UC Santa Barbara

In this thesis, we first study the action of Hubeny-Rangamani-Takayanagi (HRT) area operators on the covariant phase space of classical solutions in Einstein-Hilbert gravity. We find that this action is a boundary-condition-preserving kink transformation, which introduces a relative boost between the entanglement wedges on either side of the HRT-surface but preserves the asymptotically Anti-de Sitter (AdS) boundary conditions. We then perform a similar analysis for the ''geometric entropy", i.e. the bulk dual to boundary entanglement entropy, in topologically massive gravity (TMG). Here, the geometric entropy is given by the HRT-area plus an anomalous contribution. We find that the action of this geometric entropy on the covariant phase space of classical solutions agrees precisely with the action of HRT-area operators in Einstein-Hilbert gravity.

In Einstein-Hilbert gravity, we show that HRT-areas do not generally commute. This poses an obstruction to constructing random tensor networks (RTNs), which are most analogous to fixed-area states of the bulk quantum gravity theory, with mutually commuting HRT-areas. We probe the severity of such obstructions in pure AdS$_3$ Einstein-Hilbert gravity by constructing networks whose links are codimension-2 extremal-surfaces and by explicitly computing semiclassical commutators of the associated link-areas. We find a simple 4-link network for which all link-areas commute, but the algebra generated by the link-areas of more general networks tends to be non-Abelian.

In the final chapter, we switch gears and explore perturbative quantum gravity around de Sitter space, where gauge-invariant observables cannot be localized and, instead, local physics can arise only through certain relational constructions. In particular, we describe a class of gauge-invariant observables which, under appropriate conditions, provide good approximations to certain algebras of local fields. Our results suggest that, near any minimal $S^d$ in $dS_{d+1}$, this approximation can be accurate only over regions in which the corresponding global time coordinate $t$ spans an interval of order $\Delta t \lesssim \ln G^{-1}$. In contrast, however, we find that the approximation can be accurate over arbitrarily large regions of global dS$_{d+1}$ so long as those regions are located far to the future or past of such a minimal $S^d$. This in particular includes arbitrarily large parts of any static patch.