  This work starts from the premise that sinusoidal plane waves cease to be
solutions of field theories when turning on an interaction. A nonlinear
interaction term generates harmonics analogous to those observed in nonlinear
optical media. This calls for a generalization to anharmonic waves in both
classical and quantum field theory. Three simple requirements make anharmonic
waves compatible with relativistic field theory and quantum physics. Some
non-essential concepts have to be abandoned, such as orthogonality, the
superposition principle, and the existence of single-particle energy
eigenstates. The most general class of anharmonic waves allows for a zero
frequency term in the Fourier series, which corresponds to a quantum field with
a non-zero vacuum expectation value. Anharmonic quantum fields are defined by
generalizing the expansion of a field operator into creation and annihilation
operators. This method provides a framework for handling exact quantum fields,
which define exact single particle states.
