### 3.4 Path integral quantization

An alternative quantization, that has been successfully employed in standard quantum field theories,
consists in using a path integral to represent relevant physical amplitudes and then develop perturbative
techniques to extract the physical information as some kind of expansion (usually asymptotic) in terms of
coupling constants. The main idea is to represent transition amplitudes as integrals over a set of
“interpolating configurations” (trajectories for particle systems or, more generally, field histories). For
example, the expression
would represent the probability amplitude to go from a state with metric and matter fields on a
3-surface to a state with metric and matter fields on a 3-surface . Here is the
classical action and is the “measure” on the space of field configurations determined by the phase-space
measure. The integral has to be computed over the field configuration on the region of the
spacetime manifold, which has and as boundaries. One of the advantages that is usually
attributed to the path integral is that it provides a “covariant” approach to quantum field
theory. However, it is important to notice that only the phase-space path integral can be shown
to be formally equivalent to canonical quantization [113]. If the integral in the momenta can
be performed in a closed algebraic form, one gets a configuration space path integral whose
integrand can be seen to be, in some cases but not always, just the action expressed in terms of
configuration variables and their derivatives. In the case of reduced phase-space quantization the
correct writing of the path integral requires the introduction of the Fadeev–Popov terms that
take into account the fact that the integration measure is the pullback of the formal Liouville
measure to the hypersurface defined by the first-class constraints and gauge fixing conditions (or
alternatively to the space of gauge orbits [113]). The path integral method can be rigorously
defined in some instances, for example in the quantum mechanics for systems with a finite
number of degrees of freedom and some field theories, such as topological models and lower
dimensional scalar models. In other cases, though, it is just a formal (though arguably very useful)
device.
The first proposals to use path integrals in quantum gravity go back to the fifties (see, for
example, the paper by Misner [163]) and were championed by Hawking, among many other
authors, in the study of quantum cosmology, black hole physics and related problems. Path
integrals are also useful in other approaches to quantum gravity, in particular Regge calculus, spin
foams and causal dynamical triangulations (see the Living Review by Loll [147] and references
therein). Finally they establish a fruitful relationship between quantum field theory and statistical
mechanics.

Although the majority of the work on quantum midisuperspaces uses the canonical approach, there are
nonetheless some papers that use standard perturbative methods based on path integrals to deal with some
of these models, for example the Einstein–Rosen waves [183, 27]. The results obtained with these methods
suggest that this model, in particular, is renormalizable in a generalized sense and compatible with the
asymptotic safety scenario [184].