Kiefer C. Quantum gravity

PATH-INTEGRAL QUANTIZATION 55
formalism ‘covariant’. Donoghue and Torma (1999) have shown that the one-loop
calculations of graviton–graviton scattering yield a finite result in the infrared
(IR) limit, independent of any parameters such as c 1 or c 2 . The cancellation of
IR divergences with the emission of soft gravitons is needed and shown in fact to
occur (as e.g. in QED). This yields again a definite result from quantum gravity.
There is, in fact, a huge literature about IR-effects from quantum gravity. One
example is the dynamical relaxation of the cosmological constant and its possible
relevance for the dark-matter problem (Tsamis and Woodard 1993). Another
way of addressing this issue is the investigation of renormalization-group equations,
which can be applied also to effective theories; cf. Section 2.2.5. At least
in principle, one could understand from this method the occurrence of a small
positive cosmological constant in agreement with observations because it would
arise as a strong IR quantum effect.
The idea that non-renormalizable theories can be treated as ordinary physical
theories from which phenomenological consequences can be drawn was also discussed
in other contexts. Kazakov (1988), for example, generalized the standard
formalism of the renormalization group to non-renormalizable theories. Barvinsky
et al. (1993) developed a version of the renormalization-group formalism for
non-renormalizable theories, which is particularly convenient for applications to
GR coupled to a scalar field.
2.2.4 Semiclassical Einstein equations
In this subsection we shall give a general introduction into the concept of effective
action, which is of central importance for quantum field theory. We shall then
apply this to quantum gravity and present in particular a derivation of the
semiclassical Einstein equations (1.35). More details can be found in Barvinsky
(1990) and Buchbinder et al. (1992). 22
For a general quantum field ϕ (with possible components ϕ i ), the generating
functional W [J] is defined by the path integral
∫
〈out, 0|in, 0〉 J ≡ Z[J] ≡ e iW [J] = Dϕ e iS[ϕ]+iJ kϕ k , (2.102)
where J is an external current and J k ϕ k is an abbreviation for ∫ d 4 xJ i (x)ϕ i (x).
This is also known as ‘DeWitt’s condensed notation’; cf. DeWitt (1965). We
write the index of ϕ usually only in expressions where more than one index
occurs. If ϕ is a gauge field, the measure in (2.102) is understood as including
gauge-fixing terms and Faddeev–Popov ghosts, see (2.85) above. Later ϕ will
be the gravitational field. W [J] is called the generating functional because one
can calculate from it Green functions of the theory. More precisely, W generates
connected Green functions 23 according to
22 In the preparation of this subsection, I have benefited much from discussions with Andrei
Barvinsky.
23 A connected Green function is a Green function referring to a connected graph, that is, a
graph for which any two of its points are connected by internal lines.