Abstract
Following a recently developed approach to the problem of divergences in field theory, the author computes the second-order correction to the graviton Green function arising from insertion of the graviton loop. As was the case with the calculations reported earlier, no subtractions, regularizations or renormalizations are required. His corrected graviton Green function depends only logarithmically on a constant l and satisfies the appropriate Slavnov-Ward identity (SWI). The absence of divergences means that he does not need to introduce counter terms into the Einstein Lagrangian. It also allows a straightforward and expeditious evaluation of the self-energy insertion. In addition, he reports the calculation of the correction to the graviton Green function arising from the insertion of a scalar meson loop. Again there are no divergences. However, in this case he finds that the result violates the SWI. On the other hand, if the gravitons are coupled to the scalar field through the improved energy tensor of Callan, Coleman and Jackiw (1970), the corrected graviton Green function does satisfy the SWI. The correction as well as the self-energy is traceless. Thus there is no conformal anomaly.