Whittaker Quantum

We show that closed, radiation-filled Friedmann-Robertson-Walker quantum universes of arbitrary factor ordering obey the Whittaker equation. We also present the formal Witten factorization as well as the double Darboux strictly isospectral scheme for the Whittaker equation. If in the " time-time " component of Einstein's equations for Friedmann-Robertson-Walker (FRW) universes ˙ a 2 + k = 8πG 3 ρa 2 one substitutes a canonical momentum π a conjugate to the scale factor and quantizes the system according to the common procedure π a → −i(d/da), one will obtain the corresponding Wheeler-DeWitt (WDW) equation − a −p d da a p d da + 3π 2G 2 1 k 3 ka 2 − 8πG 3 ρa 4 u(a) = 0. (1) The parameter p enters as a consequence of the ambiguity in the ordering of a and d/da. Amongst the most used orderings, the p = 1 is the so-called Laplacian one, which has been used for the two-dimensional mini-superspace describing a massive scalar field in a closed universe, i.e., a model obtained through the substitution 8πG 3 ρ → m 2 φ 2 , k = 1 in Eq. (1) and adding 1 a 2 ∂ 2 ∂φ 2 to the a derivatives turned into partial ones too, while the dependent variable becomes u(a, φ) [1]. On the other hand, the p = 2 ordering is convenient for the semiclassical approximation [2].