Gravitons from a loop representation of linearised gravity

TL;DR

This paper develops a loop representation for linearized gravity using a real U(1) connection, linking Fock space gravitons to distributional states and providing a new mathematical framework for understanding gravitons in loop quantum gravity.

Contribution

It generalizes previous work on photons to gravitons, establishing a connection between Fock space states and loop representation states using real U(1) connections.

Findings

Fock space gravitons correspond to distributional states in the loop representation.

The real U(1) loop representation can reproduce complex connection results.

Provides a mathematically precise language for loop quantum gravity formulations.

Abstract

Loop quantum gravity is based on a classical formulation of 3+1 gravity in terms of a real SU(2) connection. Linearization of this classical formulation about a flat background yields a description of linearised gravity in terms of a {\em real} $U(1)\times U(1)\times U(1)$ connection. A `loop' representation, in which holonomies of this connection are unitary operators, can be constructed. These holonomies are not well defined operators in the standard graviton Fock representation. We generalise our recent work on photons and U(1) holonomies to show that Fock space gravitons are associated with distributional states in the $U(1)\times U(1)\times U(1)$ loop representation. Our results may illuminate certain aspects of the much deeper (and as yet unkown,) relation between gravitons and states in nonperturbative loop quantum gravity. This work leans heavily on earlier seminal work by Ashtekar, Rovelli and Smolin (ARS) on the loop representation of linearised gravity using {\em complex} connections. In the last part of this work, we show that the loop representation based on the {\em real} $U(1)\times U(1)\times U(1)$ connection also provides a useful kinematic arena in which it is possible to express the ARS complex connection- based results in the mathematically precise language currently used in the field.