Selection of physically meaningful solutions of the Wheeler-DeWitt equation
for the wavefunction in quantum cosmology, can be attained by a reduction of
the theory to the sector of true physical degrees of freedom and their
canonical quantization. The resulting physical wavefunction unitarily evolving
in the time variable introduced within this reduction can then be raised to the
level of the cosmological wavefunction in superspace of 3-metrics. We apply
this technique in several simple minisuperspace models and discuss both at
classical and quantum level physical reduction in {\em extrinsic} time -- the
time variable determined in terms of extrinsic curvature. Only this extrinsic
time gauge can be consistently used in vicinity of turning points and bounces
where the scale factor reaches extremum. Since the 3-metric scale factor is
canonically dual to extrinsic time variable, the transition from the physical
wavefunction to the wavefunction in superspace represents a kind of the
generalized Fourier transform. This transformation selects square integrable
solutions of the Wheeler-DeWitt equation, which guarantee Hermiticity of
canonical operators of the Dirac quantization scheme. Semiclassically this
means that wavefunctions are represented by oscillating waves in classically
allowed domains of superspace and exponentially fall off in classically
forbidden (underbarrier) regions. This is explicitly demonstrated in flat FRW
model with a scalar field having a constant negative potential and for the case
of phantom scalar field with a positive potential. The FRW model of a scalar
field with a vanishing potential does not lead to selection rules for solutions
of the Wheeler-DeWitt equation, but this does not violate Hermiticity
properties, because all these solutions are anyway of plane wave type and
describe cosmological dynamics without turning points and bounces.Comment: final version, to appear in Physical Review
