I Witten’s paper: "A Note On The Canonical Formalism for Gravity" LQG

[kodama]

TL;DR Summary

Hamiltoninan constraint and Wheeler de Witt equation

Submitted on 16 Dec 2022 (v1), last revised 26 Jun 2023 (this version, v2)]
A Note On The Canonical Formalism for Gravity
Edward Witten
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We describe a simple gauge-fixing that leads to a construction of a quantum Hilbert space for quantum gravity in an asymptotically Anti de Sitter spacetime, valid to all orders of perturbation theory. The construction is motivated by a relationship of the phase space of gravity in asymptotically Anti de Sitter spacetime to a cotangent bundle. We describe what is known about this relationship and some extensions that might plausibly be true. A key fact is that, under certain conditions, the Einstein Hamiltonian constraint equation can be viewed as a way to gauge fix the group of conformal rescalings of the metric of a Cauchy hypersurface. An analog of the procedure that we follow for Anti de Sitter gravity leads to standard results for a Klein-Gordon particle.

Comments: 55 pp, minor corrections and clarifications in this version
Subjects: High Energy Physics - Theory (hep-th)
Cite as: arXiv:2212.08270 [hep-th]
(or arXiv:2212.08270v2 [hep-th] for this version)

Contents
1 Introduction
In this article, we will re-examine the canonical formalism for quantum gravity [1], focusing
on the case of an asymptotically Anti de Sitter (AAdS) spacetime X. One advantage
of the AAdS case is that, because of holographic duality, it is possible to explain in a
straightforward way what problem the canonical formalism is supposed to solve, thereby
circumventing questions like what observables to consider and what is a good notion of
“time.” In holographic duality, there is a straightforward notion of boundary time, and
there is no difficulty in defining local boundary observablesHamiltoninan constraint and Wheeler de Witt equation and Canonical Formalism for Gravity is also loop quantum gravity

any overlays of Witten and loop quantum gravity

if you think loop quantum gravity is wrong, does Witten's paper offer better ideas on how to canonically quantize gravity?

what would happen if you loop quantize n asymptotically Anti de Sitter spacetime then answer questions via holographic duality

 

[Demystifier]

Loop quantum gravity is a particular realization of canonical quantum gravity, not necessarily equivalent to other realizations. A simple analogy is canonical quantization of nonrelativistic particle in one dimension via
$$\left( \frac{p^2}{2m} + V(x) \right)\psi=i\hbar\partial_t\psi$$
The standard realization of $p$ is
$$p=-i\hbar\partial_x$$
but there is also a different realization, analogous to the loop quantum gravity realization, which is not equivalent to the standard realization: https://arxiv.org/abs/hep-th/0409182v1 (Sec. 3)

So if Witten accepts canonical quantum gravity as an appropriate effective theory, it doesn't mean that he accepts loop quantum gravity.

 

[mitchell porter]

I looked at this paper last year, and was eventually led to discover the "Sparling 3-form", which has connections to the Ashtekar variables and to the issue of quasi-local energy conservation in general relativity, and which was discussed in connection with Anti de Sitter space in 2017. But I haven't had time to put it all together coherently.

 

[kodama]

I looked at this paper last year, and was eventually led to discover the "Sparling 3-form", which has connections to the Ashtekar variables and to the issue of quasi-local energy conservation in general relativity, and which was discussed in connection with Anti de Sitter space in 2017. But I haven't had time to put it all together coherently.

tell us more

could Ashtekar variablesgive rise to an asymptotically Anti de Sitter (AAdS) spacetime
that, because of holographic duality, it is possible to explain in a
straightforward way what problem the canonical formalism is supposed to solve,

 

[mitchell porter]

My speculation is that twistors and the Sparling form are a clue to the "real meaning" of the Ashtekar variables, that loop quantization is a detour (at least in the form that has defined loop quantum gravity; maybe Sathiapalan's "loop variables" work), and that by combining Witten's 2022 paper with the 2017 paper by Mahdi Godazgar, one might be able to figure out the "proper method" of quantizing Ashtekar variables. Way back in the early days, there were a handful of papers considering other methods, including an "asymptotic quantization" due to Ashtekar himself. I'd look at "New variables for gravity" for guidance on how to do the classical theory in Anti de Sitter space, early papers on BRST quantization of Ashtekar variables, and maybe also the side remark here on BRST cohomology.

 

[kodama]

My speculation is that twistors and the Sparling form are a clue to the "real meaning" of the Ashtekar variables, that loop quantization is a detour (at least in the form that has defined loop quantum gravity; maybe Sathiapalan's "loop variables" work), and that by combining Witten's 2022 paper with the 2017 paper by Mahdi Godazgar, one might be able to figure out the "proper method" of quantizing Ashtekar variables. Way back in the early days, there were a handful of papers considering other methods, including an "asymptotic quantization" due to Ashtekar himself. I'd look at "New variables for gravity" for guidance on how to do the classical theory in Anti de Sitter space, early papers on BRST quantization of Ashtekar variables, and maybe also the side remark here on BRST cohomology.

by twistors do you think of Woit Euclidean Twistor Unification?

do Ashtekar variables, in asymptotically Anti de Sitter also included holographic duality?

 

[mitchell porter]

I see AdS/CFT as the context for Witten's paper. The full duality in AdS/CFT is between a conformal field theory on the boundary, and a string theory in the bulk. One often sees simplifications of this duality, e.g. in which the bulk theory is described using classical gravity. I think the significance of Witten's proposal is that it should lead to a new kind of approximation of bulk physics: a quantum gravity theory, but only using field theory, rather than the full apparatus of string theory.

Meanwhile, Godazgar's paper is one of those papers in which explores a relationship between quantum mechanics on the boundary, and classical gravity in the bulk - but it uses something very similar to Ashtekar variables! So I definitely think that "Witten + Godazgar" is a way to explore holography for Ashtekar variables.

As for the relationship to twistors, according to Mason & Skinner 2008, Ashtekar's variables have a 4-dimensional form (page 6), which is particularly appropriate for describing "anti self dual" spacetimes (page 7), whose twistor transform hosts Penrose's construction of the "nonlinear graviton" (page 15)... But actually I think the relationship between twistors and Ashtekar variables is broader and deeper than this.

Clarity on how to quantize Ashtekar variables would certainly help Woit, but his proposal has other unresolved issues too (e.g. how to obtain fermions), as we've discussed elsewhere.

 

[kodama]

so does "Witten + Godazgar" and Mason & Skinner 2008 give you any concrete ideas for how to quantize Ashtekar variables plausibly.

I recall that late PF Marcus wrote of other ways to quantize Ashtekar variables

 

[mitchell porter]

I note the recent paper

"Generalized Symmetry in Dynamical Gravity" (Clifford Cheung et al)

which is unusual in being an advanced, technical, mainstream work of quantum field theory, in which Ashtekar variables are used at one point. But note, they are not proposing a fundamental theory of quantum gravity, they are studying the properties of a low-energy approximation.

 

[kodama]

I note the recent paper

"Generalized Symmetry in Dynamical Gravity" (Clifford Cheung et al)

which is unusual in being an advanced, technical, mainstream work of quantum field theory, in which Ashtekar variables are used at one point. But note, they are not proposing a fundamental theory of quantum gravity, they are studying the properties of a low-energy approximation.

could their advanced, technical, mainstream work of quantum field theory be applied to Ashtekar variables in a new way for proposing a fundamental theory of quantum gravity

if mainstream lqg is wrong

is there a way to quantize Ashtekar variables to create a fundamental theory of quantum gravity you could endorse

 

[billtodd]

But I haven't had time to put it all together coherently.

Who said that Quantum Gravity theory should be coherent if QM or QFT themselves aren't coherent?

 

[mitchell porter]

is there a way to quantize Ashtekar variables to create a fundamental theory of quantum gravity you could endorse

The bridge between quantum field theory and a classical field state is something called a coherent state. Essentially, a coherent state is a kind of quantum field state that resembles a classical field state.

So in quantum gravity one normally says that the classical metric is a coherent state of gravitons. One still says this in string theory too.

With Ashtekar variables, one is describing gravity in terms of a connection rather than a metric, but the same principle applies. My guess is that Ashtekar gravity should arise as a coherent state of the twistor string. (This is also how I would try to realize Woit's scenario within string theory.)

 

[kodama]

The bridge between quantum field theory and a classical field state is something called a coherent state. Essentially, a coherent state is a kind of quantum field state that resembles a classical field state.

So in quantum gravity one normally says that the classical metric is a coherent state of gravitons. One still says this in string theory too.

With Ashtekar variables, one is describing gravity in terms of a connection rather than a metric, but the same principle applies. My guess is that Ashtekar gravity should arise as a coherent state of the twistor string. (This is also how I would try to realize Woit's scenario within string theory.)

any chance you could write a paper on Ashtekar variable that by combining Witten's 2022 paper with the 2017 paper by Mahdi Godazgar, one might be able to figure out the "proper method" of quantizing Ashtekar variables

would the semi classical limit to GR be easier since arise as a coherent state of the twistor string.

do you think that t the "proper method" of quantizing Ashtekar variables could create a fundamental theory of quantum gravity in 4d

 

[kodama]

The bridge between quantum field theory and a classical field state is something called a coherent state. Essentially, a coherent state is a kind of quantum field state that resembles a classical field state.

So in quantum gravity one normally says that the classical metric is a coherent state of gravitons. One still says this in string theory too.

With Ashtekar variables, one is describing gravity in terms of a connection rather than a metric, but the same principle applies. My guess is that Ashtekar gravity should arise as a coherent state of the twistor string. (This is also how I would try to realize Woit's scenario within string theory.)

what about Ashtekar gravity should arise as a coherent state of boundary theory is a d=3 N=6 Chern-Simons theory known as ABJM via the AdS4/CFT3

 

[mitchell porter]

what about Ashtekar gravity should arise as a coherent state of boundary theory is a d=3 N=6 Chern-Simons theory known as ABJM via the AdS4/CFT3

Well, the full holographic dual of ABJM is a particular AdS compactification of M-theory down to four dimensions. So the AdS4 contains branes and supergravity fields, as well as seven extra compact dimensions, all of which can in some sense be built out of combinations of the ABJM boundary fields (since that's the meaning of holographic duality).

If we work in a geometry where the extra dimensions have a small enough radius that they can be neglected, and at low energies where the branes don't appear (since they are heavy objects), then you're in a regime where it's just a supergravity field theory in AdS4. That means you have gravity coupled to some assortment of gauge, scalar, and fermion fields resulting from the compactification. There is a standard holographic "dictionary" describing how each of these fields corresponds to combinations of ABJM field operators on the boundary. Basically, some quantum state (not necessarily a coherent state, it's the bulk state which must be coherent if it is a classical geometry) of the ABJM theory corresponds holographically to an AdS4 geometry, and then ABJM field operators describe the asymptotic behavior of the supergravity fields within that AdS4 geometry.

For a 4d gravitational field, in this discussion, we have two ways of describing it, via metric (Einstein) or via connection (Ashtekar). As far as I know, all work in AdS/CFT has described the gravitational field using the metric variables. Incidentally, the boundary dual of gravitation is the same in all examples of AdS/CFT, not just ABJM; gravity comes from the "stress-energy tensor" of the CFT.

Work in AdS/CFT involves redescribing physics in AdS in terms of a CFT, or vice versa. As I mentioned, the full duality should involve string or M theory on the AdS side, but often one just looks at fields in AdS (which, I emphasize, are different from the fields in the CFT; the AdS fields correspond to combinations of the CFT fields). This could involve a classical supergravity theory in AdS, or a quantum supergravity in AdS (i.e. a quantum field theory of gravity).

On the one hand, if we have a particular classical supergravity theory in AdS4, we probably know how to rewrite the equations in terms of Ashtekar's connection rather than Einstein's metric. It's the quantum theory for Ashtekar variables which is an open question.

On the other hand, we know how to write a quantum theory for supergravity expressed in terms of metric variables; that's just the standard perturbative quantum gravity, in which you have gravitons. In Witten's 2022 paper that started this thread, he appears to be working with the metric, but only the "conformal class" of the metric matters.

OK, back to the question of how to quantize gravity when we use Ashtekar variables. Loop quantum gravity is one such proposal, and I can see that there are LQG papers out there, in which they talk about quantizing "conformal classes" of Ashtekar connections, e.g. I see one that uses spin foams. So presumably you could look at this AdS4 fields-only scenario, do the time-slicing and gauge-fixing that Witten used, and then follow an LQG procedure on top of that, and see if you can relate that to the expected ABJM dual. Maybe someone could work on it for Loops 2025.

Then there's the BRST quantization of Ashtekar variables from a Russian paper that I mentioned in other threads, which I assume leads to the standard perturbative quantum gravity derived from the metric. Just on the grounds that this standard quantum gravity embeds into string theory, I would expect the BRST quantization (if it does work) to be compatible with ABJM duality, but the LQG quantization possibly not (though spin foams I feel do have a chance, and of course it would be epically interesting if it did work out).

All this would be hard work, and as a research program it might always run into some unanticipated barrier, but I think it's a well-defined set of questions. However, there is one question that's already clear, and to which I don't know the answer. In reality, the dual of ABJM has four non-compact dimensions and seven compact dimensions. We have just assumed, for the sake of discussion, that the seven compact dimensions are small enough to be neglected. The question is: if you are working in Ashtekar variables, and you transition to a regime in which the seven compact dimensions are large enough to matter, you now have to concern yourself with 11-dimensional gravity not 4-dimensional gravity. But the Ashtekar connection, and its neat properties, are specific to 4 dimensions. What happens as you transition back to 11 dimensions? Do you use some kind of product of 4-dimensional connection with a 7-dimensional metric? There's probably relevant ideas somewhere in the voluminous literature of LQG, which must have considered extra dimensions alongside other attempted generalizations of the original idea.

So that develops in a little more detail this concept of "quantizing Ashtekar variables in the context of 4-dimensional AdS/CFT". Whether ABJM is the best place to start, I don't know; there are other examples of AdS4/CFT3. And one might hope to ultimately switch to flat-space ("celestial") holography or dS/CFT, which are closer to the real world, and with a set of fields including the standard model, and so on.

Now, let me mention something which might be more important than all of the above. I had another look at the paper mentioned in #9, and what do you know, on page 26 they say they have reinvented a modified version of the area operator from loop quantum gravity! I emphasize that this was a mainstream theory paper in which they weren't doing any exotic form of quantization. But they are studying gravity using a form of Ashtekar's connection (they call their theory "tetradic Palatini gravity"), and applying some concepts that are new, but still mainstream QFT ("generalized symmetry"). And a construct similar to something from LQG, falls out naturally from that recipe! This is surely an excellent clue on the "proper way" to quantize Ashtekar's connection, though I haven't yet tried to understand it in context.

 

[kodama]

Well, the full holographic dual of ABJM is a particular AdS compactification of M-theory down to four dimensions. So the AdS4 contains branes and supergravity fields, as well as seven extra compact dimensions, all of which can in some sense be built out of combinations of the ABJM boundary fields (since that's the meaning of holographic duality).


All this would be hard work, and as a research program it might always run into some unanticipated barrier, but I think it's a well-defined set of questions. However, there is one question that's already clear, and to which I don't know the answer. In reality, the dual of ABJM has four non-compact dimensions and seven compact dimensions. We have just assumed, for the sake of discussion, that the seven compact dimensions are small enough to be neglected. The question is: if you are working in Ashtekar variables, and you transition to a regime in which the seven compact dimensions are large enough to matter, you now have to concern yourself with 11-dimensional gravity not 4-dimensional gravity. But the Ashtekar connection, and its neat properties, are specific to 4 dimensions. What happens as you transition back to 11 dimensions? Do you use some kind of product of 4-dimensional connection with a 7-dimensional metric? There's probably relevant ideas somewhere in the voluminous literature of LQG, which must have considered extra dimensions alongside other attempted generalizations of the original idea.

is there a cft dual to Ashtekar variables in 4-dimensional gravity?

perhaps rewrite qcd as a cft then double copy it ?

 

[kodama]

mitchell porter

have you seen

arXiv:2404.18220 (hep-th)
[Submitted on 28 Apr 2024]
Asymptotically safe -- canonical quantum gravity junction
Thomas Thiemann

"The canonical (CQG) and asymptotically safe (ASQG) approach to quantum gravity share to be both non-perturbative programmes...The purpose of the present work is to either overcome actual differences or to explain why apparent differences are actually absent. Thereby we intend to enhance the contact and communication between the two communities."

arXiv:2503.22474 (hep-th)
[Submitted on 28 Mar 2025]
Asymptotically safe canonical quantum gravity: Gaussian dust matter
Renata Ferrero, Thomas Thiemann

"In a recent series of publications we have started to investigate possible points of contact between the canonical (CQG) and the asymptotically safe (ASQG) approach to quantum gravity"

what are your thoughts about Asymptotically safe -- canonical quantum gravity junction