All-orders wormhole vertex operators from the Wheeler-deWitt equation

We discuss the calculation of semi-classical wormhole vertex operators from wave functions which satisfy the Wheeler-deWitt equation and momentum constraints, together with certain `wormhole boundary conditions'. We consider a massless minimally coupled scalar field, initially in the spherically symmetric `mini-superspace' approximation, and then in the `midi-superspace' approximation, where non-spherically symmetric perturbations are linearized about a spherically symmetric mini-superspace background. Our approach suggests that there are higher derivative corrections to the vertex operator from the non-spherically symmetric perturbations. This is compared directly with the approach based on complete wormhole solutions to the equations of motion where it has been claimed that the semi-classical vertex operator is exactly given by the lowest order term, to all orders in the size of the wormhole throat. Our results are also compared with the conformally coupled case.