  We discuss the connection between the Fock space introduced by Ashtekar and
Pierri for Einstein-Rosen waves and its perturbative counterpart based on the
concept of particle that arises in linearized gravity with a de Donder gauge.
We show that the gauge adopted by Ashtekar and Pierri is a generalization of
the de Donder gauge to full (i.e. non-linearized) cylindrical gravity. This
fact allows us to relate the two descriptions of the Einstein-Rosen waves
analyzed here by means of a simple field redefinition. Employing this
redefinition, we find the highly non-linear relation that exists between the
annihilation and creation-like variables of the two approaches. We next
represent the particle-like variables of the perturbative approach as
regularized operators, introducing a cut-off. These can be expanded in powers
of the annihilation and creation operators of the Ashtekar and Pierri
quantization, each additional power being multiplied by an extra square-root of
the three-dimensional gravitational constant, $\sqrt{G}$. In principle, the
perturbative vacuum may be reached as the limit of a state annihilated by these
regularized operators when the cut-off is removed. This state can be written as
the vacuum of the Ashtekar and Pierri quantization corrected by a perturbative
series in $\sqrt{G}$ with no contributions from particles with energies above
the cut-off. We show that the first-order correction is in fact a state of
infinite norm. This result is interpreted as indicating that the Fock
quantizations in the two approaches are unitarily inequivalent and, in any
case, proves that the perturbative vacuum is not analytic in the interaction
constant. Therefore a standard perturbative quantum analysis fails.
