  In minisuperspace quantum cosmology, the Lorentzian path integral
formulations of the no-boundary and tunneling proposals have recently been
analyzed. But it has been pointed out that the wave function of linearized
perturbations around a homogeneous and isotropic background is of an inverse
Gaussian form and thus that their correlation functions are divergent. In this
paper, we revisit this issue and consider the problem of perturbations in
Lorentzian quantum cosmology by modifying the dispersion relation based on
trans-Planckian physics. We consider two modified dispersion relations, the
generalized Corley-Jacobson dispersion relation with higher momentum terms and
the Unruh dispersion relation with a trans-Planckian mode cut-off, as examples.
We show that the inverse Gaussian problem of perturbations in Lorentzian
quantum cosmology is hard to overcome with the trans-Planckian physics
modifying the dispersion relation at short distances.
