  We perform the canonical quantization of a general scalar-tensor theory and
derive the first quantum gravitational corrections following from a
semi-classical expansion of the Wheeler-DeWitt equation. The non-minimal
coupling of the scalar field to gravity induces a derivative coupling between
the scalar field and the gravitational degrees of freedom, which prevents a
direct application of the expansion scheme. We address this technical
difficulty by transforming the theory from the Jordan frame to the Einstein
frame. We find that a large non-minimal coupling can have strong effects on the
quantum gravitational correction terms. We briefly discuss these effects in the
context of the specific model of Higgs inflation.
