Quantum coherent dynamics of quasiclassical spacetimes

Abstract

In a wide range of quantum gravity theories, quasiclassical geometries, which are solutions to the Einstein field equations approximately, are described by "coherent states." Here we propose a Hamiltonian formalism for gravitational dynamics with respect to this coherent state basis, which generates time evolution of the spacetime with respect to a clock at infinity. Since the coherent states are not orthogonal, an initial quasiclassical geometry is dynamically driven into a superposition of different amplitudes. Our framework provides a dynamical mechanism for tunneling between geometries that is ubiquitous in a number of approaches to quantum gravity, from loop quantum gravity to the Euclidean path integral. We apply our framework to the problem of black hole evaporation, providing a hint at how unitarity may be preserved with the inclusion of quantum corrections to the semiclassical evolution of the black hole.

Figures (2)

• Figure 1: Schematic diagram of the phase space evolution of an initially localised quasiclassical geometry, $| g_n \rangle$, into an admixture of $| \bar{g}_n \rangle ,~ \varepsilon | \bar{g}_{n\pm 1} \rangle$ (and as time progresses, higher-order states). Each ring with mean energy $\bar{E}_n$ represents a quasiclassical geometry, up to diffeomorphisms.
• Figure 2: (a) Time-evolution of the fidelity $F_n(t) = | \langle \bar{g}_n | \psi_G(t) \rangle |^2$ for initial state $| \bar{g}_{15} \rangle$ and different values of $n$ for the projected state. Here, $N = 15$. (b) $F_n(t)$ plotted as a function of $(n,t)$, where individual black curves correspond to a single value of $n$. We assumed the linear spectrum $\bar{E}_n = \bar{E}_0 + \alpha n$, truncated the Hilbert space at $N = 40$, and chose the initial state to be $| \bar{g}_{20} \rangle$. (c) Toy model of black hole evaporation, with the spectrum $\bar{E}_n = \bar{E}_0 + \alpha n^{1/3}$ and $N = 32$. The white curve is the semiclassical prediction $\bar{E}_n(t) = ( \bar{E}^3_0 - \eta t)^{1/3}$ with $\eta$ a proportionality constant. The semiclassical curve matches the most probable trajectory of the full unitary dynamics (i.e. the ridge followed by the blue-white surface). In all plots $\bar{E}_0 /\alpha = 1/1000$ and $\varepsilon = 1/10$.