Since photons and phonons all carry linear momenta their interactions among themselves or with other quantum fields of space-time, matter, or energy can be described properly by a quantum theory of elasticity. Classical elasticity is defined as the property of a substance by which it tends to restore its shape or size after being subjected to deforming stresses. The deformation is called the strain, of which there are three simplest classifications: (1) linear strain (longitudinal), the change in length per unit length; (2) volume strain (bulk), the change in volume per unit volume (e.g. when hydrostatic pressure is applied as felt by deep-sea divers); and (3) shear strain as angular displacements without change in volume. There are also three classifications of simplest stresses: (1) tension or compression (i.e. normal stress), e.g. the force per unit area of cross section applied to each end of an elongated “rigid” substance to extend or to compress it; (2) hydrostatic pressure, e.g. the force per unit area applied to a “solid” by immersion in a fluid; and (3) shear stress, e.g. the system of four tangential forces applied to the surfaces of a rectangular “solid.” Hence, the ratio of stress to strain for a substance following the application of a restoring force (e.g. Hooke’s law) is then called the modulus of elasticity of which there are also three classifications: (1) Young modulus, (2) bulk modulus, and (3) shear modulus.

Fortunately, as a course in theoretical physics, the classical theory of elasticity was already published by Lev Davidovich Landau, et al. in 1959. Landau (1908-68 ) was the Soviet physicist who received the Nobel Prize for Physics in 1962 for his works on liquid helium. He also predicted the existence of neutron stars and developed a theory of superfluidity. The linearized Hooke’s law, discovered by the English physicist Robert Hooke (1635-1703) worked with the English chemist Robert Boyle at Oxford University, was later applied to the quantum theory of harmonic oscillators as formulated in non-relativistic quantum mechanics. Nonetheless, the theory of simple harmonic motion (SHM) is applicable to many other periodic motions. The Hooke’s restoring force is given by ��=-���� where �� is known as the restoring constant indicating the stiffness of the oscillating medium. The independent variable �� is the linear displacement. The equilibrium equation for a given system is usually expressed as ��₁+��₁��₁=0. SHMs vanish if ��₁+��₁��₁≠0, either ��₁+��₁��₁<0 or ��₁+��₁��₁>0. Since at equilibrium both �� and ���� can have either the same or opposite direction, in fact any different arbitrary directions, it is also true that ��₂-��₂��₂=0 The product (��₁+��₁��₁)(��₂-��₂��₂) also vanishes. For self-restoring media, ��₁=��₂=��, ��₁=��₂=��, and ��₁=��₂=��, and the product becomes ��˛-��˛��˛=0 where �� is now defined as the restoring constant of the quantum vacuum, the ratio of the thermal force divided by Planck length.