  We apply a type of background independent "polymer" quantization to a free
scalar field in a flat spacetime. Using semi-classical states, we find an
effective wave equation that is both nonlinear and Lorentz invariance
violating. We solve this equation perturbatively for several cases of physical
interest, and show that polymer corrections to solutions of the Klein-Gordon
equation depend on the amplitude of the field. This leads to an effective
dispersion relation that depends on the amplitude, frequency and shape of the
wave-packet, and is hence distinct from other modified dispersion relations
found in the literature. We also demonstrate that polymer effects tend to
accumulate with time for plane-symmetric waveforms. We conclude by discussing
the possibility of measuring deviations from the Klein-Gordon equation in
particle accelerators or astrophysical observations.
