  The Hamiltonian constraint operator in Loop Quantum Gravity acts
ultralocally. Smolin has argued that this ultralocality seems incompatible with
the existence of a quantum dynamics which propagates perturbations between
macroscopically seperated regions of quantum geometry. We present evidence to
the contrary within an LQG type `polymer' quantization of two dimensional
Parameterised Field Theory (PFT). PFT is a generally covariant reformulation of
free field propagation on flat spacetime. We show explicitly that while, as in
LQG, the Hamiltonian constraint operator in PFT acts ultralocally, states in
the joint kernel of the Hamiltonian and diffeomorphism constraints of PFT
necessarily describe propagation effects. The particular structure of the
finite triangulation Hamiltonian constraint operator plays a crucial role, as
does the necessity of imposing (the continuum limit of) its kinematic adjoint
as a constraint. Propagation is seen as a property encoded by physical states
in the kernel of the constraints rather than that of repeated actions of the
finite triangulation Hamiltonian constraint on kinematic states. The analysis
yields robust structural lessons for putative constructions of the Hamiltonian
constraint in LQG for which ultralocal action co-exists with a description of
propagation effects by physical states.
