  We argue that two-dimensional dilaton gravity models can all be derived from
an analog of Jacobson's covariant version of the first law of thermodynamics.
We then specialize to the JT gravity model and couple it to massless fermions.
This model is exactly soluble in quantum field theory, and we present a new
derivation of that result. The field theory model violates two principles one
might want to impose on a quantum theory of gravity describing the near horizon
region of an extremal charged black hole in four dimensions: finiteness of the
entropy for finite causal diamonds, and the absence of global conservation
laws. It preserves an infinite number of conservation laws that one would have
expected to be violated, since the fermion state on each side of the $AdS_2$
wormhole is unavoidably thermal. We describe a cutoff version of the model,
with extra interactions, which cures these difficulties. Our UV completion of
the model depends on the AKK map of non-relativistic fermions in an inverted
oscillator potential to Weyl fermions in Minkowski space. We argue that gauging
the $Z_2$ symmetry of the oscillator model, using a density matrix with
temperature that depends on the oscillator coordinates, and inserting chaotic
interactions at (almost) infinite oscillator coordinate, we obtain a model with
properties expected of quantum gravity in the near horizon region of an
extremal charged black hole in four dimensions.
