The graviton vacuum as a distributional state in kinematic Loop Quantum Gravity

TL;DR

This paper explores how the graviton vacuum state can be represented as a distributional state within kinematic Loop Quantum Gravity, providing an approximate correspondence that could inform solutions to quantum gravity constraints.

Contribution

It identifies the graviton vacuum as a distributional state in LQG and demonstrates an approximate relation to linearised operators, advancing the understanding of quantum gravitational states.

Findings

Identifies graviton vacuum as a distributional state in LQG.

Shows the approximation is valid up to small distributions in a semi-norm.

Suggests a scheme for solving full constraints order by order.

Abstract

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 linearised 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 non-normalizable, distributional states in LQG by demanding that relations between linearised operator actions on the former are mirrored by those of their non-linear counterparts on the latter. We define a semi- norm on the space of kinematical distributions and show that the identification is approximate upto distributions which are small in this semi-norm. We argue that our candidate states are annihilated by the linearised constraints (expressed as operators in the full theory) to leading order in the parameter characterising 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.