The graviton vacuum as a distributional state in kinematic loop quantum gravity

The quantum behaviour of weak gravitational fields admits an adequate, albeit approximate, description by those graviton states in which the expectation values and fluctuations of the linearized gravitational field are small. Such states must approximate corresponding states in full quantum gravity. We analyse the nature of this approximation for the graviton vacuum state in the context of kinematical loop quantum gravity (LQG) wherein the constraints are ignored. We identify the graviton vacuum state with kinematically normalizable, distributional states in LQG by demanding that relations between linearized operator actions on the former are mirrored by those of their nonlinear counterparts on the latter. We define a semi-norm on the space of kinematical distributions and show that the identification is approximate up to distributions which are small in this semi-norm. We argue that our candidate states are annihilated by the linearized constraints (expressed as operators in the full theory) to leading order in the parameter characterizing the approximation. This suggests the possibility, in a scheme such as ours, of solving the full constraints order by order in this parameter. The main drawback of our considerations is that they depend on certain auxilliary constructions which, though mathematically well defined, do not arise from physical insight. Our work is an attempt to implement an earlier proposal of Iwasaki and Rovelli.