  We consider a massive quantum test Klein-Gordon field probing an isotropic
quantum cosmological space-time in the background. The result obtained is
surprising. It turns out, that despite the isotropy of the quantum
gravitational field, the semi-classical metric experienced by a mode of the K-G
field is non-isotropic. The anisotropy depends on the direction of the momentum
of the mode. Specifically, what we do is to derive a semi-classical space-time
which emerges to a mode of the field. The method amounts to a comparison
between QFT on a quantum background and QFT on a classical curved space-time,
giving rise to an emergent metric tensor. The components of the semi-classical
metric tensor are calculated from the equation of propagation of the quantum
K-G field in the test field approximation. The anisotropies are of a quantum
nature: they are proportional to Planck constant and "dress" the isotropic
classical space-time obtained in the classical limit.
