  We study the voter model on $\mathbf{Z}$ with long range interactions, as
proposed by Hammond and Sheffield. We show the associated random field
converges to a Gaussian random field, which can be seen as a one parameter
family of evolving fractional Brownian motions. As a consequence we obtain that
in equilibrium the voter model on $\mathbf{Z}$ with suitable long range
interactions rescales to fractional Gaussian noise. The argument uses the
Lindeberg swapping technique and heat kernel estimates for random walks with
jump distributions in the domain of attraction of a stable law.
