Cosmic Wave Functions with the Brans-Dicke Theory

Using the standard Wentzel-Kramers-Brillouin method, the Wheeler-DeWitt equation for the Brans-Dicke theory is solved under three kinds of boundary conditions (proposed by Hartle-Hawking, Vilenkin and Linde, respectively). It is found that, although the gravitational and cosmological “constants”are dynamical and time-dependent in the classical models, they will acquire constant values when the universe comes from the quantum creation, and that in particular, the amplitude of the resulting wave function under Linde or Vilenkin boundary conditions reaches its maximum if the cosmological constant is the minimum.