  The tunneling wave function of the universe is calculated exactly for a de
Sitter minisuperspace model with a massless conformally coupled scalar field,
both by solving the Wheeler-DeWitt equation and by evaluating the Lorentzian
path integral. The same wave function is found in both approaches. The
back-reaction of quantum field fluctuations on the scale factor amounts to a
constant renormalization of the vacuum energy density. This is in contrast to
the recent suggestion of Feldbrugge $et$ $al.$ that the back-reaction should
diverge when the scale factor gets small, $a \to 0$. Similar results are found
for a massive scalar field in the limit of a large mass. We also verified that
the tunneling wave function can be expressed as a transition amplitude from a
universe of vanishing size with the scalar field in the state of Euclidean
vacuum, as it was suggested in our earlier work.
