Quantum Gravity

40 COVARIANT APPROACHES TO QUANTUM GRAVITYa footing is lacking (there is no measure-theoretical foundation). Still, the fieldtheoreticalpath integral is of great heuristic value and plays a key role especiallyin gauge theories (see e.g. Böhm et al. (2001) among many other references). Consider,for example, a real scalar field φ(x). 12 Then one has instead of (2.66), theexpression (setting again =1)∫Z[φ] = Dφ(x) e iS[φ(x)] , (2.68)where Z[φ] is the usual abbreviation for the path integral in the field-theoreticalcontext (often referring to in-out transition amplitudes or to partition sums, seebelow). The path integral is very useful in perturbation theory and gives a concisepossibility to derive Feynman rules (via the notion of the generating functional,see below). Using the methods of Grassmann integration, a path integral suchas (2.68) can also be defined for fermions. For systems with constraints (suchas gauge theories and gravity), however, the path-integral formulation has to begeneralized, as will be discussed in the course of this section. It must also be notedthat the usual operator ordering ambiguities of quantum theory are also presentin the path-integral approach, in spite of integrating over classical configurations:The ambiguities are here reflected in the ambiguities for the integration measure.Instead of the original formulation in space–time, it is often appropriate toperform a rotation to four-dimensional Euclidean space via the Wick rotationt →−iτ. In the case of the scalar field, this leads to∫iS[φ] = i∫t→−iτ−→ −dt d 3 xdτ d 3 x(12( ) )2 ∂φ− 1 ∂t 2 (∇φ)2 − V (φ)+L int( ) )2 ∂φ+ 1 ∂τ 2 (∇φ)2 + V (φ) −L int(12≡ −S E [φ] , (2.69)where V (φ) is the potential and an interaction L int to other fields has beentaken into account. This formal rotation to Euclidean space has some advantages.First, since S E is bounded from below, it improves the convergence propertiesof the path integral: instead of an oscillating integrand one has an exponentiallydamped integrand (remember, however, that e.g. the Fresnel integrals used inoptics are convergent in spite of the e ix2 -integrand). Second, for the extremizationprocedure, one has to deal with elliptic instead of hyperbolic equations, whichare more suitable for the boundary problem of specifying configurations at initialand final instants of time. Third, in the Euclidean formulation, the path integralcan be directly related to the partition sum in statistical mechanics (e.g. for the,12 The notation x is a shorthand for x µ , µ =0, 1, 2, 3.