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Starting with the Hamiltonian formulation for spacetimes with two commutingspacelike Killing vectors, we construct a midisuperspace model for linearlypolarized plane waves in vacuum gravity. This model has no constraints and itsdegrees of freedom can be interpreted as an infinite and continuous set ofannihilation and creation like variables. We also consider a simplified versionof the model, in which the number of modes is restricted to a discrete set. Inboth cases, the quantization is achieved by introducing a Fock representation.We find regularized operators to represent the metric and discuss whether thecoherent states of the quantum theory are peaked around classical spacetimes.It is shown that, although the expectation value of the metric on Killingorbits coincides with a classical solution, its relative fluctuations becomesignificant when one approaches a region where null geodesics are focused. Inthat region, the spacetimes described by coherent states fail to admit anapproximate classical description. This result applies as well to the vacuum ofthe theory.Comment: 11 pages, no figures, version accepted for publication in Phys. Rev.

Plane waves in quantum gravity: Breakdown of the classical spacetime