ON A QUANTUM MINIUNIVERSE FILLED WITH YANG-MILLS RADIATION

The Wheeler-DeWitt equation for a Robertson-Walker closed universe fil led with an SU(2) Yang-Mills spatially uniform field is separable; we are then led to impose on the wave function the usual rules for a quan tum bound system: square integrability and orthogonality. We investiga te the correspondence principle for large gravitational quantum number s. The classical probability depends on the gauge chosen for the time, and we examine lapse functions of the form N = N(alpha) (having in mi nd cosmic time and conformal time). The correspondence principle holds for these gauges if the differential representation of the momentum i s gauge-dependent. This gives a prescription for the ordering of opera tors in the Wheeler-DeWitt equation. We also discuss the measure in th e minisuperspace needed to ensure invariance of matrix elements of gau ge-invariant operators.