# M-theory and loop quantization of higher dimensional SUGRA

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## Main Question or Discussion Point

A new duality between Topological M-theory and Loop Quantum Gravity
(Submitted on 17 Jul 2017)
Inspired by the long wave-length limit of topological M-theory, which re-constructs the theory of 3+1D gravity in the self-dual variables' formulation, we conjecture the existence of a duality between Hilbert spaces, the H-duality, to unify topological M-theory and loop quantum gravity (LQG). By H-duality non-trivial gravitational holonomies of the kinematical Hilbert space of LQG correspond to space-like M-branes. The spinfoam approach captures the non-perturbative dynamics of space-like M-branes, and can be claimed to be dual to the S-branes foam. The Hamiltonian constraint dealt with in LQG is reinterpreted as a quantum superposition of SM-brane nucleations and decays.
Subjects: High Energy Physics - Theory (hep-th)
Cite as: arXiv:1707.05347 [hep-th]

New Variables for Classical and Quantum Gravity in all Dimensions I. Hamiltonian Analysis
Norbert Bodendorfer, Thomas Thiemann, Andreas Thurn
(Submitted on 18 May 2011 (v1), last revised 12 Feb 2013 (this version, v2))
Loop Quantum Gravity heavily relies on a connection formulation of General Relativity such that 1. the connection Poisson commutes with itself and 2. the corresponding gauge group is compact. This can be achieved starting from the Palatini or Holst action when imposing the time gauge. Unfortunately, this method is restricted to D+1 = 4 spacetime dimensions. However, interesting String theories and Supergravity theories require higher dimensions and it would therefore be desirable to have higher dimensional Supergravity loop quantisations at one's disposal in order to compare these approaches. In this series of papers, we take first steps towards this goal. The present first paper develops a classical canonical platform for a higher dimensional connection formulation of the purely gravitational sector. The new ingredient is a different extension of the ADM phase space than the one used in LQG, which does not require the time gauge and which generalises to any dimension D > 1. The result is a Yang-Mills theory phase space subject to Gauss, spatial diffeomorphism and Hamiltonian constraint as well as one additional constraint, called the simplicity constraint. The structure group can be chosen to be SO(1,D) or SO(D+1) and the latter choice is preferred for purposes of quantisation.
Comments: 28 pages. v2: Journal version. Minor clarifications
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
Journal reference: Class. Quantum Grav. 30 (2013) 045001
DOI: 10.1088/0264-9381/30/4/045001
Cite as: arXiv:1105.3703 [gr-qc]

Towards Loop Quantum Supergravity (LQSG)
Norbert Bodendorfer, Thomas Thiemann, Andreas Thurn
(Submitted on 6 Jun 2011 (v1), last revised 12 Jun 2012 (this version, v2))
Should nature be supersymmetric, then it will be described by Quantum Supergravity at least in some energy regimes. The currently most advanced description of Quantum Supergravity and beyond is Superstring Theory/M-Theory in 10/11 dimensions. String Theory is a top-to-bottom approach to Quantum Supergravity in that it postulates a new object, the string, from which classical Supergravity emerges as a low energy limit. On the other hand, one may try more traditional bottom-to-top routes and apply the techniques of Quantum Field Theory. Loop Quantum Gravity (LQG) is a manifestly background independent and non-perturbative approach to the quantisation of classical General Relativity, however, so far mostly without supersymmetry. The main obstacle to the extension of the techniques of LQG to the quantisation of higher dimensional Supergravity is that LQG rests on a specific connection formulation of General Relativity which exists only in D+1 = 4 dimensions. In this Letter we introduce a new connection formulation of General Relativity which exists in all space-time dimensions. We show that all LQG techniques developed in D+1 = 4 can be transferred to the new variables in all dimensions and describe how they can be generalised to the new types of fields that appear in Supergravity theories as compared to standard matter, specifically Rarita-Schwinger and p-form gauge fields.
Comments: 9 pages. v2: minor improvements in presentation, virtually identical to published version
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
MSC classes: 83C05, 83E50, 83E15
Journal reference: Phys. Lett. B 711: 205-211 (2012)
DOI: 10.1016/j.physletb.2012.04.003
Cite as: arXiv:1106.1103 [gr-qc]

A note on quantum supergravity and AdS/CFT
Norbert Bodendorfer
(Submitted on 7 Sep 2015)
We note that the non-perturbative quantisation of supergravity as recently investigated using loop quantum gravity techniques provides an opportunity to probe an interesting sector of the AdS/CFT correspondence, which is usually not considered in conventional treatments. In particular, assuming a certain amount of convergence between the quantum supergravity sector of string theory and quantum supergravity constructed via loop quantum gravity techniques, we argue that the large quantum number expansion in loop quantum supergravity corresponds to the 1/N2c expansion in the corresponding gauge theory. In order to argue that we are indeed dealing with an appropriate quantum supergravity sector of string theory, high energy (α′) corrections are being neglected, leading to a gauge theory at strong coupling, yet finite Nc. The arguments given in this paper are mainly of qualitative nature, with the aim of serving as a starting point for a more in depth interaction between the string theory and loop quantum gravity communities.
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
Cite as: arXiv:1509.02036 [hep-th]

M-theory claims to unite 5 10D string theories, and 11D SUGRA

eleven-dimensional supergravity is the lower energy limit of M-theory.

there is no non-pertubative formulation of M-theory, it remains a conjecture.

since there is no non-pertubave forumlation of M-theory, and M-theory has
eleven-dimensional supergravity is the lower energy limit of M-theory,
why not define M-theory nonpertubative loop quantization of eleven-dimensional supergravity
Norbert Bodendorfer, Thomas Thiemann, Andreas Thurn have applied loop quantization of higher dimensional SUGRA, including eleven-dimensional supergravity

loop quantization of higher dimensional SUGRA, including eleven-dimensional supergravity is nonpertubative, it describes physics at the Planck scale

is there a reason it cannot serve as a nonpertubative basis of M-theory?

also worth noting

if this research program is successful,

M-theory contains 11D supergravity as its low energy limit,

at the UV limit it is described by loop quantization of 11D supergravity

loop quantization of 11D supergravity has 11D supergravity as its semiclassical limit, and therefore via M-theory also contains 5 10D string theories

Last edited:

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By now it's pretty clear that "loop quantization" is some sort of technical mistake. This procedure has been applied to dozens of starting points and it has never yet been able to make contact with either classical general relativity or with nongravitational quantum field theory. String theory makes contact with both of those. So whatever the right way to define M-theory is, loop quantization is not the answer.

By now it's pretty clear that "loop quantization" is some sort of technical mistake. This procedure has been applied to dozens of starting points and it has never yet been able to make contact with either classical general relativity or with nongravitational quantum field theory. String theory makes contact with both of those. So whatever the right way to define M-theory is, loop quantization is not the answer.

Einstein Equation from Covariant Loop Quantum Gravity in Semiclassical Continuum Limit
Muxin Han
(Submitted on 25 May 2017 (v1), last revised 11 Jul 2017 (this version, v2))
In this paper we explain how 4-dimensional general relativity and in particular, the Einstein equation, emerge from the spinfoam amplitude in loop quantum gravity. We propose a new limit which couples both the semiclassical limit and continuum limit of spinfoam amplitudes. The continuum Einstein equation emerges in this limit. Solutions of Einstein equation can be approached by dominant configurations in spinfoam amplitudes. A running scale is naturally associated to the sequence of refined triangulations. The continuum limit corresponds to the infrared limit of the running scale. An important ingredient in the derivation is a regularization for the sum over spins, which is necessary for the semiclassical continuum limit. We also explain in this paper the role played by the so-called flatness in spinfoam formulation, and how to take advantage of it.
Comments: 12+3 pages, no figure, presentation improved
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th)
Cite as: arXiv:1705.09030 [gr-qc]
(or arXiv:1705.09030v2 [gr-qc] for this version)

The author seems to be using a given classical solution to general relativity to zoom into the part of the spin-foam path integral which follows that solution, rather than showing that the unrestricted spin-foam path integral intrinsically peaks around the classical solution. He considers (section VI) a series of triangulations which by design converges on a classical geometry, and then argues that for each such triangulation, the spin foams reproduce something from Regge calculus, a known approximation of general relativity.

But what about spin foams that deviate from classicality? We need some argument that they won't drown out the semiclassical part, and I don't see it. I could be missing it. Someone like @Haelfix or @atyy would probably be a better critic... They certainly know the subject much better than me.

The author seems to be using a given classical solution to general relativity to zoom into the part of the spin-foam path integral which follows that solution, rather than showing that the unrestricted spin-foam path integral intrinsically peaks around the classical solution. He considers (section VI) a series of triangulations which by design converges on a classical geometry, and then argues that for each such triangulation, the spin foams reproduce something from Regge calculus, a known approximation of general relativity.

But what about spin foams that deviate from classicality? We need some argument that they won't drown out the semiclassical part, and I don't see it. I could be missing it. Someone like @Haelfix or @atyy would probably be a better critic... They certainly know the subject much better than me.
But here, the author does show "loop quantization" makes "contact with either classical general relativity" via Regge calculus.

But here, the author does show "loop quantization" makes "contact with either classical general relativity" via Regge calculus.
A path integral is a sum over all possibilities. So that necessarily includes both classical histories and highly nonclassical ones. To show that the theory has a classical limit, you would need to show that the overall sum is dominated by the contribution from the classical histories, or something similar. I didn't see, in the paper, any attempt to put the analysis of classical histories into such a context.

Urs Schreiber
Gold Member
By now it's pretty clear that "loop quantization" is some sort of technical mistake. This procedure has been applied to dozens of starting points and it has never yet been able to make contact with either classical general relativity or with nongravitational quantum field theory.
This point deserves to be expanded on a bit:

To recall, the starting point of LQG is to encode the Riemannian metric in terms of the parallel transport of the affine connection that it induces. This parallel transport is an assignment to each smooth curve in the manifold between points $x$ and $y$ of a linear isomorphism $T_x X \to T_y Y$ between the tangent spaces over these points.

This assignment is itself smooth, as a function on the smooth space of smooth curves, suitably defined. Moreover, it satisfies the evident functoriality conditions, in that it respects composition of paths and identity paths.

It is a theorem that smooth (affine) connections on smooth manifolds are indeed equivalent to such smooth functorial assignments of parallel transport isomorphisms to smooth curves. This theorem goes back to Barrett, who considered it for the case that all paths are taken to be loops. For the general case it is discussed in arxiv.org/abs/0705.0452, following suggestion by John Baez.

So far so good. The idea of LQG is now to use this equivalence to equivalently regard the configuration space of gravity as a space of parallell transport/holonomy assignments to paths (in particular loops, whence the name "LQG").

But now in the next step in LQG, the smoothness condition on these parallel transport assignments is dropped. Instead, what is considered are general functions from paths to group elements, which are not required to be smooth or even to be continuous, hence plain set-theoretic functions. In the LQG literature these assignments are then called "generalized connections". It is the space of these "generalized connections" which is then being quantized.

The trouble is that there is no relation left between "generalized connections" and the actual (smooth) affine connections of Riemanniann geometry. The passage from smooth to "generalized connections" is an ad hoc step that is not justified by any established rule of quantization. It effectively changes the nature of the system that is being quantized.

Removing the smoothness and even the continuity condition on the assignment of parallel transport to paths loses all contact with how the points in the original spacetime manifold "cohere", as it were, smoothly or even continuously. The passage to "generalized connections" amounts to regarding spacetime as just a dust of disconnected points.

Much of the apparent discretization that is subsequently found in the LQG quantization is but an artifact of this dustification. Since it is unclear what (and implausible that) the generalized connections have to do with actual Riemannian geometry, it is of little surprise that a key problem that LQG faces is to recover smooth spacetime geometry in some limit in the resulting quantization. This is due to the dustification of spacetime that happened even before quantization is applied.

When we were discussing this problem a few years back, conciousness in the LQG community grew that the step to "generalized connections" is far from being part of a "conservative quantization" as it used to be advertized. As a result, some members of the community started to investigate the result of applying similar non-standard steps to the quantization of very simple physical systems, for which the correct quantization is well understood. For instance when applied to the free particle, one obtains the same non-separable Hilbert spaces that also appear in LQG, and which are not part of any (other) quantization scheme. Ashtekar tried to make sense of this in terms of a concept he called "shadow states" https://arxiv.org/abs/gr-qc/0207106 . But the examples considered only seemed to show how very different this shadowy world is from anything ever seen elsewhere.

Some authors argued that it is all right to radically change the rules of quantization when it comes to gravity, since after all gravity is special. That may be true. But what is troubling is that there is little to no motivation for the non-standard step from actual connections to "generalized connections" beyond the fact that it admits a naive quantization.

julian
Gold Member
See page 258 of "String Gravity and Physics at the Planck Scale":

"While we can restrict ourselves to suitably smooth fields in the classical theory, in quantum field theory, we are forced to allow distributional field configurations. Indeed, even in the free field theories in Minkowski space, the Gaussian measure that provides the inner product is concentrated on genuine distributions. This is the reason why in quantum theory fields arise as operator-valued distributions."

These so-called "quantum configuration spaces" that include distributional fields are not something unique to LQG.

Urs Schreiber
Gold Member
See page 258 of "String Gravity and Physics at the Planck Scale":

"While we can restrict ourselves to suitably smooth fields in the classical theory, in quantum field theory, we are forced to allow distributional field configurations. Indeed, even in the free field theories in Minkowski space, the Gaussian measure that provides the inner product is concentrated on genuine distributions. This is the reason why in quantum theory fields arise as operator-valued distributions."

These so-called "quantum configuration spaces" that include distributional fields are not something unique to LQG.
Hey Julian,

ah, no, that's some misunderstanding.

First, there is a big difference between a distribution (which is a well-behaved limit of smooth functions) and a plain discontinuous function (which is what is used in LQG).

To see this, consider the simple example of 1d field theory, i.e. a particle moving in some smooth manifold $X$. The ordinary field configurations are smooth functions $\mathbb{R}^1 \longrightarrow X$. Passing to "generalized field configurations" in the sense of LQG means to allow discontinuous functions. This means that we allow particles to jump around at will, they may sit in your lab for $t \in (-\infty,0)$ and at $t = 0$ they may be behind the moon, or behind the cosmic horizon, or anywhere.

Second, the operator valued distributions in field theory are quantum observables on the phase space, reflecting the fact that not all observables on phase space may be quantized, only those that are what is called "smeared with a test function", that's why they become genuine operators only after being fed a test function, and that's what makes them operator valued distributions.

For the simplest example take free scalar field theory. The phase space is the space $C^\infty_{sol}(X)$ of smooth functions on spacetime that solve the wave equation (often expressed equivalently in terms of their initial value data, i.e. field coordinates and momenta, on a Cauchy surface). The operator-valued distributions appear when computing the Poisson algebra of functions _on_ that phase space, because the Poisson bracket of two point-evaluation functions $\Phi(x) : C^\infty_{sol}(X) \to \mathbb{R}$ is the causal Green's function, which needs to be smeared with a test function in order to become a smooth function. But the phase space is $C^\infty_{sol}(X)$, or otherwise we would be talking about a different theory.

Third, notice that there is no disagreement among the LQG researchers anymore that what they do is not the established way of quantization. Check out the "shadow state" article by Ashtekar that I cited above. It's not arguing that the procedure is secretly the usual one. It is accepting that the "polymer state" quantization used in LQG is not the established procedure, it shows this vividly for simple examples such as the free particle, and instead it is trying to argue that "polymer" quantization can nevertheless make sense physically.

Urs any thoughts on the theory that LQG is related to M-theory via H-duality

A new duality between Topological M-theory and Loop Quantum Gravity
(Submitted on 17 Jul 2017)
Inspired by the long wave-length limit of topological M-theory, which re-constructs the theory of 3+1D gravity in the self-dual variables' formulation, we conjecture the existence of a duality between Hilbert spaces, the H-duality, to unify topological M-theory and loop quantum gravity (LQG). By H-duality non-trivial gravitational holonomies of the kinematical Hilbert space of LQG correspond to space-like M-branes. The spinfoam approach captures the non-perturbative dynamics of space-like M-branes, and can be claimed to be dual to the S-branes foam. The Hamiltonian constraint dealt with in LQG is reinterpreted as a quantum superposition of SM-brane nucleations and decays.
Subjects: High Energy Physics - Theory (hep-th)
Cite as: arXiv:1707.05347 [hep-th]

Induced loop quantum cosmology on a brane via holography
C. A. S. Silva
(Submitted on 20 Jul 2017)
Based on the holographic principle, it is demonstrated that loop quantum Friedmann equations can be induced on a brane, corresponding to a strongly coupled string regime in the bulk, and have braneworld cosmology equations as its low energy limit. Such result can establish a possible connection between loop quantum gravity and string theory.
Subjects: General Relativity and Quantum Cosmology (gr-qc)
Cite as: arXiv:1707.07586 [gr-qc]

Urs Schreiber
Gold Member
Urs any thoughts on the theory that LQG is related to M-theory via H-duality
A new duality between Topological M-theory and Loop Quantum Gravity
Weird stuff. Are you just going by the title, or did you try to read this and see if it makes sense to you?

MathematicalPhysicist
Gold Member
Weird stuff. Are you just going by the title, or did you try to read this and see if it makes sense to you?
Yes, but is it weirder than your stuff? :-D

Your theory is crazy, the question is whether it's crazy enough to be true.

Nils Bohr.

Urs Schreiber
Gold Member
Yes, but is it weirder than your stuff?
I suppose you are referring to arXiv:1611.06536 and arXiv:1702.01774 which I was recently talking about at PF-Insights here and which I presented at StringMath17 here. These articles contain a sequence of mathematical propositions and proofs, the assumptions and the laws of inference are stated clearly. To any educated reader the statements ought to be unambiguous and all details to check them are provided, in the established style of rigorous mathematics. These results clearly suggest a conjecture about physics which is far-reaching (as on the second but last slide of the talk), but you are not asked to buy into this conjecture until it is proven.

My trouble with the articles mentioned by Kodama above is not that their conclusion is weird, rather I find the conclusion, as advertized in the title, does not quite parse. Instead, my trouble with these articles is that their reasoning seems incoherent to me. That's why I was asking if Kodama actually tried reading them, or if he just went by the flashy title.

The phrase "not even wrong", much abused these days, originally referred to a statement which does not parse, so that one cannot start checking whether it is right or wrong. In formal logic this is well understood: a proposition may be true or false, but to be a proposition in the first place, it needs to be what mathematicians call a "well-formed sentence". Gibberish is not even wrong.

But if I am missing something, let me know what you think might be a well-formed claim that these authors are meaning to make. Then I can try to reply to that, if desired.

MathematicalPhysicist
Gold Member
I am sure you believe your theory is consistent and mathematically sound, but does it describe reality?

More than 3+1 dimensions which we "sense" is weird stuff no matter how you wrapped it around your head.

There are a lot of toy models in the literature of theoretical physics and mathematical physics, I don't see why take one work any more serious than the others if eventually all papers only suggest more conjectures and don't provide proof for the earlier work.

For example has anyone provided proof for the BFSS conjectures in the following paper:
https://arxiv.org/abs/hep-th/9610043

It seems people in this field shoot for more conjectures and less proof, perhaps the propositions are indeed so much hard to prove mathematically.

Urs Schreiber
Gold Member
I am sure you believe your theory is consistent and mathematically sound, but does it describe reality?
Let's try to concentrate and exercise some sober intellectual discipline. The topic of this thread is whether two mathematical models are related, to the extent that they are defined. Due the power of mathematics, such a question makes sense independently of what these models have to do with reality. In principle this could be worth a technical analysis. But that will be more tedious than throwing around buzzwords. I repeat that I am willing to try to react to any well-formed statement that somebody can extract from these articles. Otherwise I'll call it quits on this thread here.

MathematicalPhysicist
Gold Member
The question is what I need to know before reading those articles?

What are the prerequisites for reading this stuff?

I can say that I am quite versed in QFT and Statistical Mechanics, and in maths I have basic courses in Algebraic Topology and Algebraic Geometry under my belt; I also read Griffiths and Harris' book (long and tough), also read Krengel's Ergodic Theorems (and know quite a lot of analysis stuff, though this stuff in the articles seems more related to algebra).

The question is what one needs to know before reading this stuff and actually rigorously criticize the models?

Urs Schreiber
Gold Member
What are the prerequisites for reading this stuff?
It's pretty elementary. You need to know what a super-Lie algebra is, and how real spin representations work, and what a differential graded-super-commutative algebra is. Expository lecture notes are here

By now it's pretty clear that "loop quantization" is some sort of technical mistake. This procedure has been applied to dozens of starting points and it has never yet been able to make contact with either classical general relativity or with nongravitational quantum field theory. String theory makes contact with both of those. So whatever the right way to define M-theory is, loop quantization is not the answer.
ok, so you and Urs don't care for "loop quantization" or "polymer quantization" as completely nonstandard
why not re-write 11 dimensional supergravity in an equivalent form, something like Ashketar's variables, or a theory that reproduces 11 dimensional supergravity but written as a gauge theory, then perform a standard canonical quantization as an approach to nonpertubative definition of M-theory

The problem with those methods of quantization is not just that they are "nonstandard", the problem is that they don't connect with reality in any way! These authors may start classically with known field theories, but when they construct the quantum theory, they do it differently; and the way they do it does not reproduce anything from known physics, not even qualitatively. String theory may usually predict a lot of things we aren't seeing, and a lot of quantities we would like to test remain impossible to calculate; but at least it exhibits qualitative continuity with established physics. All the new phenomena that came with the revival of quantum field theory in the era of the standard model, like anomalies, instantons, you name it, have their counterparts in string theory. On the gravity side, string theory has a classical limit, and it also reproduces theoretical phenomena of semiclassical gravity.

Loop quantum gravity, on the other hand, seems to provide a recipe where you start with a set of fields that includes general relativity, then you follow their special quantization procedure, and you end up with various equations that the resulting wavefunctions have to satisfy, and maybe you can prove one or two things. But these results are entirely abstract and algebraic, and do not give you back anything like quantum fields in space-time, despite the starting point. So this recipe can certainly produce research papers, but the resulting papers are radically disconnected from ordinary quantum field theory, from classical gravity - and from string theory.

Once I concluded that the divide is really that great, I became perplexed by the size and persistence of the loop quantum gravity literature. I can see one person, or a handful of people, stubbornly persevering in a research program that disdains lots of established physics, due to an idiosyncratic investment in particular ideas. But loop quantum gravity is dozens of people over decades. I was especially troubled by the lack of historical precedent for this.

But I'm happier now because I did find a historical analogy: algebraic quantum field theory. It's not an exact analogy, but it is another example of a research community developing over decades a mathematical formalism which is largely disconnected from what physicists were actually doing. Algebraic quantum field theory starts with a few ideas drawn from real physics, but describes either fields that don't interact, or a few special interacting field theories in lower dimensions. Meanwhile, real physicists were using effective field theory, the renormalization group, and the lattice. Similarly, loop quantum gravity started with some real things, but its subsequent development has diverged from everything in gauge theory and quantum gravity that actually works.

Returning to the idea that some sort of standard but nonperturbative canonical quantization of supergravity might help with the formulation of M-theory, well, it might help just because it would be in that same mainstream of quantization methods, and would therefore actually be relevant to M-theory. But I think that at best it would still only give you a new perspective on what's missing in supergravity. In the end you should find that you need new heavy states, the branes. The best guide might be AdS/CFT in the case of AdS4/CFT3 and AdS7/CFT6, where known CFTs are believed to be entirely equivalent to M-theory in the dual AdS spaces.

haushofer
Funny. I've tried to read Rovelli's Covariant Loop Quantum Gravity for the last few weeks, and with my background I thought I'd be able to understand the basics just fine. Nevertheless I tried a few times and quit because I couldn't get my head around much of it, and even things I was supposed to know were explained in such a way it confused me quite a bit. Maybe I'll open a "basic LQG questions"-topic some time :P

But I'm intruiged by the posts of Urs and Mitchell about the consistency of the whole framework, and maybe it's a sign that I'm not dumb after all :P

MathematicalPhysicist
Gold Member
Funny. I've tried to read Rovelli's Covariant Loop Quantum Gravity for the last few weeks, and with my background I thought I'd be able to understand the basics just fine. Nevertheless I tried a few times and quit because I couldn't get my head around much of it, and even things I was supposed to know were explained in such a way it confused me quite a bit. Maybe I'll open a "basic LQG questions"-topic some time :P

But I'm intruiged by the posts of Urs and Mitchell about the consistency of the whole framework, and maybe it's a sign that I'm not dumb after all :P
I would only say that LQG might not be consistent, but it doesn't mean QFT is necessarily consistent either.

Anyway, you cannot prove the consistency of a mathematical theory, let alone physical theory.

You can only show the inconsistency of a theory by showing that a both a theorem and its negation are contained in this theory.

So did you find such a contradiction?

T
Once I concluded that the divide is really that great, I became perplexed by the size and persistence of the loop quantum gravity literature. I can see one person, or a handful of people, stubbornly persevering in a research program that disdains lots of established physics, due to an idiosyncratic investment in particular ideas. But loop quantum gravity is dozens of people over decades. I was especially troubled by the lack of historical precedent for this.

But I'm happier now because I did find a historical analogy: algebraic quantum field theory. It's not an exact analogy, but it is another example of a research community developing over decades a mathematical formalism which is largely disconnected from what physicists were actually doing. Algebraic quantum field theory starts with a few ideas drawn from real physics, but describes either fields that don't interact, or a few special interacting field theories in lower dimensions. Meanwhile, real physicists were using effective field theory, the renormalization group, and the lattice. Similarly, loop quantum gravity started with some real things, but its subsequent development has diverged from everything in gauge theory and quantum gravity that actually works.
.
Couldn't the same be said for string/M-theory? what if Nature is neither supersymmetric nor extra dimensional? there are indications from LHC, dark matter experiments, non - WIMP miracle, electron EDM experiments, and results such as the 126 Gev higgs that there is no supersymmetry nor extra dimensions as required by string theory.

and loop quantum cosmology may make contact with established physics and observation

Urs Schreiber
Gold Member
What Mitchell Porter writes is right to the point. Allow me just to add some details on this aspect here:

Once I concluded that the divide is really that great, I became perplexed by the size and persistence of the loop quantum gravity literature. I can see one person, or a handful of people, stubbornly persevering in a research program that disdains lots of established physics, due to an idiosyncratic investment in particular ideas. But loop quantum gravityconst is dozens of people over decades. I was especially troubled by the lack of historical precedent for this.

But I'm happier now because I did find a historical analogy: algebraic quantum field theory.
While I agree with the broad impetus of the analogy, hence while I agree that parts of the AQFT community got stuck with an axiomatics which they couldn't connect to known QFT, it should be highlighted that the situation for AQFT is really not as bad. The axioms in themselves are physically reasaonable, if maybe in need of slight modification (see below) and the problem is mainly that the axioms of local nets of C-star algebras of observables are over-ambitious, in that they aim to axiomatize non-perturbative QFTs. The failure to construct interacting examples of such in dimensions 4 or higher remains a failure, but given that for the archetypical case of interest, Yang-Mills theory, this is a famous open question, one in the list of the "Millenium Problems", this is not really a shame.

That the axioms of AQFT in themselves are good is witnessed by what they achieve in 2d, specifically for 2d conformal field theory. The mathematical physics literature on constructions and classification of (rational) 2d CFTs has two pillars, one is vertext operator algebras, the other is AQFT in the guise of conformal nets of observables, notably in the hands of Kawahigashi and Longo and their school. Ironically, thereby AQFT has probably done so far more for the rigorous construction of models in string theoy than for 4d QFT..

This problem has not remained unobserved in the community. Notice that in the last years things have drastically changed with the development of what is now called "perturbative AQFT" due to Brunetti and Fredenhagen. Where Haag-Kastler originally demanded a local net to take values in C-star algebras, aiming for the non-perturbative quantum theory and thus making it hard to construct examples, Brunetti-Fredenhagen make the obvious observation that one may leave the axioms intact while asking not for C-star algebras but just for formal power series algebras. These are the kind of algebras of quantum operators that are produced by standard pertrubative QFT methods (Feynman diagram expansion).

A key observation of these authors is now that the "causal perturbation theory" of the Epstein-Glaser renormalization scheme naturally produces not just global formal power series algebras of interacting fields, but a local net of these (in the sense of AQFT): The notorious "adiabatic swtiching" of the Epstein-Glaser approach, which is traditionally thought of as being something that needs to be taken away at the end of the computation (the "adiabatic limit" of Epstein-Glaser, the source of infrared divergences) is precisely what gives a construction of perturbative observables on each causal local patch, and the causality axiom on the S-matrix in the Epstein-Glaser approach is precisely what makes this assignment "Einstein causally local" in the original sense of Haag-Kastler. Now the framework of AQFT serves to show that we don't actually need to take the adiabatic limit, since the QFT is already encoded in its local net.

This is in fact an old insight due to

V. A. Il’in and D. S. Slavnov, "Observable algebras in the S-matrix approach", Theor. Math. Phys. 36 (1978) 32.

but this work was ignored and forgotten until it was rediscovered in

Romeo Brunetti, Klaus Fredenhagen, "Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds", Commun. Math. Phys. 208 : 623-661, 2000 (math-ph/9903028)

This unification of traditional perturbative QFT (in the Epstein-Glasser renormalization scheme) and AQFT methods now goes by the name locally covariant perturbative quantum field theory (or some variant term) and is arguably the state of the art of modern mathematically precise perturbative QFT. For example this is the framework in which Hollands proves the renormalizability of Yang-Mills theory on general curved backgrounds

Stefan Hollands, "Renormalized Quantum Yang-Mills Fields in Curved Spacetime", Rev.Math.Phys.20:1033-1172,2008 (arXiv:0705.3340)

Finally, and this brings us back to the core topic of this thread, locally covariant perturbative quantum field theory is the natural theoretical context in which to precisely formulate perturbative quantum gravity in what readers here might want to call the "canonical quantization" approach, but done right (albeit just perturbatively), see the references here.

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Urs Schreiber
Gold Member
why not re-write 11 dimensional supergravity in an equivalent form, something like Ashketar's variables, or a theory that reproduces 11 dimensional supergravity but written as a gauge theory, then perform a standard canonical quantization as an approach to nonpertubative definition of M-theory
There is a whole list of reasons why this is overly naive.

First, the idea that a choice of coordinates on phase space (e.g. Ashtekar variables) should affect the outcome of quantization is in contradiction with basic facts of physics. On the contrary, everything ought to be independent of artificial choices of parameterizing phase space, even if maybe one choice of coordinates may make some aspects more transparent than others. But if you find yourself with a would-be quantization that only works in one set of "variables" but not in another, then something went wrong.

Second, gravity is not fundamentally a gauge theory as Yang-Mills theory is, even if you write it in first-order language in terms of a vielbein and a spin connection. The spin connection is an auxiliary field that serves to implement the torsion freeness constraint, but the genuine field of gravity is encoded in the vielbein. Mathematically the statement is that a field configuration of gravity is equivalently encoded in a variant of an affine connection, yes, but the full structure is that of a Cartan connection This is a plain affine connection plus extra data and constraints. Glossing over this point is the source of much confusion in the literature.

Third, the implicit suggestion that it would be easy to write down a "canonical quantization" once we have realized a theory as a gauge theory is in stark contrast to the observed reality: the mathematically precise nonperturbative quantization of Yang-Mills theory is a famous open problem, dubbed a "Millenium Problem". That LQG sweepingly claims to alread have easily solved the much harder problem of non-perturbative quantization of gravity shows the disconnect between LQG and the realities as perceived by the rest of the scientific community. It is similar to claims of simple proofs of the Riemann hypothesis: apart from the "proofs" being overly naive, their statement displays ignorance of the real nature of the problem.

Fourth, the idea that "canonical quantization" is something one applies without much ado in field theory is naive. First of all, what is really meant by "canonical quantization" in mathematically precise terms is that one computes the phase space of the field theory as a (pre-)symplectic (infinite-dimensional) manifold, and then applies either algebraic deformation quantization or else geometric quantization to this (pre-)symplectic space. Nothing about this is particularly "canonical" as the whole subtlety of quantization in field theory rests in the fact that various ambiguities are encountered. These manifest themselves ultimately in what in the perturbative description appears as counter-terms in the renormalization procedure.

The key reason why the quantization of gravity is more subtle than that of Yang-Mills theory is that there is an infinite number of counter-terms to be chosen in the (perturbative) quantization. This is what it means to make the famous statement that gravity is not renormalizable. There are many ways to exhibit this ambiguity, but the most vivid is that this corresponds to the choice of coefficients for all higher-order terms that could be added to the action functional which one starts out thinking one is quantizing. (The entire lack alone of appearance of any manifestation of this infinite ambiguity in the quantization of gravity is what makes LQG dubious without even looking at any details.)

This is by the way where perturbative string theory and also asymptotic safety comes in. Both approaches may be regarded as parameterizing these infinite arrays of possible counterterms by certain algorithms.

In string theory the string perturbation series comes out term-wise finite, hence normalized, and one may trace this back to the presence of the higher string modes, which hence serve as the counter-terms. There is still ambiguity in this (the choice of perturbative string background, hence of point in the "landscape of perturbative vacua") but now it is repackaged in a algebro/geometric way that reveals further structure, notably a dynamics which may choose among different such backgrounds.

In asymptotic safety instead one requires certain lower dimensional submanifolds in the space of counter terms (coupling constants) and the specification of such a submanifold in an infinite-dimensional space absorbes an infinite number of choices, leaving only finitely many to be still specified.

Notice that in principle any finite-dimensional submanifold would do, it need not be the one including a UV-fixed point of the RG flow. It's all about finding some conceptual means of picking among an infinite set of counter-terms by some principle.

Anyway, be that as it may, here it just serves to highlight that without further ideas or principle, there is nothing "canonical" about "canonical quantization" in field theory This is not a new insight, this is the heart of field theory and its RG flow since the days of the founding fathers. The complete absence of any reflection of this in the purported "canonical quanization" of gravity is another way to deduce that somethng is wrong here, without even looking at the details. Of course once we do look at the details it is also easy to see where the problem stems from