Gravitons from a loop representation of linearized gravity

Loop quantum gravity is based on a classical formulation of $3+1$ gravity in terms of a real $\mathrm{SU}\left(2\right)\mathrm{}$ connection. Linearization of this classical formulation about a flat background yields a description of linearized gravity in terms of a real $U\left(1\right)\mathrm{}\times U\left(1\right)\mathrm{}\times U\left(1\right)\mathrm{}$ 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 generalize our recent work on photons and $U\left(1\right)\mathrm{}$ holonomies to show that Fock space gravitons are associated with distributional states in the $U\left(1\right)\mathrm{}\times U\left(1\right)\mathrm{}\times U\left(1\right)\mathrm{}$ 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 linearized gravity using complex connections. In the last part of this work we show that the loop representation based on the real $U\left(1\right)\mathrm{}\times U\left(1\right)\mathrm{}\times U\left(1\right)\mathrm{}$ 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.