Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
1996-10-09
J.Math.Phys. 39 (1998) 4499-4512
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
Substantially improved version. Some new observations and nine figures are added. An explicit example is worked out
Scientific paper
10.1063/1.532522
The Wheeler-DeWitt equation for the minimally coupled FRW-massive-scalar-field minisuperspace is written as a two-component Schr\"odinger equation with an explicitly `time'-dependent Hamiltonian. This reduces the solution of the Wheeler-DeWitt equation to the eigenvalue problem for a non-relativistic one-dimensional harmonic oscillator and an infinite series of trivial algebraic equations whose iterative solution is easily found. The solution of these equations yields a mode expansion of the solution of the original Wheeler-DeWitt equation. Further analysis of the mode expansion shows that in general the solutions of the Wheeler-DeWitt equation for this model are doubly graded, i.e., every solution is a superposition of two definite-parity solutions. Moreover, it is shown that the mode expansion of both even and odd-parity solutions is always infinite. It may be terminated artificially to construct approximate solutions. This is demonstrated by working out an explicit example which turns out to satisfy DeWitt's boundary condition at initial singularity.
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