Quantization of the Closed Mini-Superspace Models as Bound States

TL;DR

This paper applies the Wheeler-DeWitt equation to closed FRW models, showing that matter-dominated universes without a cosmological constant behave as bound states with quantized parameters, while universes with a persistent positive cosmological constant do not.

Contribution

It demonstrates that the Wheeler-DeWitt equation can be formulated as a bound state problem for certain cosmological models, providing a natural quantization condition and resolving boundary condition ambiguities.

Findings

Bound state solutions exist for matter-dominated universes without cosmological constant.

Quantization relates spatial curvature to the universe's energy density.

Presence of a positive cosmological constant leads to non-bound states with complex wave functions.

Abstract

Wheeler-DeWitt equation is applied to $k > 0$ Friedmann Robertson Walker metric with various types of matter. It is shown that if the Universe ends in the matter dominated era (e.g., radiation or pressureless gas) with zero cosmological constant, then the resulting Wheeler-DeWitt equation describes a bound state problem. As solutions of a non-degenerate bound state system, the eigen-wave functions are real (Hartle-Hawking) and the usual issue associated with the ambiguity in the boundary conditions for the wave functions is resolved. Furthermore, as a bound state problem, there exists a quantization condition that relates the curvature of the three space with the energy density of the Universe. Incorporating a cosmological constant in the early Universe (inflation) is given as a natural explanation for the large quantum number associated with our Universe, which resulted from the quantization condition. It is also shown that if there is a cosmological constant $\Lambda > 0$ in our Universe that persists for all time, then the resulting Wheeler-DeWitt equation describes a non-bound state system, regardless of the magnitude of the cosmological constant. As a consequence, the wave functions are in general complex (Vilenkin) and the initial conditions for wave functions are a free parameters not determined by the formalism.