  The wave function of the universe is usually taken to be a functional of the
three-metric on a spacelike section, Sigma, which is measured. It is sometimes
better, however, to work in the conjugate representation, where the wave
function depends on a quantity related to the second fundamental form of Sigma.
This makes it possible to ensure that Sigma is part of a Lorentzian universe by
requiring that the argument of the wave function be purely imaginary. We
demonstrate the advantages of this formalism first in the well-known examples
of the nucleation of a de Sitter or a Nariai universe. We then use it to
calculate the pair creation rate for sub-maximal black holes in de Sitter
space, which had been thought to vanish semi-classically.
