Quantum Tree Graphs and the Schwarzschild Solution

TLDR: In this paper, it is shown 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.

Abstract: 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, ${m}_{0}$, and invariant radius, ${\ensuremath{\epsilon}}_{r}$, by the Newtonian-like formula, $m={m}_{0}\ensuremath{-}\frac{3Gm_{0}^{}{}_{}{}^{2}}{5{\ensuremath{\epsilon}}_{r}}+O({G}^{2})$, and the infinities in the quantum theory are seen to be a manifestation of the divergent self-energy problem encountered in classical mechanics.