Diffeomorphism symmetry is an intrinsic difficulty in gravitational theory, which appears in almost all of the questions in gravity. As is well known, the diffeomorphism symmetries in gravity should be interpreted as gauge symmetries, so only diffeomorphism invariant operators are physically interesting. However, because of the non-linear effect of gravitational theory, the results for diffeomorphism invariant operators are very limited.

In this work, we focus on the Jackiw-Teitelboim gravity in classical limit, and use Peierls bracket (which is a linear response like computation of observables’ bracket) to compute the algebra of a large class of diffeomorphism invariant observables. With this algebra, we can reproduce some recent results in Jackiw-Teitelboim gravity including: traversable wormhole, scrambling effect, and $SL(2)$ charges. We can also use it to clarify the question of when the creation of an excitation deep in the bulk increases or decreases the boundary energy, which is of crucial importance for the “typical state” version of the firewall paradox.

In the talk, I will first give a brief introduction of Peierls bracket, and then use the Peierls bracket to study the brackets between diffeomorphism invariant observables in Jackiw-Teitelboim gravity. I will then give two applications of this algebra: reproducing the scrambling effect, and studying the energy change after creating an excitation in the bulk.

Work based on https://arxiv.org/pdf/2108.04841.pdf