Canonical quantization of cylindrical gravitational waves with two polarizations

The canonical quantization of the minisuperspace model describing cylindrically symmetric gravitational waves with two polarizations is presented. A Fock space-type representation is constructed. It is based on a complete set of quantum observables. Physical expectation values may be calculated in arbitrary excitations of the vacuum. Our approach provides a nonlinear generalization of the quantization of the collinearly polarized Einstein-Rosen gravitational waves. [S0031-9007(97)04861-8] The quantization of dimensionally reduced models of 4D Einstein gravity serves as interesting testing ground for many issues of quantum gravity. The physical output of this approach to an understanding of characteristic features of the full theory, however, strongly depends on the complexity of the model under consideration. Probably, the simplest and best understood examples are the minisuperspace models [1] which contain only a finite number of physical degrees of freedom and thus hide the field effects of quantum gravity. A more complicated example of steady interest is given by the minisuperspace model of cylindrically symmetric gravitational waves with one polarization [2,3]. This model already involves an infinite number of degrees of freedom. It becomes treatable with the methods of flat space quantum field theory, because the Einstein field equations essentially reduce to the axisymmetric 3D wave equation. This underlying linearity, on the other hand, may conceal typical nonlinear features of quantum gravity. It is the purpose of this Letter to generalize the results of Refs. [2,3] to cylindrical gravitational waves with two polarizations, where the Einstein equations become nonlinear. We achieve a consistent canonical quantization in terms of a complete set of quantum observables. Creation and annihilation operators are constructed in a type of Fock space representation of these observables. The full space of physical quantum states is then built from excitations of the vacuum. This allows one to calculate, in principle, all physical expectation values in arbitrary quantum states. We start from a general space-time with cylindrical symmetry, i.e., we assume the existence of two commuting Killing vector fields, one of which has closed orbits. We choose coordinates such that the Killing vector fields are given by ›z and ›w associated with the axis of symmetry z and the azimuthal angle w, respectively. Further gauge fixing brings the metric into the standard form [4], ds 2 › e Gsr, td s2dt 2 1 dr 2 d 1r ˜ g absr, tddx a dx b ,