Quantum Tree Graphs and the 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, ${\epsilon }_r$, by the Newtonian-like formula, $m=m_0-\frac{3Gm_0^{}{}_{}{}^2}{5{\epsilon }_r}+O\left(G^2\right)$, and the infinities in the quantum theory are seen to be a manifestation of the divergent self-energy problem encountered in classical mechanics.