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It is verified explicitly to second order in Newton's constant, G, that the quantum-tree-graph contribution to the vacuum expectation value of the gravitational field produced by a spherically symmetric c-number source correctly reproduces the classical Schwarzschild solution. If the source is taken to be that of a point mass, then even the tree diagrams are divergent, and it is necessary to use a source of finite extension which, for convenience, is taken to be a perfect fluid sphere with uniform density. In this way both the interior and exterior solutions may be generated. A mass renormalization takes place; the total mass of the source, m, being related to its bare mass, m₀, and invariant radius, ₑ, by the Newtonian-like formula, m=m₀-3Gm₀^{_^2}5{ₑ}+O (G^2), and the infinities in the quantum theory are seen to be a manifestation of the divergent self-energy problem encountered in classical mechanics.
M. J. Duff (Sun,) studied this question.