Show description

Moreover, a sum-over-histories approach in quantum gravity is necessary if one wants to allow and study processes involving topology change, either spatial topology change or change in spacetime topology[47]; if the topology itself has to be considered as a dynamical variable then the only way to do it is to include a sum over topologies in the path integral defining the quantum gravity theory. Also, the study of topological geons [48] seems to lead to the conclusion that a dynamical metric requires a dynamical topology, so, indirectly, requires a sum over histories formulation for quantum gravity.

E. g. it can be generalised to include spacetime configurations in which the system moves back and forth in a given coordinate time. Let us stress that the reason for this impossibility does not reside in the use of path integrals versus operators per se, but in the fundamental spacetime nature of the variables considered (the histories). e. that are not globally hyperbolic. Of course a sum-over-histories formulation should be completed by a rule for partitioning the whole set of spacetime histories into a set of exhaustive alternatives to which the theory can consistently assign probabilities, using some coarse-graining procedure, and this is where the mechanism of decoherence is supposed to be necessary; such a mechanism looks particularly handy in a quantum gravity context, at least to some people [46], since it provides a definition of physical measurements that does not involve directly any notion of observer, and can thus apply to closed system as the universe as a whole.

Consider the space E of test functions f (X, T ) and the linear map P : E → H sending each function f into the state | f defined as above, 29 whose image is dense in H, being the linear space of solutions of the Schroedinger equation; of course the scalar product for the states | f can be pulled back to E, by means of the propagator W ; therefore the linear space E, equipped with such a scalar product, quotiented by the zero norm states, and completed in this norm, can be identified with the Hilbert space H of the theory, and we see that the propagator W (X, T ; X ′, T ′ ) contains the full information needed to reconstruct H from E.