Spinors are defined, their basic properties and relation to tensors are derived. The spinor image of the Weyl tensor is derived and it is shown that it is symmetric in all four of its spinor indices. From this, the classification of Weyl tensors equivalent to Petrov’s (by the Penrose method) is derived. The equivalence of these two approaches is proved. The third (Debever’s) method of classification of Weyl tensors is derived, and its equivalence to those of Petrov and Penrose is demonstrated. Extended hints for verifying the calculations (moved to the exercises section) are provided.

In this chapter we construct the closed-time-path (CTP) two-particle-irreducible (2PI) effective action to two-loop order. The CTP formalism introduced in Chapter 3 is needed to track the dynamics of expectation values and to produce real and causal equations of motion. The composite particle or 2PI formalism introduced in Chapter 6 is needed to treat critical phenomena, because the correlation function and the mean field act as independent variables, instead of the former being a derivative of the latter, as in the 1PI formulation. The large N expansion has the advantage of yielding nonperturbative evolution equations in the regime of strong mean field and a covariantly conserved stress-energy tensor. To leading order in large N, the quantum effective action for the matter fields can be interpreted as a leading-order term in the expansion of the full matter plus gravity quantum effective action, which produces equations of motion for semiclassical gravity and, at the next-to-leading order in large N, stochastic gravity. Two types of quantum fields are treated: (a) O(N) self-interacting Phi-4 fields, and (b) Yukawa coupling between scalar and spinor fields, as an example of dealing with fermions in curved spacetime.