in terms of a reductionist "theory space" of all possible life forms. Such an idea simply lacks predictive and adaptive power for reasons on information capacity limits and computational times and stability against truncation (deterministic chaos).

/Fredrik

tom.stoer
The theory space is well defined: you write down all possible Lorentz scalars which can be derived from the metric plus matter fields. Within this space you have a RG flow defining something like a fibration; a theory is not represented by a point in this space but by trajectory connecting different scales (this is something like the concept of equivalence classes).

The 'only' problem is to explore this space - but this is always the problem in science! We restrict (artficially) our model of nature (we fix a framework) and we hope that this framework (nearly) fits to the observed phenomena, i.e. that the restriction (model building) is sufficient. The difference between 'F=ma' and the RG approach is only that the later one is less restrictive :-)

And how about the so call ''Canonical Quantum Gravity''?

tom.stoer
it is expected that this theory is subject to renormalization, too, but this (renormalization) is still in its infancy

Fra
Tom, I am quite sure we don't share the same visions here and I expect no agreement so I'll just make a short comment.

With the more general framwork I men the general contect of relating two inferred theories(as encoded by two observers), which is not necessarily sensibly described by "renormalisation" in the sense of tracetories in theory space. Thus this applies to all interactions, not just gravity.
a theory is not represented by a point in this space but by trajectory connecting different scales (this is something like the concept of equivalence classes).
My point is that in the most general cases, I think a simple "scaling" of the theory in terms of parameter scalings just doesn't work in principle. I see the theory - theory interactions as physical. Not just an abstraction. This means that the theory space itself sort of also needs "rescaling" except that it doesn't work with the litteral SCALING of parameters. I think a more complex "translation" is needed in the general case.

I see the path in theory space as an evolution path, and this evolution process can't be described simply as a scaling process, except in special cases.

As I see it this is not ONLY a technical problem, it's a problem of how theory - theory translation is understood, and it applies to all interactions and to unification.

But here I'm formally speculative indeed, I have not seen much published on this unfortunately.

The key distinction is that normally in RG, what we scale is the energy scale where we probe, but what also needs scaling (and is completely ignored) is the energy scale of the observer! Now if everything is referring back to a scattering matrix at infinity noone is ever going to see the difference here. So there are many more things to vary in a full theory - theory translation beyond just the energy of the probe. This is why RG to me in principle is a special case. I think this may be needed to look at so solve the open issues.

/Fredrik

tom.stoer
I understand why you mean; your theory space is larger; it does not only contain QCD and the scaling of its coupling constants, but it contains MSSM, SUGRA, string, ... as well. This is clear to me from several discussions, but in the context of this thread where it is about the physical meaning of renormalization within one given theory (not about constructing theories) I think the theory space as I have described it reasonable.

Fra
Yes, maybe I misunderstood the OP as usual

I'm not aware of what mainstream "quantum gravity theory" he refers to in that case, so I interpreted it generically as QG = the open quest for unification of gravity and _QM framework (which means QFT; and then more or less the SM).

I noticed though in several thread that sometimes when the terms QG is used implicitly means more or less LQG or other "pure gravity" quantum theories?

"QG" to me is just a label of the open problem on howto incorporate gravitational interaction into measurement theory in which the other forces are already descirbed.

/Fredrik

tom.stoer
We started to talk about renormalization in some xQG theory, but my post was meant to explain the 'asymptotic safety' approach which is usually (!) applied to the continuum, metric formulation of GR. But in principle renormalization has to be investiated in other approaches xQG as well.

Afaik there are a few attempts within a continuum formulation using continuum Ashtekar variables i.e. no spin networks or foams.

Fra
I understand why you mean; your theory space is larger
Yes, but my main concern is not just that it's bigger, my concern is that the concept of fixed theory space itself may be wrong. I don't that way of thinking works. What happens is that somehow the spaces get so big that you can't even define proper measures on them, and in particular no obsever "sees this full space".

String theory seems to have a bit of that symptom. Each "supposed cure" just adds more of the same medicine which just makes the theory space YET larger. I think this entire concept is flawed.

Actually it's almost a reasonable point that all this that I keep mentioning in several discussions is not quite "traditional physics", it's rather like artificial intelligence or theory of theory building. But I sincerely think that the discussion of open problems in physics (which is all about building and extendting theorues) becomes a little shallow if we don't see this context. I felt this was relevant as you question the foundations of renormalisation, whose points is really just to generally related two theories that are inferred in different ways (at different scales). Then you can of course rename things and call it "representations" and that the observer invariant theory is then just the scaling transofmrations. But I think we can't escape these questions.

/Fredrik

So,if we can not put a measure in this theory space,then the space is not a mathematical space?So it might be we could not complete the String Theory?

tom.stoer
Fredrik,

why not restricting to the initial question why renormalization is required in non-perturbative QG? let's wait for ndung200790's response

All your answers broaden my knownledge very very much!I owe all your helping very much.By the way,is the asymptotic safety the simply applying of Renormalization Group in Quantum Gravity?

Please teach me the skeleton of asymptotic safety in QG.Is it being the applying of normal Renormalization Group to Quantum Gravity?

tom.stoer