# String Field Theory and Background Independence?

• asimov42

#### asimov42

Hi all,

I've recently been reading about string field theory (note: I'm a novice). As I understand, the string field is an infinite collection of classical fields. But I'm uncertain as to why this formulation leads to background independence?

Thanks all.

The word "background independance" doesn't really mean what you might think it does. In general, be careful about how people define it and in what context you are in, b/c there is a lot of fog (particularly on the internet) about what people really mean. In general proffessionals rarely use the terminology unless they're pitching an advertisement, mostly b/c its ambiguous.

Now, SFT is background independant in the same sense that Maxwells equations are. Everything is dynamical, there is no fixed structure that cannot be varied, everything can be written in terms of forms, potentials and field strengths.

In another sense, SFT is no more or no less background independant or dependant than regular string theory, b/c it essentially just reproduces perturbation series and *might* miss some (but not all) of the nonperturbative 'backgrounds' that also must exist in a full theory.

Be aware there are several versions of string field theory, like covariant SFT or another that is confusingly called 'background independant open SFT'. Its best to think about what exactly the objects are that they are quantizing (second quantization, third quantization etc) eg string fields, spaces of 2dim conformal field theories, etc. Its also an advanced subject that I don't recommend to novices until you have a lot of ... wait for it.. background material.

In another sense, SFT is no more or no less background independant or dependant than regular string theory, b/c it essentially just reproduces perturbation series and *might* miss some (but not all) of the nonperturbative 'backgrounds' that also must exist in a full theory.
To write the string action, you have to pick up some background spacetime metric (usually taken to be flat). To write the string-field action (e.g., Witten's cubic bosonic open-string-field action), you don't have to pick up some background spacetime metric. Doesn't it mean that string field theory is more background independent than regular string theory?

Yea, see point 1, but they're also shown to be equivalent theories. Again it depends how you define BI. You could say SFT is manifestly BI, and regular ST manifestly BD, but in terms of what it tells you about the full nonperturbative theory (where the metric tensor is emergent), it does no better or worse (in fact, sometimes it does better and sometimes it does worse). Even that last statement I wrote is confused, b/c of dualities and other technical details.

The point is its really just semantics at the end of the day and what you want to compute.

Yea, see point 1, but they're also shown to be equivalent theories.
Perhaps they are equivalent perturbatively, but not non-perturbatively. For example, tachyon condensation (which is a non-perturbative effect) can be obtained from Witten's cubic string field theory, but not from "regular" string theory.

The way I understand it, picking a background is the same as picking a guage, if you were working in a run of the mill Yang-Mills theory. So it seems odd that physical things that you can compute should turn out differently in one case vs. the other.

Unless I am mistaken, which is probably the case :)

Reflection

I see the logic in Ben's statement, and I think it focuses on the general issue (regardless if we're talking about string theory or some other theory). This "issue" and sometimes paradox of BI vs BD is I think a general problem and not a problem just for string theory. There IMO are problems also in other theories. In a certain sense, to do away with ALL backgrounds is a certain sense also prolbemativ, because what are you referring to? I think the problem is more difficult thta these extrems.

Background independence vs background dependence
Fundamental symmetry vs emergent symmetry

From the point of view of the "fundamental symmetry" and the equivalence class of all distinctions generate by a set of transformations, the choice of element(gauge choice) is completely arbitrary and should not change the "physics" determined by the symmetry itself.

I think a problem is that it seems (to me) common to view the existence of equivalence classes in a realist sense.

Ie. the question is thus, IS the actual physics determined by a fundamental symmetry? and how is that conceptually consistent with the modern non-realist idea of quantum interactions which usually suggest that physical interactions correspond to relative information?

Could it instead me that one can, due to constraints of nature, not pick a gauge AND maintain an undistorted view of the fundamental symmetry at the same time? Ie. perhaps there is a kind of uncertaintly principle here, where from the gauge fixed point of view, the symmetry is not fundamental after all, it's only emergent.

/Fredrik

I see the logic in Ben's statement,

Be careful!

I'm not sure it's right!

Be careful!

I'm not sure it's right!

Ben, even if I got your intention wrong (I don't mean to suggest you said something you didn't!), I would still raise the issue I tried to described as important, even outside of string theory. Give or take some details I made an association to your comment.

The issue exists also in classical GR and it's diff.symmetry. But there the issues is usually a non-issue since it's a realist type of theory. But in a measurement theory of it, the question is what parts that really are the physical observables. In the classical theory this isn't an issue. Because there is realism. But then the problem becomes a conceptual one.

/Fredrik

I guess my original question was whether string field theory can be considered to be background independent in a general relativistic sense? If string field theory is string theory reformulated in the language of quantum field theory, and QFT is background dependent, how does the background independence in SFT arise?

If we consider the string field as an infinite sum of classical fields, can the classical fields be considered background independent? I'm somewhat confused by Haelfix's comment that Maxwell's equations are background independent - Haelfix, could you expand on this?

Sorry for all the questions! The forums have been very helpful so far.

Maxwells equations without sources reads like 0 = dF = d*F. Where F is the two form F = dA, A is the gauge connection and * is the hodge star operator.

These field equations are manifestly background independant, no background is specified or involved in the dynamics and the domain of validity is arbitrary up to that which is induced by the hodge star (technically it adds structure to the dual space).

This particular formalism can live in curved space, flat space etc etc You are free as well to write Maxwells equations in a manifestly background dependant way as well in terms of the more familiar fields B(X,t) and E(X,t) where this choice spontanously breaks the diffeomorphism symmetry of the manifold where we want the field equations to live in.

In a sense, BI is completely trivial here. Varying the manifolds base space metric does not change the validity, form and universality of the solution.

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Note that in the Feynman lectures he notes that any interaction can be written in a rotationally invariant form, whether the interaction is rotationally invariant or not.

ian

And that the equations can be written in a rotationall invariant manner does not imply rotations are a symmetry - solutions of these equations related by a rotation are NOT physically equivalent - that is, rotations are not symmetry.

Any equations can be be put into a generally covariant form but that does not mean that general covariance are a symmetry. The point about GR, as apposed to other theories, is that its solutions, related by diffomorphisms, are physically equivalent.

ian

p.s. diffeomorphisms are not the same as coordinate transformations.

I don't think there is a lot of fog around the defintion of BI. I think there is just fog around people's understanding of it. I wrote a section on BI for wiki on the subject of in the hope to clar up the understanding.

ian

"on the subject of LQG" is what I meant to say

On the issue of the BI of string theory, appealing to authority:

In the words of Ed Witten:

“Finding the right framework for an intrinsic, background independent formulation
of string theory is one of the main problems in string theory, and so far has remained out
of reach.” ... “This problem is fundamental because it is here that one really has to address the question of what kind of geometrical object the string represents.”

I'm sure he wouldn't have made this statement if ST was only non-manifestly BI but BI all the same

ian

When String theorists talk about background independance, they're referring to their configuration space. Eg every part or corner is ideally described using the same language. The point being, that language does not exist yet, and that's what Witten is talking about.

Now, in one corner of the configuration space of string theory, AdS/CFT lives and is valid. In that regime, it is often said to be completely BI (where here we have chosen a new definition for the word background, different than the former and closer in spirit to the GR meaning), but in the more general context it is still BD.

You have to understand, in no sense is what string theorists talking about, the same as the backgrounds the LQG people are talking about. Its really just a statement of formalism more so than a physical statement.

Furthering the confusion are differing definitions (not just of what is or is not a background). So for instance, people will often say QFT is background dependant. But that's misleading too, it is trivial for me to write down a topological field theory, or a 2+1 dim QFT that is exactly soluble, either using perturbation series (resumming) or without. So then we have to expand what we are talking about to 'manifest' vs 'nonmanifest' and somehow it seems to simply be a statement about 'perturbative' vs 'nonperturbative' or 'before calculation' and 'after calculation'.

Yet another, completely different definition that I've seen used in the past is more akin to the difference between (active/passive) diffeomorphisms.

Bottomline, the whole thing is horribly vague, and context dependant and sensitively depends on the definition. Where it becomes really illdefined is the second people start comparing completely different classes of theories.

With the risk of getting to philosophical again I wanted to connect to this key.

Bottomline, the whole thing is horribly vague, and context dependant and sensitively depends on the definition.

I'm just an amateur, but I agree with this.

The following is my own opinion and it does not necessarily relate to anthing Haelfix meant to say, but I personally think this relates to some deeper reflections of background independence and the quest for new logic, that smolin (appears to me) to have sniffed. In a certain sense one could also refer to the CONTEXT as a "background", and part of the implicit references, such as the choice of logic and languages.

In this for admittedly more obscure and abstract version of "background", not even GR in background independent IMHO, because there are still background references, such as the manifold etc. This is one conceptual objection I have to classical GR and classical physics in general. These background "structures", manifolds or whatever, does not fit in the grand vision of information processing, which does not allow for "hidden background information", and ultimately this seems to trace down to logic itself.

It's easy to get a feeling that your head is spinning and this nothing but circular reasoning - there seems to be no static acceptable reference. Now, the question is if that is such a bad thing? If we look around us? I think it's a hint. Maybe we just need to find out how to tame the circular madness, and turn it into evolutionary progression?

This is what I hope, and it's the spirit that I personally see in some of Smolins crazy papers. And there the observation that the notion of BI seems fundamentally context-relative, may be one of the key observations to progress? If even physical law is evolving, we seem to have lost all connection to solid ground.

/Fredrik

Not trying to be funny but in short I think a good way of expressing what I tried to expand upon is that in this generalise sense, paradoxally, the notion of background-independence is itself background dependent. This is a nutshell describes IMHO at least, the problem.

I *think* this is pretty much what Healfix said in his last paragraph, if you extend the concept of background to include ALL backgrounds (including "definitions", which are rarely as innocent as they sound, since the involve choices).

/Fredriik

Not trying to be funny but in short I think a good way of expressing what I tried to expand upon is that in this generalise sense, paradoxally, the notion of background-independence is itself background dependent...

What I find of primary interest is what researchers are trying to achieve, it is only of secondary importance what words they use to refer to their goal.

For many years, nonstring quantum gravity researchers used the words "background independence" as a kind of flag to help distinguish their work from string and to identify one of the main aims of their research. But lately the phrase has become involved in a verbal tug-of-war, with string thinkers adopting it and giving it a special meaning within stringy context. So instead of promoting communication, the phrase can now lead to sterile discussions---about who gets to control this bit of verbal turf and impose their own meaning on it, wear it as a badge of honor, etc.---and about what is the "real meaning" of the words, as if they stood for some permanent abstract concept.

For the quantum gravity researchers, one solution to the verbal tug-of-war has simply been to stop relying on the words "background independence", and find other ways of stating their overall goal.

So for example when Carlo Rovelli was invited to come to the annual Strings-2008 conference in August and give an overview talk on Loop Quantum Gravity (and related approaches) he did not once use the phrase "background independence". Here is the video:
http://cdsweb.cern.ch/record/1121957?ln=en

When, at the beginning of his talk, he needed to state the main motivation of the LQG program and address the question Why Loops? he used different words. He said something like this: The central problem LQG addresses is how to describe the fundamental degrees of freedom of a QFT when there is no fixed background spacetime.

Abhay Ashtekar has made a similar departure. See for example his October 2008 31-page overview of Quantum Space-times to be published in a book commemorating the Minkowski centennial. http://arxiv.org/abs/0810.0514
He describes the main aim of his community's research program at the start without using the debased terminology.

Once near the end, at the top of page 27 (last page of text) he uses the phrase, but there it is merely shorthand for what has already been described and discussed at length. In his statement of purpose at the beginning he avoids the corrupted abstraction and spells things out this way

==quote Ashtekar, page 3==
Over the last 2-3 years several classically singular space-times have been investigated in
detail through the lens of loop quantum gravity (LQG) [2–4]. This is a non-perturbative
approach to the unification of general relativity and quantum physics in which one takes
Einstein’s encoding of gravity into geometry seriously and elevates it to the quantum level.
One is thus led to build quantum gravity using quantum Riemannian geometry [5–8]. Both
geometry and matter are dynamical and described quantum mechanically from the start.
In particular, then, there is no background space-time. The kinematical structure of the
theory has been firmly established for some years now. There are also several interesting
and concrete proposals for dynamics (see, in particular [2–4, 9]). However, in my view there
is still considerable ambiguity and none of the proposals is fully satisfactory. Nonetheless,
over the last 2-3 years, considerable progress could be made by restricting oneself to subcases
where detailed and explicit analysis is possible [10–15]. These ‘mini’ and ‘midi’ superspaces
are well adapted to analyze the deep conceptual tensions discussed above. For, they consider
the most interesting of classically singular space-times —Friedman-Robertson-Walker
(FRW) universes with the big bang singularity and black holes with the Schwarzschild-type
singularity— and analyze them in detail using symmetry reduced versions of loop quantum
gravity. In all cases studied so far, classical singularities are naturally resolved and the
quantum space-time is vastly larger than what general relativity had us believe. As a result,
there is a new paradigm to analyze the old questions.
==endquote==

As you can probably guess, Lee Smolin is not relying heavily on the phrase either. For instance he gave a seminar talk at the ILQGS on 21 October which did not use the phrase at all.
http://relativity.phys.lsu.edu/ilqgs/
Instead, in the first part of the talk he chose to discuss three different levels or meanings of the emergence of spacetime. I will quote just his first two:
==quote Smolin 21 October slide #3==
What do we mean by emergence of space‐time?

Emergence of the manifold: The fundamental description
of nature does not involve fields (quantum or classical)
on a differential manifold.

Emergence of the classical metric: The fundamental
description of nature does not involve a classical metric field.
...
...
==endquote==
Here the main issue seems to me a clear and practical one: does your description use differential manifolds or not? And if it does use a manifold, do you or don't you specify a classical metric on it, giving it a fixed geometry?

=======================
EDIT: As an afterthought, prompted in part by Fra's next post, I should say that I gave these links primarily as examples to illustrate the point that representative QG people (Rovelli, Ashtekar, Smolin) were not using the phrase "background independence", or were depending on it less these days.
Rovelli's talk at Strings-2008:
http://cdsweb.cern.ch/record/1121957?ln=en
Ashtekar's October 2008 survey overview essay "Quantum Space-times":
http://arxiv.org/abs/0810.0514
Smolin's October 2008 seminar talk:
http://relativity.phys.lsu.edu/ilqgs/
I just wanted to provide these links as evidence, in case any reader had trouble believing the point I was making.

I didn't want to make extra work, if you didn't need convincing, or perhaps already knew about that shift in vocabulary. In fact the talks and the essay are interesting in their own right, I think, but the links were just brought in as evidence to corroborate.

Three or four years ago, the vocabulary was different. B.I. meant "does not use a fixed metric" or "does not use a differential manifold". And Ashtekar would write a survey called "The Status of Background Independent Quantum Gravity". That was the flag the community waved to identify itself and distinguish itself from string.

But then string theorists basically grabbed the other guys' flag, and gave it a different meaning, and made a big noise about it, so there was enough confusion that it was no longer useful as a concise identifier any more. So QG people now use the term less---or depend on it less. They still use the words on occasion, but they use other words as well, so that they no longer rely on B.I. to have a clear meaning without further explanation. That's my take on it anyway.

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I totally agree that the choice of words are not relevant. I could easily replace the word background for many any words. I tend to for myself think in terms of prior information, to which statements are conditional. Sometimes I had the impression that those that argue in favour of BI (or we can choose another word) appeal to a principle of indifference, that there is no reason to choose one background over the other, and thus it seems that if the fundamental equations of nature depends on an arbitrary choice, then something is wrong.

While that argument is not wrong, I think an additional complexity is that in the distinction of the symmetry class of arbitrary choices itsel - there is an implicity choice still, although containing more information.

IMHO, regardless of the words we choose, this is somewhat analogous to the problem of the ergodic hypothesis. We appeal to a principle of indifference, and argue that all microstates are equally possible - but, how is the microstructure itself selected? the ergodic hypothesis is implicit in the choice of microstructure (to me the microstructure is close to synonmous to background structure, but is more general, it applies (in my mind at least:) to an arbitrary state spce, not just 4D spacetime)

But I agree that the interesting point is the problem one tries to solve.

I'll skim those recent smoling lectures later, I don't thikn I've seen them.

/Fredrik

When, at the beginning of his talk, he needed to state the main motivation of the LQG program and address the question Why Loops? he used different words. He said something like this: The central problem LQG addresses is how to describe the fundamental degrees of freedom of a QFT when there is no fixed background spacetime.

I agree this is a good question. But if I understand rovelli's reasoning (I started to read his book), he tries to use normal QM, and I can't motivate myself for that. To me the problems are mixed up with the foundations QM - which ultimately regards the notion of probability, which (correct me if I am wrong) Rovelli more or less explicitly says on a footnote that he doesn't want to question. I love rovelli's reasoning, and the initial reasoning in his RQM! But then the lost me.

Here the main issue seems to me a clear and practical one: does your description use differential manifolds or not? And if it does use a manifold, do you or don't you specify a classical metric on it, giving it a fixed geometry?

In my personal imagination, to assume the existence of a 4D-manifold from start is not acceptable, simply because the starting point isn't innocent. I expect a fundamental model to explain the emergence of the manifold as well. Emergence of the geometry should also be emergent. But any attempt to fix another background, say a space of manifolds is subject to the same critics. This is why I like the evolutionary ideas, which I mentally picture as a "drifting windows" in this relational hierarchy, so that no inside observer can ever see anything more than a "window". I think this includes the Earth based human community and their model building.

/Fredrik

Hello all

I think background independence has a clear meaning and what is needed is for Edward Witten to make a clear statement on the issue of the status of BI of String theory and LQG!

ian

Hi Fra

You say

"this is somewhat analogous to the problem of the ergodic hypothesis. We appeal to a principle of indifference, and argue that all microstates are equally possible - but, how is the microstructure itself selected? the ergodic hypothesis is implicit in the choice of microstructure (to me the microstructure is close to synonmous to background structure, but is more general, it applies (in my mind at least:) to an arbitrary state spce, not just 4D spacetime)"

The process of going from one microstate to another is a physical process, whereas two spacetimes considered as "one is no better than the other" are related by a gauge transformation! -i.e are physically equivalent not mere starting points

ian

Can I just say to anyone who claims that BI is semantics...

In BI theories the distanc between two points DEFINED in terms of coordinate points is GAUGE INVARIANT. That is small and large distances are gauge equivalent! Hence the reason why BI theories like

lqg are manifestly UV finite! A very important point to understand!

ian

No that's incorrect. Back to definitons again.

LGQ has absolutely not been shown *yet* to be 'UV finite' however string theory is completely UV finite. The proof thereof more or less started the string theory craze back in the early 80s.

A theory that is UV finite is sort of like any theory with a cutoff, except that no renormalization is necessary and divergences are not present. It preserves all global symmetries and all local diffeomorphisms as well (as long as you can show BRST invariance for the latter)

This is to be contrasted with any other quantum theory you might have met with in the past (where you have in principle quadratic or higher UV divergences that require counterterms and special treatment)

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You say

"this is somewhat analogous to the problem of the ergodic hypothesis. We appeal to a principle of indifference, and argue that all microstates are equally possible - but, how is the microstructure itself selected? the ergodic hypothesis is implicit in the choice of microstructure (to me the microstructure is close to synonmous to background structure, but is more general, it applies (in my mind at least:) to an arbitrary state spce, not just 4D spacetime)"

The process of going from one microstate to another is a physical process, whereas two spacetimes considered as "one is no better than the other" are related by a gauge transformation! -i.e are physically equivalent not mere starting points

Hello ian!

Perhaps you see this from a more specific context than I do so I am not sure I got your message so I can't argue. But it seems you say that the evolution of the microstate (of a microstructure) is physics, and the CHOICE of microstructure is not.

I both see your point, and partially disagree with it.

In a classic sense I see your point and it's clear. However I think there is more to this, in particular when you ponder the merging of dynamical microstructures with a measurement theory. If the processing of choosing a microstructure is not physics, then what is it? What is it even doing here? I think that some of these reasonings make use of a realist view of certain things, for example symmetries. The problem I see is, how the inside observer can deduct a symmetry in the external world. THIS process is IMO physics, and it might suggest (I think so, but work is in progress of course) that there is an uncertainty principle between the symmetry itself, and the "choice" of microstructure. I think this originates from a information constraint of the overall view, in which also our theories live.

Many theories to day are of a realist type of theory, in different ways. The physicists equations are considered somehow not truly interacting with reality. I think they should.

The particular connection I imagine is that this "tension" and uncertainty of external symmetry, is partly related to the evolution and emergence of the "arbitrary" choices. So the choices might in a sense be arbitrary and non-physical, but their persistence and manifestation in nature is I think not.

I tried to analyse rovelli's logic in his book on QG, but I have to say that he lost me. He seems to make, somewhere in his exploration of RQM, some assumptions and omissions of analysing problems that I simply can't buy. But my general impression is that my disagreement in reasoning regards the nature of symmetries.

I think that while external symmetries are consistent with the observers information, this is something different from certainty. And in a measurement theory, the action of an observer is usually thought to be a function of information at hand. I think the lack of perfect symmetry does impact the actions of the observer - the inside uncertainty of the CHOICE of symmetry(note here the picture is inverted!) - might be reflected in the action taken by this observer/system.

This is IMO one key point in the problem of foundations of QM in the context of a background independent way. I think the two problems are related. Rovelli's RQM reasoning started very good, but I think there might be an optional end.

What I tries to say, that the external SYMMETRY, is itself a background, in the space of possible symmetries. So the fundamental problem persists. It's just that we move/transform the background around without solving the real problem.

This is admittedly just my personal conjecturd view though.

/Fredrik

What I tries to say, that the external SYMMETRY, is itself a background, in the space of possible symmetries. So the fundamental problem persists. It's just that we move/transform the background around without solving the real problem.

And the point was, that this is the observation I hade personally made, and it has caused me to change the focus. To the search for "the perfect symmetry", to the PROCESS of evolving symmetries. The reason is that nothing at all suggest to me that this perfect symmetry is likely to be nailed. But instead, perhaps physics is all about this process? This still doesn't contradict the existence of meta-stable symmetries. It could rather (hopefully!) explain them.

/Fredirk

LGQ has absolutely not been shown *yet* to be 'UV finite' however string theory is completely UV finite.
Even though I personally prefer strings over LQG, I disagree with this statement.
First, as far as I know, in string theory UV finiteness is shown rigorously only for 1 and 2 loops, while a rigorous proof for arbitrary number of loops is still lacking.
Second, as far as I know, UV finiteness of LQG is shown rigorously (LQG is defined non-perturbatively) to be a consequence of compactness of SU(2) or SO(3).

If LQG was UV finite, it would be a big deal. Last I check, no one had shown that.

What is true is that some (but maybe not all --open question) diagrams in the spin foam model (and afaik, only the spin foam model) is claimed to be UV finite. This is technically different than what you can show (at least heuristically) in models like string theory or even certain supergravity theories where you have supersymmetry present that affords you powerful nonrenormalization theorems that go beyond order by order checks.

It is true that finitiness has only been rigorously shown up to genus 2 in st, though I gather there isn't much discussion about going further atm and there exist tentative handwavey proofs of finiteness to all orders (depending on who you talk too)

Haelfix, sometimes it seems your statements can cause confusion just because you use words differently from other people. Not necessarily with any intent to confuse the issues. Maybe it would help if you gave written online sources from known people.

No that's incorrect. Back to definitons again.

LGQ has absolutely not been shown *yet* to be 'UV finite' however string theory is completely UV finite.

...
First, as far as I know, in string theory UV finiteness is shown rigorously only for 1 and 2 loops, while a rigorous proof for arbitrary number of loops is still lacking.

Second, as far as I know, UV finiteness of LQG is shown rigorously (LQG is defined non-perturbatively) to be a consequence of compactness of SU(2) or SO(3).

If LQG was UV finite, it would be a big deal. Last I check, no one had shown that...

With whom did you check?

Here's an example where Rovelli was discussing UV finiteness in an invited talk at Strings 2008.

The string theoreticians at the conference didn't seem to have any trouble understanding him. They asked a number of questions about other things but didn't bother to challenge or question about UV finiteness, as might have been expected if it were controversial.

Here are quotes. Slides 4, 14, 22, 27, 28 and 40 mention UV finiteness. I've tried to preserve Rovelli's emphasis. He used colored arrows, which I've tried to copy. One statement was outlined for special graphic emphasis---I simply bolded it.

(Slide 4) said:
- In gravity, (unrenormalizable) UV divergences are consequences of a perturbation expansion around a wrong vacuum. => Confirmed a posteriori in LQG.
...
Main result
=> Definition of Diffeomorphisms invariant quantum field theory (for gauge fields plus fermions), in canonical and in covariant form.

Slide 14 said:
...
Result:
=>A (separable) Hilbert space H of states, and an operator algebra A .
=> Basis of H: abstract spin network states: graph labelled by spins and intertwiners.
=> A well defined UV-finite dynamics.

(Slide 22) said:
Dynamics:

Given by a Wheeler-deWitt operator H in H: H Psi = 0
• H is defined by a regularization of the classical Hamiltonian constraint. In the limit in which the regularization is removed.
=> H is a well defined self-adjoint operator, UV finite on diff-invariant states.
...

(Slide 27) said:
The limit alpha -> 0 is trivial because
there is no short distance structure at all in the theory!

• The theory is naturally ultraviolet finite

(Slide 28) said:
Matter:
• YM, fermions
• Same techniques: The gravitational field is not special
=>UV finiteness remains
YM and fermions on spin networks = on a Planck scale lattice!
Notice: no lattice spacing to zero!
...

(Slide 40) said:
IV. Summary

• Loop quantum gravity is a technique for defining Diff-invariant QFT. It offers a radically new description of space and time by merging in depth QFT with the diff-invariance introduced by GR.
• It provides a quantum theory of GR plus the standard model in 4d, which is naturally UV finite and has a discrete structure of space at Planck scale.
• Has applications in cosmology, black hole physics, astrophysics; it resolves black hole and big bang
singularities.

- Unrelated to a natural unification of the forces (we are not at the “end of physics”).
- Different versions of the dynamics exist.
- Low-energy limit still in progress.
+ Fundamental degrees of freedom explicit.
+ The theory is consistent with today’s physics.
+ No need of higher dimensions (high-d formulation possible).
+ No need of supersymmetry (supersymmetric theories possible).
+ Consistent with, and based on, basic QM and GR insights.

The reference to discrete structure is to the discrete spectrum of geometrical measurements, not to a division of space into little chunks. The highlighting and italics here are Rovelli's: I tried to preserve the sense of what he considered important to get across to the audience in the 30-minute talk, and therefore emphasized.

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Some of the statements that Rovelli wrote down there are to put it bluntly, stretching the truth a little, probably for simplification or for motivational purposes. For instance statements like this will leave people scratching their head.

"In gravity, (unrenormalizable) UV divergences are consequences of a perturbation expansion around a wrong vacuum"

Trivial and modern counterexample: N=8 Supergravity.

Further there is hyperbole going on a bit. Like I said, they've shown UV finiteness in some, but not all of the Barret-Crane diagrams. What they have not shown is that the entire theory (describing Einstein Hilbert gravity) perse is UV finite. If they did, believe me all of HEP would jump right on the bandwagon. This has been one of the main sticking points for years now against them, summed up in eg Nicolai's paper or anyone of L Motls endless rants on the subject. Its also why they really require gravity to have a nontrivial fixed point, which is perfectly plausible but so far controversial.

Now there is a bit of a subtlety going on here and the main source of the confusion, and it again goes back to definitions. When LQG people talk about UV finiteness, they don't mean it in the same sense HEP physicists usually do (or what I mean).

See hep-th/0501114
To quote:
"This question is in part answered by the fact that the notions of ‘finiteness’ and ‘regulator independence’ as currently used in LQG on the one hand, and in conventional quantum field theory and perturbative quantum gravity on the other, are not the same; see section 4.5."

...
Now there is a bit of a subtlety going on here and the main source of the confusion, and it again goes back to definitions. When LQG people talk about UV finiteness, they don't mean it in the same sense HEP physicists usually do (or what I mean).

See hep-th/0501114
To quote:
"This question is in part answered by the fact that the notions of ‘finiteness’ and ‘regulator independence’ as currently used in LQG on the one hand, and in conventional quantum field theory and perturbative quantum gravity on the other, are not the same; see section 4.5."

Heh heh. It sounds as if those LQG people are being a bit sneaky, using a different definition of UV finiteness! But here's what Hermann Nicolai says---he's a leading European HEP physicist whose work is primarily in string. Has co-authored the only two scholarly critiques of LQG that I know of by a string theorist. You just quoted an older article of his. In his more recent one he says explicitly:

At least in its present incarnation, the canonical formulation of LQG does not encounter any UV divergences,...

Here is one prominent HEP physicist---string theorist---who is quite clear about what LQG people mean by "UV finite".

That doesn't mean he isn't critical of LQG! He immediately points to the struggle to find the right Hamiltonian. Things have progressed on that front since Nicolai's 2006 critique and I believe that is probably one reason Rovelli was invited to speak at Strings 2008.

But the Hamiltonian (more broadly QG dynamics) is a tough problem and Rovelli devoted a substantial part of his talk to it: to current work on n-point functions and the semiclassical limit. He was quite frank about it.

I would say there was nothing devious or obscure about Rovelli's UV finiteness statements. I would guess that the string audience understood exactly what he was saying.
They were certainly clever enough to realize that since the approach is non-perturbative, UV divergences if present would be manifested outside of perturbation series context. There were 400 smart people in the audience---if they had any doubts about UV finiteness they could have asked. Indeed they asked about plenty of other stuff! Rovelli said he got more questions than most of the other speakers and he was very pleased by the response.

It would be interesting to see a 2009 version of Hermann Nicolai's critique! You quoted what he said in 2005, I quoted from his 2006 paper. A great deal has happened since then (particularly in the spinfoam department). One way to read Rovelli's talk at String 2008 is as responding to points in Nicolai's 2006 paper. Now I would like to see how Nicolai replies--an updated version.

Anyway, Haelfix, please give some more sources, hopefully more recent. You mentioned something about Barrett-Crane diagrams. (Does that mean spinfoams?) You said something had been proved in a few cases of B-C diagrams. Could you give an arxiv link?
=====================

For more context, here is an extended passage from Nicolai's 2006 critique:
==quote==
At least in its present incarnation, the canonical formulation of LQG does not encounter any UV divergences, but the problem reappears through the lack of uniqueness of the canonical Hamiltonian. For spin foams (or, more generally, discrete quantum gravity) the problem is no less virulent. The known finiteness proofs all deal with the behaviour of a single foam, but, as we argued, these proofs concern the infrared rather than the ultraviolet. Just like canonical LQG, spin foams thus show no signs of ultraviolet divergences so far, but, as we saw, there is an embarras de richesse of physically distinct models,...
==endquote==
http://arxiv.org/abs/hep-th/0601129

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Sigh, this is very basic and uncontroversial. Read section 4.5 of that HEP paper I linked too. It spells out what's going on explicitly and the redefinitions that are taking place. Part of the whole premise of Nicolai's set of papers is to get regular physicists in tune with the language the LQG people use, rather than to have discussions like this.

Either way we are talking about different things.

"Just like canonical LQG, spin foams thus show no signs of ultraviolet divergences so far"

This is true, but its similar to the string in the sense that they haven't shown this in generality (eg for the string, past 2 loops) and is still open question at least in the usual lore that I've listened too in conferences/lectures etc. Its strongly suspected that they'll oneday be able to prove it in generality. But this is still not the same thing as what regular physicists mean when they're talking about UV divergences of gravity (back to the first point)

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Sigh, this is very basic and uncontroversial. Read section 4.5 of that HEP paper I linked too. .

I read that 2005 paper when it came out, and have just re-read section 4.5. It does not say what you seem to think it says. Here is a quote:

==quote section 4.5 of Nicolai's earlier paper==
From what has just been said, it is evident that infinities can never appear in the LQG
regularisation procedure, and in this sense the resulting theory is ‘finite’, at least as far as the kinematical operators are concerned. LQG nevertheless requires the regularisation of the area and volume operators in order even to be able to define the quantum counterparts of the classical constraints via Thiemann’s trick. The outlined regularisation is, therefore, not introduced to remove divergences. Standard short distance, QFT divergences ‘disappear’ in the LQG approach by the very construction of the theory: all states are discrete, and at any step of the calculation one deals only with a finite number of objects.

The price one pays are ambiguities of the type encountered above, some of which can only be eliminated by making ad hoc choices... We shall encounter more such ambiguities when we attempt to define the Hamiltonian constraint operator.
==endquote==

Clearly LQG is UV finite and you might as well grant that---it doesn't have UV divergences, doesn't develop infinities. This UV finiteness is obtained at a price: and that price must be paid---a satisfactory dynamics must be developed and they are working on that. How Nicolai describes the price may be out of date. Ambiguities may have been eliminated, work with spinfoams has made considerable progress since 2006. We can't say if Nicolai is correct in detail about the state of LQG dynamics. But that is a different issue.

"Just like canonical LQG, spin foams thus show no signs of ultraviolet divergences so far"

This is true, but its similar to the string in the sense that they haven't shown this in generality (eg for the string, past 2 loops) and is still open question at least in the usual lore that I've listened too in conferences/lectures etc..

Haelfix, I'm glad you recognize the truth of that statement by Nicolai in his 2006 paper!
That is great! It almost concludes our discussion. But you draw an incorrect analogy with string. There is nothing more to be shown in the UV finiteness department. That is built in.
What the so far refers to is the work proceeding in the dynamics department, the spinfoam path integral, the canonical Hamiltonian.

The game isn't over until the dynamics is settled, and shown to have the correct limiting behavior. We all know that and Rovelli was extremely frank about it in his Strings 2008 talk.
That is what the so far refers to. It's not like in perturbative analysis where you have to add on another level and go up from 2 loops to 3 loops. LQG UV finiteness is a done deal---essentially a consequence of diffeo-invariance (Rovelli explained this in his talk).
The flip side is that diffeo-invariance is a demanding requirement and makes finding the correct dynamics tough.