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
∫ (g2,ϕ2,Σ2 |g1,ϕ1, Σ1) = exp (iS(g,ϕ ))μ(dg,d ϕ) (2 )
would represent the probability amplitude to go from a state with metric g1 and matter fields ϕ1 on a 3-surface Σ1 to a state with metric g2 and matter fields ϕ2 on a 3-surface Σ2. Here S is the classical action and μ 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 Σ1 and Σ2 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 [113Jump To The Next Citation Point]. 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 [183Jump To The Next Citation Point, 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 [184Jump To The Next Citation Point].

  Go to previous page Go up Go to next page