An Introduction to Covariant Quantum Gravity and Asymptotic Safety

The following sections are included:

• Preface
• Acknowledgments
• Contents

The following sections are included:

• Early years
• DeWitt’s era
• Loop quantum gravity
• The standard model
• GUTs, supergravity and superstrings
• Recent developments and future prospects

The following sections are included:

• The linear field equations
  • The relativistic spin-2 field equation
  • Linearizing Einstein’s equations
  • Plane waves
  • Quantization
  • Spin projectors
• Four derivative theories
  • Actions
  • Linearized equations
• Power counting
• Appendix: Topological invariants

The main goal of this chapter is to calculate the one-loop divergences in Einstein’s theory. This requires a number of technical steps. By using the background field method, the calculation can be reduced to the case of a spin-2 quantum field propagating in an external non-dynamical gravitational field. We review a very general method to compute the divergent part of the one-loop effective action in any Euclidean QFT, based on the early time asymptotic expansion of the heat kernel. In Minkowskian signature, this is known as the Schwinger-DeWitt method [63–65]. As a warmup we apply this method to scalar and gauge theories, then we come to gravity. In order not to interrupt the flow of the arguments, some necessary results on the heat kernel are left to Section 3.7.

The following sections are included:

• Options
• Emergent gravity
• Higher derivative gravities
• Special matter choice: Supergravity
• The Effective Field Theory approach
  • The general idea
  • Example: Chiral perturbation theory
  • Gravity
  • The leading corrections to the Newtonian potential

In this and in the next chapter we will work our way towards the discussion of asymptotic safety in gravity. This requires the use of additional techniques that have not been necessary so far. Such techniques have wider applicability, and it is pedagogically useful to introduce them in the simpler and more familiar setting of one-loop QFT in curved spacetime, introduced in Chapter 3. As a concrete application, we shall use these techniques to define the EA in a more general gauge than the one used in chapter 3, and prove independence of the gauge parameters on shell. The calculation will be continued in section 7.3.3, where the one-loop divergences will be obtained.

In chapter 3 we introduced the (Euclidean) Schwinger-de Witt method to calculate one-loop divergences in QFT. It depends on a genuine mass cutoff Λ, but is nevertheless well suited to discuss gauge theories, because one can retain background gauge invariance. One of the main results of this type of calculation is the beta function, which, for dimensionless couplings, coincides with the coefficient of the logarithmic divergence. For example, in a scalar theory the beta function of the quartic coupling is the coefficient of (log Λ)ϕ⁴ in Γ_Λ, and in Yang-Mills theory the beta function can be read from the coefficient of (log Λ).

In QFT the action typically contains only a small number of terms and a theory is said to be “renormalizable” if the perturbative treatment produces only divergences that are proportional to such terms. One can then absorb the divergences in renormalizations of the respective couplings. This terminology is slightly misleading, in the sense that non-renormalizable theories can also be renormalized. The problem is that one has to introduce infinitely many counterterms and accordingly infinitely many couplings have to be determined by resorting to experiment. The real issue is not the divergences, but a lack of predictivity.

The calculations in the preceding chapter illustrate various aspects of the search for a gravitational fixed point. These examples cover only a small part of the existing literature and have been presented without much commentary and with minimal references. Here we shall partly make up for these omissions by reviewing the existing literature and then discussing the main results, open problems and possible future developments.

The following sections are included:

• Units
• Notations
  • Conventions
  • Acronyms

The following sections are included:

• Bibliography
• Subject Index
• Author Index