Lorentzian Vacuum Transitions: Open or Closed Universes?
  We consider the generalisation of quantum tunneling transitions in the WKB
approximation to the time-independent functional Schr\"odinger and
Wheeler-DeWitt equations. Following a Lorentzian approach, we compute the
transition rates among different scalar field vacua and compare with those
performed by Coleman and collaborators using the Euclidean approach. For
gravity, we develop a general formalism for computing transition rates in
Wheeler's superspace. This is then applied to computing decays in flat space
and then to transitions in the presence of gravity. In the latter case we point
out the complexities arising from having non-positive definite kinetic terms
illustrating them in the simplified context of mini-superspace. This
corresponds to a generalisation of the well-known `tunneling from nothing'
scenarios. While we can obtain the leading term for the transitions obtained by
Euclidean methods we also point out some differences and ambiguities. We show
that there is no obstruction to keeping the spherically ($SO(4)$) symmetric
closed slicing for the new vacuum after a de Sitter to de Sitter transition. We
argue that this is the natural Lorentzian realisation of the Coleman-De Luccia
instanton and that a closed universe is also obtained if the mini-superspace
assumption is relaxed. This is contrary to the open universe predicted by
Coleman-De Luccia which relies on an analytic continuation performed after
bubble nucleation. Our findings may have important cosmological implications
related to the origin of inflation and to the string landscape. In particular,
they question the widespread belief that evidence for a closed universe would
rule out the string landscape.
