Quantum cosmology with a positive-definite action

We argue that the indefiniteness of the Euclidean Einstein action is more serious in the cosmological context than in the asymptotically Euclidean context. To correct this, we consider a positive-definite action containing quadratic curvature terms. The physical states Ψ are now functions of both a three-metric $q_{\mathrm{ab}}$ and extrinsic curvature $K_{\mathrm{ab}}$, and satisfy a differential equation analogous to the Wheeler-DeWitt equation. This equation has the form of a Schrödinger equation with $q_{\mathrm{ab}}$ playing the role of ‘‘time.’’ By adopting Hartle and Hawking’s boundary condition on the Euclidean function integral, we obtain a ‘‘preferred’’ solution to this equation. It is shown that in a simple minisuperspace model this wave function describes an inflationary universe.