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
Download PDF

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

 

[weirdoguy]

any chance you could write a paper on

What for?