General Relativity from Quantum Field Theory

TL;DR

This thesis explores deriving classical general relativity from quantum field theory, analyzing the Schwarzschild-Tangherlini metric and one-loop quantum corrections, while examining gauge dependence and invariance in various dimensions.

Contribution

It provides a detailed quantum field theoretic derivation of the Schwarzschild-Tangherlini metric and computes one-loop corrections, including gauge dependence analysis.

Findings

One-loop correction to the metric is gauge-independent.

Derived Feynman rules for graviton interactions.

Logarithmic behavior in five-dimensional space-time.

Abstract

The quantum field theoretic description of general relativity is a modern approach to gravity where gravitational force is carried by spin-2 gravitons. In the classical limit of this theory, general relativity as described by the Einstein field equations is obtained. This limit, where classical general relativity is derived from quantum field theory is the topic of this thesis. The Schwarzschild-Tangherlini metric, which describes the gravitational field of an inertial point particle in arbitrary space-time dimensions, $D$, is analyzed. The metric is related to the three-point vertex function of a massive scalar interacting with a graviton to all orders in $G_N$, and the one-loop contribution to this amplitude is computed from which the $G_N^2$ contribution to the metric is derived. To understand the gauge-dependence of the metric, covariant gauge is used which introduces the parameter, $\xi$, and the gauge-fixing function $G_\sigma$. In the classical limit, the gauge-fixing function turns out to be the coordinate condition, $G_\sigma=0$. As gauge-fixing function a novel family of gauges, which depends on an arbitrary parameter $\alpha$ and includes both harmonic and de Donder gauge, is used. Feynman rules for the graviton field are derived and important results are the graviton propagator in covariant gauge and a general formula for the n-graviton vertex in terms of the Einstein tensor. The Feynman rules are used both in deriving the Schwarzschild-Tangherlini metric from amplitudes and in the computation of the one-loop correction to the metric. The one-loop correction to the metric is independent of the covariant gauge parameter, $\xi$, and satisfies the gauge condition $G_\sigma=0$ where $G_\sigma$ is the family of gauges depending on $\alpha$. In space-time $D=5$ a logarithm appears in position space and this phenomena is analyzed in terms of redundant gauge freedom.