Description of Friedmann observables in quantum universe

The solution of the problem of describing the Friedmann observables (the Hubble law, the redshift, etc.) in quantum cosmology is proposed on the basis of the method of gaugeless Hamiltonian reduction in which the gravitational part of the energy constraint is considered as a new momentum. We show that the conjugate variable corresponding to the new momentum plays the role of the invariant time parameter of evolution of dynamical variables in the sector of the Dirac observables of the general Hamiltonian approach. Relations between these Dirac observables and the Friedmann observables of the expanding Universe are established for the standard Friedmann cosmological model with dust and radiation, The presented reduction removes an infinite factor from the functional integral, provides the normalizability of the wave function of the Universe and distinguishes the conformal frame of reference where the Hubble law is caused by the alteration of the conformal dust mass. (C) 1997 Elsevier Science B.V.