  In this paper we use the covariant Peierls bracket to compute the algebra of
a sizable number of diffeomorphism-invariant observables in classical
Jackiw-Teitelboim gravity coupled to fairly arbitrary matter. We then show that
many recent results, including the construction of traversable wormholes, the
existence of a family of $SL(2,\mathbb{R})$ algebras acting on the matter
fields, and the calculation of the scrambling time, can be recast as simple
consequences of this algebra. We 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"
versions of the firewall paradox. Unlike the "Schwarzian" or "boundary
particle" formalism, our techniques involve no unphysical degrees of freedom
and naturally generalize to higher dimensions. We do a few higher-dimensional
calculations to illustrate this, which indicate that the results we obtain in
JT gravity are fairly robust.
