Large quantum gravity effects: Cylindrical waves in four dimensions

Linearly polarized cylindrical waves in four-dimensional vacuum gravity are mathematically equivalent to rotationally symmetric gravity coupled to a M axwell (or Klein-Gordon) field in three dimensions. The quantization of thi s latter system was performed by Ashtekar and Pierri in a recent work. Empl oying that quantization, we obtain here a complete quantum theory which des cribes the four-dimensional geometry of the Einstein-Rosen waves. In partic ular, we construct regularized operators to represent the metric. It is sho wn that the results achieved by Ashtekar about the existence of important q uantum gravity effects in the Einstein-Maxwell system at large distances fr om the symmetry axis continue to be valid from a four-dimensional point of view. The only significant difference is that, in order to admit an approxi mate classical description in the asymptotic region, states that are cohere nt in the Maxwell field need not contain a large number of photons anymore. We also analyze the metric fluctuations on the symmetry axis and argue tha t they are generally relevant for all of the coherent states.