Minisuperspaces with conformally and minimally coupled scalar fields

One may perform a local field redefinition to transform between gravity minimally coupled to a free scalar field and gravity conformally coupled. However, the allowed field values differ in the two cases. For a minisuperspace consisting of a Friedman–Robertson–Walker geometry and a homogeneous scalar field, the conformal coupling allows a more general class of solutions of the Wheeler–DeWitt equation than does the minimal coupling. Nevertheless, there is a one-to-one correspondence between the bounded solutions in the two cases for k=1. This correspondence exploits an isomorphism between harmonic oscillator wavefunctions and solutions of the massive Klein–Gordon equation in the 1+1 dimensional Rindler wedge.