Canonical Quantization of Cylindrical Gravitational Waves

Abstract

The Einstein-Rosen cylindrical gravitational waves are quantized by the canonical methods due to Dirac (the constraint formalism) and to Arnowitt, Deser, and Misner (the ADM deparametrized formalism). The general reduction of geometrodynamical phase space to a mini-phase-space by intransitive groups of motion with spacelike Killing vectors is presented. The ADM classical canonical formalism restricted to the infinite-dimensional mini-phase-space generated by the cylindrical group of motions is built up. Six invariantly defined functions of one coordinate label are introduced as new canonical variables by a canonical transformation in mini-phase-space. Two canonical coordinates are identified with the Einstein-Rosen time and cylindrical radius. Canonically conjugate to them are $C$-energy density and $C$-energy flux. The third pair of canonical variables carries the ${\ensuremath{\infty}}^{1}$ dynamical degrees of freedom of the cylindrical wave. The canonical transformation mixes superspace with momentum space, the Einstein-Rosen time being constructed from the extrinsic curvature of the spacelike hypersurface rather than from its intrinsic geometry. The ADM classical canonical formalism for cylindrical gravitational waves is proved to be identical with the parametrized formalism for the cylindrical massless scalar waves propagating in Minkowskian spacetime. The identity of the quantum formalisms follows. The extrinsic time representation, with the Einstein-Rosen time and cylindrical radius as two of the three basic variables, is used instead of the metric representation. The Dirac contraints are imposed on the state functional. If the hypersurfaces are labeled by the Einstein-Rosen cylindrical radius, the constraints are reduced to a functional differential equation of the Schr\"odinger type. This equation is further reduced to a single partial differential equation by integrability conditions which ensure that the evolution of the state functional between two hypersurfaces is path-independent. The inner product of two state functionals conserved by the deformation of the hypersurface is defined. Under the coordinate conditions restricting the allowable hypersurfaces to those of a constant Einstein-Rosen time, the Dirac formalism is deparametrized into the ADM quantum formalism.