  We describe a background independent quantization of the scalar field that
provides an explicit realization of Fock-like states and associated operators
in a polymer Hilbert space. The vacuum expectation values of the commutator and
anti-commutator of the creation and annihilation operators become energy
dependent, and exhibit a surprising transition to fermionic behavior at high
energy. Furthermore the approach yields a modified dispersion relation with a
leading correction proportional to the momentum cubed. These results suggests a
fundamental change in the ultraviolet properties of quantum fields.
