# Mini-course for ISQG

## General Relativity as a Perturbative Quantum Field Theory

This is the web page for John Donoghue’s lectures as part of the series on The Basics of Quantum Gravity sponsored by the International Society for Quantum Gravity,. It is also presently the welcome page for this web site devoted to this topic – much more information can be found elsewhere on the site. An overview page describes these resources in more detail . If you are looking for the notes for the TRISEP 2023 summer school go here.

Lecture 1 – Thursday June 8. Here are the notes to the first lecture. Here is a link to the video of the lecture . Preliminaries. Constructing GR as a QFT. Failure of scalar exchange. Energy and momentum as a source. Construction of gauge theories. Need to gauge spacetime translations, Why this could work. Equivalence principle. Coordinate changes and the metric. Describing the symmetry. A scalar field. First success – energy momentum tensor as source. Equation of motion. Preview of second success – Schroedinger equation in gravitational field and the force law. Preview of the third success – real QFT – graviton exchange and the Newtonian potential. Also some material added on some technical details

Lecture 2 Tuesday June 13. Here are the notes for the second lecture. Here is a link to the video of the lecture. Review. Covariant derivatives. Review of Lorentz invariance for fermions. Local Lorentz invariance and vierbein. Symmetries of Dirac equation. Covariant derivative and the spin connection. Field strength tensor. Invariants. Getting to GR – metricity or Palatini. Variations: torsion and metric-affine. Gravitational action and Einstein’s equation. Weak field expansion. Harmonic gauge. Static solution. Expanding the action. Gauge fixing. Propagator. Feynman rules. The gravitational potential again.

Lecture 3 Thursday June 15 Here are the notes for this lecture. Here is a link to the video of the lecture. Important Path Integral results. Background field method example for QED. Divergences and renormalization. Generalize to get “heat kernal coefficient”, Background field expansion for General Relativity. Gauge invariance. Gauge Fixing constraint. Ghost stories using Feynman’s paper. PI formalism for gauge fixing in QFT. Evaluating the determanent. The ghost Lagrangian for harmonic gauge gravity. Summary of the gravitational path integral.

Lecture 4, Sept 18 Notes for this lecture. Video of the lecture Review. Free graviton fields. Energy carried by gravitons. Intro to gravity as the square of a gauge theory. Loops of massless matter fields. Curvature squared terms and renormalization. Loops of massive matter fields. Renormalization of the cosmological constant. The cosmological constant does not run. Preview of graviton loops.

Lecture 5, Sept 20 Notes for the lecture. Video of the lecture. Effective Field Theory Day. Basic ideas of EFT. QED as an example. Explicit construction of an EFT from the linear sigma model. Haag’s Theorem. Renormalizing a non-renormalizable theory. Heat Kernel methods. Making predictions. Power counting.

Lecture 6 Sept 22 Notes for the lecture , Video of the lecture. EFT treatment of perturbative GR. The derivative/curvature expansion, matching or measuring, brief Ostrogradsky comments, Renormalization, the gravitational potential, pulling out low energy finite effects, unitarity techniques, bending of light, classical effects from quantum loops, no test particle limit, nonuniversality of geodesics, limitations of the EFT – high energy, low energy issues.

#### Some reference material

For third lecture: Here is an introductory chapter on path integrals from our book A Prelude to Quantum Field Theory. You already have access to the PI appendix in Dynamics of the Standard Model, which is somewhat more advanced. In the appendix section A-6, we discus Fadeev-Popov ghosts for gauge theories.

A question was asked about some of the details of the QED background field example. Here is a paper with some more of the details.

A reference on the background field method is Abbott

For the second lecture: Kibble and Utiyama on the construction of GR as a gauge theory.

Weinberg’s paper on infrared gravitons

Zvi Bern’s review on gravity as the square of gauge theory

Lance Dixon’s A brief introduction to modern amplitude methods

Elvang and Huang review Scattering Amplitudes also in book form

Some exercises. I hope to add to these.

Overall references:

EPFL Lectures on General Relativity as a Quantum Field Theory

Download your free open access copy of Dynamics of the Standard Model, which contains material which I will use during this mini-course.

Everyone should read Feynman’s paper: Quantum Theory of Gravitation . It is informative and also fun.