Einstein’s Theory of General Relativity as a Quantum Field Theory There is a major difference in how the Standard Model developed and how General Relativity
(GR) did, and this difference still influences how we think about them today. The Standard
Model really developed hand in hand with Quantum Field Theory (QFT). Quantum Electrodynamics
(QED) required the development of renormalization theory. Yang–Mills (YM)
theory required the understanding of gauge invariance, path integrals and Faddeev–Popov
ghosts. To be useful, Quantum Chromodynamics (QCD) required understanding asymptotic
freedom and confinement. The weak interaction needed the Brout–Englert–Higgs mechanism,
and also dimensional regularization for ’t Hooft’s proof of renormalizability. We only could
formulate the Standard Model as a theory after all these QFT developments occurred.
In contrast, General Relativity was fully formulated 100 years ago. It has been passed
down to us as a geometric theory — “there is no gravitational force, but only geodesic
motion in curved spacetime”. And the mathematical development of the classical theory has
been quite beautiful. But because the theory was formulated so long ago, there were many
attempts to make a quantum theory which were really premature. This generated a really
bad reputation for quantum general relativity. We did not have the tools yet to do the job
fully. Indeed, making a QFT out of General Relativity requires all the tools of QFT that
the Standard Model has, plus also the development of Effective Field Theory (EFT). So,
although while many people made important progress as each new tool came into existence,
we really did not have all the tools in place until the 1990s.
So, let us imagine starting over. We can set out to develop a theory of gravity from
the QFT perspective. While there are remaining problems with quantum gravity, the bad
reputation that it initially acquired is not really deserved. The QFT treatment of General
Relativity is successful as an EFT and it forms a well–defined QFT in the modern sense.
Maybe it will survive longer than will the Standard Model.
1 Constructing GR as a Gauge Theory: A QFT Point of View 5
1.1 Preliminaries
1.2 Gauge Theories: Short Reminder
1.2.1 Abelian Case
1.2.2 Non–Abelian Case
1.3 Gravitational Field from Gauging Translations
1.3.1 General Coordinate Transformations
1.3.2 Matter Sector
1.3.3 Gravity Sector
2 Fermions in General Relativity
3 Weak–Field Gravity
3.1 Gauge Transformations
3.2 Newton’s Law .
3.3 Gauge Invariance for a Scalar Field
3.4 Schr¨odinger equation
4 Second Quantization of Weak Gravitational Field 18
4.1 Second Quantization
4.2 Propagator
4.3 Feynman Rules
5 Background Field Method
5.1 Preliminaries
5.1.1 Toy Example: Scalar QED
5.2 Generalization to other interactions
5.2.1 Faddeev–Popov Ghosts
5.3 Background Field Method in GR
6 Heat Kernel Method
6.1 General Considerations
6.2 Applications
6.3 Gauss–Bonnet Term
6.4 The Limit of Pure Gravity
7 Principles of Effective Field Theory
7.1 Three Principles of Sigma–Models
7.2 Linear Sigma–Model
7.2.1 Test of Equivalence
7.3 Loops
7.4 Chiral Perturbation Theory
8 General Relativity as an Effective Field Theory
8.1 Degrees of Freedom and Interactions
8.2 Most General Effective Lagrangian
8.3 Quantization and Renormalization
8.4 Fixing the EFT parameters
8.4.1 Gravity without Tensor Indices
2
8.5 Predictions: Newton’s Potential at One Loop
8.6 Generation of Reissner–Nordstr¨om Metric through Loop Corrections
9 GR as EFT: Further Developments
9.1 Gravity as a Square of Gauge Theory
9.2 Loops without Loops
9.3 Application: Bending of Light in Quantum Gravity
10 Infrared Properties of General Relativity
10.1 IR Divergences at One Loop
10.2 Cancellation of IR Divergences
10.3 Weinberg’s Soft Theorem and BMS Transformations
10.4 Other Soft Theorems
10.4.1 Cachazo–Strominger Soft Theorem
10.4.2 One–Loop Corrections to Cachazo–Strominger Soft Theorem
10.4.3 Relation to YM Theories
10.4.4 Double–Soft Limits of Gravitational Amplitudes
11 An Introduction to Non–local Effective Actions 60
11.1 Anomalies in General
11.2 Conformal Anomalies in Gravity
11.3 Non–local Effective Actions
11.4 An Explicit Example
11.5 Non–local actions as a frontier
12 The Problem of Quantum Gravity.
