Energy and directional signatures for plane quantized gravity waves

Abstract

Solutions are constructed to the quantum constraints for planar gravity (fields dependent on $z$ and $t$ only) in the Ashtekar complex connection formalism. A number of operators are constructed and applied to the solutions. These include the familiar ADM energy and area operators, as well as new operators sensitive to intrinsic spin and directionality ($z+ct$ vs $z\ensuremath{-}\mathrm{ct}$ dependence). The directionality operators are quantum analogs of the classical constraints proposed for unidirectional plane waves by Bondi, Pirani, and Robinson (BPR). It is argued that the quantum BPR constraints will predict unidirectionality reliably only for solutions which are semiclassical in a certain sense. Schwinger has proved that a unidirectional plane electromagnetic wave is stable, even in the presence of the quantum zero point fluctuations of the vacuum. A preliminary calculation (preliminary, because not regulated) indicates that the corresponding gravitational wave may be destabilized by zero point fluctuations. The ADM energy and area operators are likely to have imaginary eigenvalues, unless one either shifts to a real connection, or allows the connection to occur other than in a holonomy. In classical theory, the area can evolve to zero. A quantum mechanical mechanism is proposed which would prevent this collapse.