We compare the path integral for transition functions in unimodular gravity
and in general relativity. In unimodular gravity the cosmological constant is a
property of states that are specified at the boundaries whereas in general
relativity the cosmological constant is a parameter of the action. Unimodular
gravity with a nondynamical background spacetime volume element has a time
variable that is canonically conjugate to the cosmological constant. Wave
functions depend on time and satisfy a Schr\”odinger equation. On the contrary,
in the covariant version of unimodular gravity with a 3-form gauge field,
proposed by Henneaux and Teitelboim, wave functions are time independent and
satisfy a Wheeler-DeWitt equation, as in general relativity. The 3-form gauge
field integrated over spacelike hypersurfaces becomes a “cosmic time” only in
the semiclassical approximation. In unimodular gravity the smallness of the
observed cosmological constant has to be explained as a property of the initial
state.
