Classically, unimodular gravity is known to be equivalent to General
Relativity (GR), except for the fact that the effective cosmological constant
Λ has the status of an integration constant. Here, we explore various
formulations of unimodular gravity beyond the classical limit. We first
consider the non-generally covariant action formulation in which the
determinant of the metric is held fixed to unity. We argue that the
corresponding quantum theory is also equivalent to General Relativity for
localized perturbative processes which take place in generic backgrounds of
infinite volume (such as asymptotically flat spacetimes). Next, using the same
action, we calculate semiclassical non-perturbative quantities, which we expect
will be dominated by Euclidean instanton solutions. We derive the entropy/area
ratio for cosmological and black hole horizons, finding agreement with GR for
solutions in backgrounds of infinite volume, but disagreement for backgrounds
with finite volume. In deriving the above results, the path integral is taken
over histories with fixed 4-volume. We point out that the results are different
if we allow the 4-volume of the different histories to vary over a continuum
range. In this "generalized" version of unimodular gravity, one recovers the
full set of Einstein's equations in the classical limit, including the trace,
so Λ is no longer an integration constant. Finally, we consider the
generally covariant theory due to Henneaux and Teitelboim, which is classically
equivalent to unimodular gravity. In this case, the standard semiclassical GR
results are recovered provided that the boundary term in the Euclidean action
is chosen appropriately.Comment: Substantially revised version, with new results and added discussio