The Wheeler-DeWitt equation as an eigenvalue problem for the cosmological constant

Abstract

The Standard Cosmological Model with energy density ρ, pressure p and cosmological term λ is reconsidered. The interdependence among the cosmological equations in different equivalent forms is pointed out with λ possibly time dependent. For constant λ and for ρ, p expressed in terms of the scalar field φ of potential V(φ), the scheme is in fact equivalent to the coupling of Einstein and scalar field equations in the Robertson-Walker metric. A Lagrangian and the corresponding Hamiltonian H, that takes the zero value, are derived directly from the equations. The Wheeler-DeWitt (WDW) equation is obtained by canonical quantization of H that is performed in two non-equivalent ways. The WDW equations are transformed into Schrödinger-like eigenvalue problems with eigenvalue λ. The equations are separated for vanishing scalar potential. The φ-separated equation results in an eigenvalue problem in the separation constant λ ₁, that must be negative, and it is easily integrated. The R-separated equation, is again an eigenvalue problem with eigenvalue λ. It is solved, in the flat space-time case, by preliminary fixing λ ₁, in terms of the Bessel functions of the first kind and implies that λ can take all possible negative values. For fixed λ ₁, λ, the wave function of the Universe vanishes in correspondence with a big-bang situation and for large R.