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- 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

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

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