Quantization of FRW universe via gauge-fixed action

This paper is devoted to investigation of the quantum Friedman-Robertson-Walker universe with matter satisfying the equation of state $p=wρ$, where $w$ is an almost arbitrary constant. The procedure starts with a reduced Lagrangian, which describes the system in a gauge fixed, so that the evolution parameter corresponds to the cosmological time. Then we construct the phase space, which is believed to correspond to the reduced phase space consisting of Dirac's observables. The physically relevant quantities are mapped into operators. We show that the operators have self-adjoint realizations and that there exist quantum states for which the evolution across singularity is well-defined.