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Quantization of closed mini-superspace models as bound states

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Abstract

The Wheeler-DeWitt equation is applied to closedk>0 Friedmann-Robertson-Walker metric with various combination of cosmological constant and matter (e.g., radiation or pressureless gas). 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 nondegenerate bound state system, the eigen-wave functions are real (Hartle-Hawking). Furthermore, as a bound state problem, there exists a quantization condition that relates the curvature of the three space with the various energy densities of the universe. If we assume that our universe is closed, then the quantum number of our universe isN∼(Gk)−1∼10122. The largeness of this quantum number is naturally explained by an early inflationary phase which resulted in a flat universe we observe today. It is also shown that if there is a cosmological constant Λ>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).

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Author notes

  1. J. H. Kung

    Present address: Department of Astronomy and Physics Bates College, 04240, Lewiston, Maine, USA

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  1. Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, 02138, Cambridge, Massachussetts, USA

    J. H. Kung

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Kung, J.H. Quantization of closed mini-superspace models as bound states. Gen Relat Gravit 27, 35–53 (1995). https://doi.org/10.1007/BF02105672

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