Relationship between Fröhlich and Davydov models of biological order

Two physical models to explain the unusual behavior of biological systems put forward by Fr\"ohlich and Davydov are reviewed. The Davydov Hamiltonian is transformed into normal coordinates and the Fr\"ohlich Hamiltonian is canonically transformed into an equivalent from within the Hartree-Fock approximation. A model Hamiltonian capable of relating both theories is proposed. The resulting analysis employs only one type of second-quantized operators which describe the effect on the active dipolar modes of vibration in biological cells. An equation of motion is derived for the creation and annihilation operators that are used. When the phenomenon of Bose condensation is taken into account, the field translation variable is shown to obey a nonlinear Schr\"odinger equation. Its solutions are provided and their significance is discussed. Some of the solutions are shown to correspond to properties already examined by Fr\"ohlich and Davydov. New solutions are found and interpreted as interfaces between either an ordered and a disordered phase or two ordered phases existing in the biosystem. The relationship between the two theories of biological order is demonstrated and discussed.