The theory of phonon-phonon interactions was formulated as soon as the quasi-particle itself was defined as the quantized lattice vibration of crystalline substances. The quantum of phonon is given as (ℏ��)², whose positive root is identical to Planck’s energy quantum. It indicates the wave-particle duality of all quantized objects of the physical universe. These include all the elementary particles of high energy physics and can be extended to include both the electrically neutral atoms and molecules. Although the wave nature of real particles is simply described by quantum mechanics or quantum field theory, the wave nature of quasi-particles are more complex. Their dynamics allows a few exceptions to the commonly accepted rules of interaction. One such exception is inelastic collision. Another is the non-conservation of quasi-momentum. Both suggest the involvement of the quantized vacuum in certain specified interactions. Hypothetically, the nature of the vacuum state |0〉 signifies the quantum vacuum operators �� and ��. These operators can create or destroy excitations of the vacuum analogous to the creation and annihilation operators of the electrons.

Since quasi-momenta are not conserved, it is theoretically possible to simply apply continuously the creation vacuum operator �� to extract more and more energy from the quantum vacuum. In this sense, �� becomes the phonon creation operator. Applying �� successively to the ��-order identity matrix is possible if and only if �� is given by a Hadamard matrix, either H-plus or H-minus. Consequently, if �� is H-plus then �� is H-minus. The product of H-pluses is always another H-plus of larger eigenvalue. The product of an H-plus and an H-minus is an H-minus of larger eigenvalue. However, the Product of H-minus and H-minus is an H-plus of larger eigenvalue. Successive products of even number of H-minuses always give an H-plus of larger and larger eigenvalue. Furthermore, both the commutative and the associative properties of matrix multiplication hold for the productions of larger and larger eigenvalues of H-pluses and H-minuses.