A Note On The Canonical Formalism for Gravity

TL;DR

<3-5 sentences>This work addresses canonical quantization of gravity in asymptotically Anti de Sitter spacetimes by exploiting a simple BRST-based gauge-fixing that reformulates the Hamiltonian constraint as a Weyl-rescaling condition on a maximal-volume Cauchy slice. It identifies the classical gravitational phase space with the cotangent bundle $T^*({\sf Conf}/{\sf Diff})$, and shows that, in perturbation theory, the physical Hilbert space can be constructed from conformal data on a maximal slice with a ghost-determinant factor $\det\Xi$ entering the inner product. The paper draws a Klein-Gordon-type analogy for the inner product and extends the formalism to include scalar and $p$-form matter, providing generalized Lichnerowicz equations that determine Weyl factors and preserving the phase-space structure under suitable energy conditions. It further discusses AdS compactifications, highlighting topological obstructions (e.g., positive scalar curvature) that can obstruct the cotangent-bundle picture, and speculates on implications for holography, topology change, and nonperturbative aspects of quantum gravity.

Abstract

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.

Figures (4)

• Figure 1: (a) The boundary $X_\infty$ of Anti de Sitter space, drawn as a cylinder, with boundary insertions ${\mathcal{O}}$ and ${\mathcal{O}}'$ to the past and future of a Cauchy hypersurface $S_\infty\subset X_\infty$. In this setup, one can compute a matrix element $\langle \Psi|{\mathcal{O}}'(t',\vec{x}'){\mathcal{O}}(t,\vec{x})|\chi\rangle$ as a sum over physical states defined on the hypersurface $S_\infty$. In the canonical formalism for gravity, one aims to find a similar formula in terms of a sum over states on a bulk Cauchy hypersurface $S$ with boundary $S_\infty$. (b) The "cutting" procedure of (a) can be iterated, with successive cuts on successive hypersurfaces.
• Figure 2: The bulk domain of dependence $\Omega$ of a Cauchy hypersurface $S_\infty$ in the boundary of an AAdS spacetime $X$. In this picture, for simplicity, $X$ is two-dimensional so its boundary $X_\infty$ is one-dimensional and $S_\infty$ consists of two points. $\Omega$ is the domain of dependence of any bulk Cauchy hypersurface $S$ with boundary $S_\infty$, or equivalently the set of bulk points that are not timelike separated from $S_\infty$.
• Figure 3: (a) Here $X_\infty$ is a Lorentz signature boundary manifold with a future boundary $S_\infty$. In the boundary theory, initial conditions in the far past, and possible boundary insertions, determine a quantum state $\Psi_\infty$ on $S_\infty$. The bulk is an AAdS manifold $X$ with future boundary $S$. The metric $h$ of $S$ is fixed and the path integral on $X$ defines a function $\Psi(h)$ which one hopes has the same physical content as $\Psi_\infty$. One can argue formally that $\Psi(h)$ satisfies the Wheeler-DeWitt equation. (b) A similar picture to (a) in Euclidean signature. The main difference is that $X_\infty$ has operator insertions but no past boundary. (c) The picture of (a) is continued into the future and some final state is specified. In the boundary one gets nice formulas for the transition amplitude between specified initial and final states involving a sum over states on $S_\infty$, but in bulk, there is a problem if the states are supposed to be solutions $\Psi(h)$ of the Wheeler-DeWitt equation. If one picks a particular bulk Cauchy hypersurface $S$ on which to cut, the Wheeler-DeWitt equation is not satisfied, but integrating over all $S$ gives a massive overcounting.
• Figure 4: An AAdS spacetime $X$ with four asymptotic regions in which asymptotic states might be specified. Such an $X$ cannot have a metric everywhere of Lorentz signature; it may have Euclidean signature or possibly a complex metric. Overlapping "cuts" of such an $X$ can be made, as sketched here, on homotopically inequivalent surfaces such as $\gamma_1$ and $\gamma_2$. No one canonical formalism is well adapted to all of the possible cuts.