# I If 11 dimensional supergravity were nonpertubative quantized

#### kodama

Edward Witten's M-theory conjecture is that there is a theory that has 5 10D superstrings and 11D supergravity as its low energy limit, and is defined nonperturbatively.

Now, I know that Urs and Mitchel are not fans of loop quantization program. fine.

Urs and Mitchell do not think that a loop quantization of 11 dimensional supergravity is physically meaningful.

fine.

But if 11 dimensional supergravity could be nonperturbatively quantized successfully, to the satisfaction of Urs and Mitchel, what are the implications to M-theory and string theory?

but conceptually speaking, if 11 dimensional supergravity were nonpertubative quantized successfully, would the resulting theory be M-theory, or what would be the relation of a nonpertubative quantized 11 dimensional supergravity and M-theory and 5 10D superstring theories?

based on current understanding of string/M-theory,

what would be some qualitative features of a successful nonpertubative quantized 11 dimensional supergravity on the planck scale to the satisfaction of Urs and Mitchell? how does it compare to current pertubative string theory? how does it compare with a loop quantization of 11 dimensional supergravity?

would it consist of wilson loops or spin networks? strings?

Ed Witten hypothesis is that M-theory is the UV completion of 11 dimensional supergravity, does not follow that a successful non-loopy nonpertubative quantized 11 dimensional supergravity is M-theory?

Are there any string theorist like Ed Witten who are attempting to nonpertubative quantized 11 dimensional supergravity? or is there any non-loop research papers and programs that attempt to directly quantize and produce a nonpertubative quantized 11 dimensional supergravity

since string theorists are not fans of loop quantization procedure, have they read loop quantization of 11 dimensional supergravity by loop theorists like Thieman and offer "corrections" to make it more physically correct?

Related Beyond the Standard Model News on Phys.org

#### Urs Schreiber

Gold Member
Now, I know that Urs and Mitchel are not fans of loop quantization program. fine.
No, please. We have explained why it is technically wrong. Science and mathematics is not about fans. Recall that "fan" is short for "fanatic" and the point of science is not to be fanatic, but to soberly check what is true and what is not.

since string theorists are not fans of loop quantization procedure, have they read loop quantization of 11 dimensional supergravity by loop theorists like Thieman and offer "corrections" to make it more physically correct?
Your questions have been going a bit in circles. I am wondering what I could do to get you out of the circle onto a trajectory on which you will eventually make progress with understanding.

Here is a suggestion: You are trying to understand what is actually known about non-perturbative effects. Now the best that actually is known follows from an approach known as resurgence theory. I suggest that you spend some time with studying introductions to resurgence theory. That might eventually provide you with some of the insights that you are after.

#### kodama

hi Urs,
thanks for writing

so you have no objection to rewriting either 4D GR or 11D-SUGRA in Ashketar' variables, as it is mathematically equivalent to standard 4D GR and 11D SUGRA.

why not then quantize this according to resurgence theory, in order to arrive at planck scale physics of 4D GR or 11D SUGRA?
https://ncatlab.org/nlab/show/resurgence+theory
What would the qualitative features of such a theory be?

#### Urs Schreiber

Gold Member
why not then quantize this according to resurgence theory, in order to arrive at planck scale physics of 4D GR or 11D SUGRA?
What would the qualitative features of such a theory be?
Resurgence theory is not a way to quantize, but a way to determine, under some assumptions, from the perturbation series of a perturbative quantization the possible effects that a would-be non-perturbative quantization should see.

If one applies such arguments to the supergravity theories that appear in string theory, one find non-perturbative effects hidden in the perturbation series which would correspond to those induced by the presence of branes. This is why people in this area think that the non-perturbative quantization of gravity cannot be purely a field theory anymore, but needs to be something else that does contain objects which can yield non-perturbative effects of this form.

If you are really interested, i urge you to pick up one of the introductions to resurgence, transseries and non-perturbative effects. It's interesting and informative.

#### kodama

Urs, thanks for this information.

What do you think is the mathematically correct way to directly nonpertubatively quantize Ashketar variables or some variation of the idea?

Would make for an interesting research paper. In the first section, explain why loop quantization is incorrect, then in the second section, provide the correct way to do it.

#### Urs Schreiber

Gold Member
What do you think is the mathematically correct way to directly nonpertubatively quantize
Quantization of Lagrangian field theory means the following:
1. Variation of the Lagrangian yields the equations of motion plus the presymplectic current.
2. After gauge fixing, these define a (graded) symplectic structure on the space of solutions of the equations of motion. This is called the covariant phase space.
3. Non-perturbative quantization means to deform the product of a suitable subalgebra of functions on this space to a C-star algebra such that the commutator of the deformed algebra is to first order given by the inverse of that symplectic form.
Most of this is being discussed in detail in the series A first Idea of Quantum Field Theory. If you are really interested in how quantum field theory works, you might want to go through that series in detail.

#### kodama

Quantization of Lagrangian field theory means the following:
1. Variation of the Lagrangian yields the equations of motion plus the presymplectic current.
2. After gauge fixing, these define a (graded) symplectic structure on the space of solutions of the equations of motion. This is called the covariant phase space.
3. Non-perturbative quantization means to deform the product of a suitable subalgebra of functions on this space to a C-star algebra such that the commutator of the deformed algebra is to first order given by the inverse of that symplectic form.
Most of this is being discussed in detail in the series A first Idea of Quantum Field Theory. If you are really interested in how quantum field theory works, you might want to go through that series in detail.
Has there been any attempts to perform steps #1-3 on Askhetar variables and if so, what were the results?
In what ways does textbook loop quantization differ from steps #1-3, and can it be corrected so it does comply with steps #1- 3 that you outline?

#### Urs Schreiber

Gold Member
Has there been any attempts to perform steps #1-3 on Askhetar variables
The closest is maybe the recent
• Alberto Cattaneo, Michele Schiavina, "The reduced phase space of Palatini-Cartan-Holst theory" (arXiv:1707.05351)
which gives a careful construction of the phase space with gauge symmetry taken properly into account.

In what ways does textbook loop quantization differ from steps #1-3
As I said before, the mistake of LQG is in discarding the phase space of gravity and replacing it by a space of "generalized connections" which has little resemblance to the original problem.

#### kodama

The closest is maybe the recent
• Alberto Cattaneo, Michele Schiavina, "The reduced phase space of Palatini-Cartan-Holst theory" (arXiv:1707.05351)
which gives a careful construction of the phase space with gauge symmetry taken properly into account.

As I said before, the mistake of LQG is in discarding the phase space of gravity and replacing it by a space of "generalized connections" which has little resemblance to the original problem.
is there a reason for
The closest is maybe the recent
• Alberto Cattaneo, Michele Schiavina, "The reduced phase space of Palatini-Cartan-Holst theory" (arXiv:1707.05351)
which gives a careful construction of the phase space with gauge symmetry taken properly into account.

As I said before, the mistake of LQG is in discarding the phase space of gravity and replacing it by a space of "generalized connections" which has little resemblance to the original problem.
thanks for the paper, do you think it deserves further research, perhaps loop theorists should drop LQG and work on this, and can it be extended to 11D SUGRA?

so LQG and loop quantization is mathematically correct implementation of Dirac quantization up until loop theorists "discarding the phase space of gravity and replacing it by a space of "generalized connections" which has little resemblance to the original problem"

why not continue developing and researching the "the phase space of gravity" rather than "generalized connections"

#### Urs Schreiber

Gold Member
perhaps loop theorists should drop LQG
Yes. It's historically unprecedented (or almost) that a whole field is based on an elementary error.

and work on this
Where "this" is the correct phase space of gravity. Yes, of course.

and can it be extended to 11D SUGRA?
Of course it can. Perturbative supergravity is an active area of research which has and is producing remarkable insights, for instance regarding the high loop finiteness of $d=4, \mathcal{N}= 8$ supergravity (see here) and the doubling KLT relations to Yang-Mills theory (see here). These insights contributed to the detection of the Higgs particle, see Mathew Strassler's story From String Theory to the Large Hadron Collider.

so LQG [...] is mathematically correct implementation of Dirac quantization
No, it's not.

up until loop theorists "discarding the phase space of gravity and replacing it by a space of "generalized connections" which has little resemblance to the original problem"
Right, so you might want to say: "it was on the right track until the first step".

why not continue developing and researching the "the phase space of gravity" rather than "generalized connections"
Yup. I hope you read out these rethorical question to yourself: There is no reason you should keep coming back to LQG. If you are interested in quantum gravity, there is plenty good stuff to learn about.

As I said before, the mistake of LQG is in discarding the phase space of gravity and replacing it by a space of "generalized connections" which has little resemblance to the original problem.
Urs, just out of curiosity, is this observation also true for the covariant spin-foam approaches to LQG or is it specific to the canonical approach?

Thanks!

#### Urs Schreiber

Gold Member
Urs, just out of curiosity, is this observation also true for the covariant spin-foam approaches to LQG or is it specific to the canonical approach?
The technical error that I have been mentioning (here) is in the "canonical" approach.

But the spin foam approach is even more disconnected from established quantum theory. It is just the speculation that some state sum models that one can write down might define quantum gravity, without these state sum models arising from any prescription of quantization. It's a guess of the kind: Let's throw everything we know about physics away and just come up with an entirely new prescription.

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#### kodama

The technical error that I have been mentioning (here) is in the "canonical" approach.
One can pinpoint the technical error in LQG explicitly:

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" role="presentation">x

and y" role="presentation">y of a linear isomorphism TxX&#x2192;TyY" role="presentation">TxX→TyY
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/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.....In the LQG literature these assignments are then called "generalized connections". It is the space of these "generalized connections" which is then being quantized.
.
This technical error in the LQG literature can be fixed by quantizing steps #1-3 you described earlier at the point where you say "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" rather than proceeding to the next step that you've identified as the error "generalized connections".

Has no researcher since Ashketar introduced his new variables suggested this?
Sounds like a good paper you could write and would set LQG on the correct direction.

#### Urs Schreiber

Gold Member
Has no researcher since Ashketar introduced his new variables suggested this?
The problem is that presently nobody has managed to quantize any interacting field theory in 4d non-perturbatively, and that there is no indication that switching to Ashtekar variables is of any help, when done correctly.

We have been at this point many time before in this discussion, you have asked this several times before, and I have replied to it several times before. I can do so yet once more, if it helps (does it ? :-):

Even the much simpler analogue problem of non-perturbative quantization in the case of Yang-Mills theory is wide open. A small fragment of this problem alone has been declared one of the "millenium problems" of our age.

Since nobody has a concrete idea for how to precisely solve these problems of non-perturbative quantization of Yang-Mills theory and of gravity (but of course there is no lack of hints and educated guesses) it might be that what it biting the community is that even the foundations of perturbative quantum field theory are traditionally left vague in the iterature, even though here everything may be made crystal clear. Maybe if we step back, and first sort out the conceptual foundations of what quantum field theory really is just a step away from full perturbation theory, for instance if we more properly understand how instanton corrections are to be included, maybe then we will also see clearer on how to do non-perturbative quantization of Yang-Mills theory, and then maybe of gravity in dimensions $\geq 4$. Work in this direction includes "homotopical AQFT".

#### kodama

Since the full non-perturbative quantization is not currently available,

Can you get a useful or working, effective predictive theory, by peforming a standard Dirac quantization on configuration space of gravity as a space of parallell transport/holonomy assignments to paths.

A useful effective theory, one that can be applied to cosmology, black holes, etc, even if standard Dirac quantization falls short of non-perturbative quantization but can be used to make calculations.

A useful effective theory in the same way semiclassical gravity is a useful effective theory, one that a true non-perturbative quantization of gravity should reproduce.

#### atyy

The technical error that I have been mentioning (here) is in the "canonical" approach.

But the spin foam approach is even more disconnected from established quantum theory. It is just the speculation that some state sum models that one can write down might define quantum gravity, without these state sum models arising from any prescription of quantization. It's a guess of the kind: Let's throw everything we know about physics away and just come up with an entirely new prescription.
There is some idea that the spin foam approach should be connected to the canonical approach: https://arxiv.org/abs/1303.4636.

I don't believe there is yet consensus over the status of the EPRL spin foam model, eg: https://arxiv.org/abs/1502.04640.

#### Urs Schreiber

Gold Member
There is some idea that the spin foam approach should be connected to the canonical approach: https://arxiv.org/abs/1303.4636.
It would be better for the viability of spin foams if they were not related to "canonical" LQG, because that is broken. But all these considerations seem to be very hand-wavy anyway.

"If 11 dimensional supergravity were nonpertubative quantized"

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