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It is remarkably difficult to reconcile unitary and Vilenkin's wave function.
For example, the natural conserved inner product found in quantum unimodular
gravity applies to the Hartle-Hawking wave function, but fails for its Vilenkin
counterpart. We diagnose this failure from different angles (Laplace transform
instead of Fourier transform, non-Hermiticity of the Hamiltonian, etc) to
conclude that ultimately it stems from allowing the connection to become
imaginary in a section of its contour. In turn this is the unavoidable
consequence of representing the Euclidean theory as an imaginary image within a
fundamentally Lorentzian theory. It is nonetheless possible to change the
underlying theory and replace the connection's foray into the imaginary axis by
an actual signature change (with the connection, action and Hamiltonian
remaining real). The structural obstacles to unitarity are then removed, but
special care must still be taken, because the Euclidean theory {\it a priori}
has boundaries, so that appropriate boundary conditions are required for
unitarity. Reflecting boundary conditions would reinstate a Hartle-Hawking-like
solution in the Lorentzian regime. To exclude an incoming wave in the
Lorentzian domain one must allow a semi-infinite tower of spheres in the
Euclidean region, wave packets travelling through successive spheres for half
an eternity in unimodular time. Such "Sisyphus" boundary condition no longer
even vaguely resembles Vilenkin's original proposal.
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