A note on the canonical formalism for gravity

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.

[1]  G. Galloway,et al.  Maximal hypersurfaces in asymptotically Anti-de Sitter spacetime , 2022, 2208.09893.

[2]  V. Chandrasekaran,et al.  An algebra of observables for de Sitter space , 2022, Journal of High Energy Physics.

[3]  Aron C. Wall,et al.  Cauchy slice holography: a new AdS/CFT dictionary , 2022, Journal of High Energy Physics.

[4]  Aitor Lewkowycz,et al.  Gravitational path integral from the T2 deformation , 2020, Journal of High Energy Physics.

[5]  C. Vafa,et al.  Cobordism Classes and the Swampland , 2019, 1909.10355.

[6]  E. Witten.  Light rays, singularities, and all that , 2019, Reviews of Modern Physics.

[7]  H. Ooguri,et al.  Symmetries in Quantum Field Theory and Quantum Gravity , 2018, Communications in Mathematical Physics.

[8]  Vasudev Shyam.  Finite cutoff AdS5 holography and the generalized gradient flow , 2018, Journal of High Energy Physics.

[9]  Thomas Hartman,et al.  Holography at finite cutoff with a T2 deformation , 2018, Journal of High Energy Physics.

[10]  T. Jacobson,et al.  Holographic complexity and volume , 2018, Journal of High Energy Physics.

[11]  E. Witten.  A Note on Boundary Conditions in Euclidean Gravity , 2018, Roman Jackiw.

[12]  M. Taylor.  TT deformations in general dimensions , 2018, 1805.10287.

[13]  R. Matzner,et al.  From path integrals to the Wheeler-DeWitt equation: Time evolution in spacetimes with a spatial boundary , 2017, 1708.07001.

[14]  F. Nogueira,et al.  The gravity dual of a density matrix , 2012, 1204.1330.

[15]  K. Krasnov,et al.  The Universal Phase Space of AdS3 Gravity , 2011, 1111.6507.

[16]  Jean-Marc Schlenker,et al.  Maximal surfaces and the universal Teichmüller space , 2009, 0911.4124.

[17]  A. Sakovich.  Constant mean curvature solutions of the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds , 2009, 0910.4178.

[18]  D. Witt.  Topological Obstructions To Maximal Slices , 2009, 0908.3205.

[19]  G. Mess.  Lorentz spacetimes of constant curvature , 2007, 0706.1570.

[20]  Michael T. Anderson.  Extension of symmetries on Einstein manifolds with boundary , 2007, 0704.3373.

[21]  Michael T. Anderson.  Boundary value problems for Einstein metrics, I , 2006, math/0612647.

[22]  D. Pollack,et al.  The constraint equations for the Einstein-scalar field system on compact manifolds , 2006, gr-qc/0610045.

[23]  Jean-Marc Schlenker,et al.  Minimal surfaces and particles in 3-manifolds , 2005, math/0511441.

[24]  R. Bartnik,et al.  The Constraint equations , 2004, gr-qc/0405092.

[25]  A. Zamolodchikov.  Expectation value of composite field $T{\bar T}$ in two-dimensional quantum field theory , 2004, hep-th/0401146.

[26]  D. Marolf.  Group Averaging and Refined Algebraic Quantization: Where are we now? , 2000, gr-qc/0011112.

[27]  J. Maldacena,et al.  Supergravity description of field theories on curved manifolds and a no go theorem , 2000, hep-th/0007018.

[28]  D. Marolf,et al.  On the generality of refined algebraic quantization , 1998, gr-qc/9812024.

[29]  I. Avramidi,et al.  Lack of strong ellipticity in Euclidean quantum gravity , 1997, hep-th/9708163.

[30]  F. Embacher.  Hand-waving Refined Algebraic Quantization , 1997, gr-qc/9708016.

[31]  Marolf.  Path integrals and instantons in quantum gravity: Minisuperspace models. , 1996, Physical review. D, Particles and fields.

[32]  R. Sorkin,et al.  Complex actions in two-dimensional topology change , 1995, gr-qc/9511023.

[33]  James Isenberg,et al.  Constant mean curvature solutions of the Einstein constraint equations on closed manifolds , 1995 .

[34]  A. Ashtekar,et al.  Quantization of diffeomorphism invariant theories of connections with local degrees of freedom , 1995, gr-qc/9504018.

[35]  A. Barvinsky.  Unitarity approach to quantum cosmology , 1993 .

[36]  Helmut Friedrich,et al.  On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations , 1992 .

[37]  A. Higuchi.  Quantum linearization instabilities of de Sitter spacetime. II , 1991 .

[38]  J. Hartle,et al.  Wave functions constructed from an invariant sum over histories satisfy constraints. , 1991, Physical review. D, Particles and fields.

[39]  S. Stolz.  Simply connected manifolds of positive scalar curvature , 1990 .

[40]  V. Moncrief.  Reduction of the Einstein equations in 2+1 dimensions to a Hamiltonian system over Teichmüller space , 1989 .

[41]  Edward Witten,et al.  (2+1)-Dimensional Gravity as an Exactly Soluble System , 1988 .

[42]  A. Achúcarro,et al.  A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories , 1986 .

[43]  Witt.  Vacuum spacetimes that admit no maximal slice. , 1986, Physical review letters.

[44]  Marc Henneaux,et al.  Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity , 1986 .

[45]  L. Romans.  Massive N=2a Supergravity in Ten-Dimensions , 1986 .

[46]  P. González-Díaz.  On the wave function of the universe , 1985 .

[47]  M. Gromov,et al.  Spin and Scalar Curvature in the Presence of a Fundamental Group. I , 1980 .

[48]  J. York,et al.  Existence and uniqueness of solutions of the Hamiltonian constraint of general relativity on compact manifolds , 1973 .

[49]  J. W. York.  ROLE OF CONFORMAL THREE-GEOMETRY IN THE DYNAMICS OF GRAVITATION. , 1972 .

[50]  B. Dewitt.  Quantum Theory of Gravity. I. The Canonical Theory , 1967 .

[51]  R. Sachs.  Gravitational waves in general relativity. VI. The outgoing radiation condition , 1961, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[52]  A. Raychaudhuri,et al.  Relativistic Cosmology. I , 1955 .

[53]  J. Cardy.  The T T deformation of quantum field theory as random geometry , 2018 .

[54]  V. S. Vladimirov.  The Cauchy Problem , 1986 .

[55]  D. Brill.  On spacetimes without maximal surfaces. , 1983 .

[56]  Y. Choquet-bruhat.  The Problem of Constraints in General Relativity: Solution of the Lichnerowicz Equation , 1976 .

[57]  M. Atiyah,et al.  The Index of Elliptic Operators: IV , 1971 .

[58]  K. Kuchař.  Ground state functional of the linearized gravitational field , 1970 .

[59]  O. Nachtmann.  Dynamische Stabilität im de-Sitter-Raum. , 1968 .