Generalizing the ADM Computation to Quantum Field Theory

TL;DR

This paper develops a novel expansion method for quantum field theory in gravity, incorporating nonanalytic effects and bound states, to better understand ultraviolet divergences and low-energy quantum gravitational phenomena.

Contribution

It introduces an alternative expansion for quantum gravitational systems based on stationary phase, including infinite diagrams, and demonstrates the existence of bound states with numerical mass results.

Findings

Bound states of charged, gravitating scalars exist.

The new expansion captures nonanalytic dependence on Newton's constant.

Numerical masses for bound states are obtained.

Abstract

The absence of recognizable, low energy quantum gravitational effects requires that some asymptotic series expansion be wonderfully accurate, but the correct expansion might involve logarithms or fractional powers of Newton's constant. That would explain why conventional perturbation theory shows uncontrollable ultraviolet divergences. We explore this possibility in the context of the mass of a charged, gravitating scalar. The classical limit of this system was solved exactly in 1960 by Arnowitt, Deser and Misner, and their solution does exhibit nonanalytic dependence on Newton's constant. We derive an exact functional integral representation for the mass of the quantum field theoretic system, and then develop an alternate expansion for it based on a correct implementation of the method of stationary phase. The new expansion entails adding an infinite class of new diagrams to each order and subtracting them from higher orders. The zeroth order term of the new expansion has the physical interpretation of a first quantized Klein-Gordon scalar which forms a bound state in the gravitational and electromagnetic potentials sourced by its own probability current. We show that such bound states exist and we obtain numerical results for their masses.