ON THE LONG-STANDING ISSUE OF AN R-PHI(2) COUPLING BETWEEN GRAVITY AND A MASSLESS FREE SCALAR FIELD

Using 2-body trees on a flat space background, it is shown that the ac tions A[g,phi]=integral d(4)x root-g [(R/2 kappa)+(1/2)(g(mu nu)partia l derivative(mu)phi partial derivative(mu)phi+lambda R phi(2))] and (A ) over bar[(g) over bar,<(phi)over bar>]=integral d(4)x root-(g) over bar[((R) over bar/2 kappa)+(1/2)(g) over bar(mu nu)partial derivative( mu)<(phi)over bar>partial derivative(nu)phi] describe the same theory at the tree-level in this case. We also demonstrate the quantum equiva lence (at one-loop) of the barred and unbarred systems for lambda=-1/6 (conformal coupling).