On statistical theory of electromagnetic waves in a fluctuating medium (II). Mathematical basis of the analogies to quantum field theory (a digest)

Abstract

Many analogies to quantum field theory exist inhere ntly in th e stati s tical theory of waves; for in stance, the problem of getting the effect iv e medium constant and the effective coupli ng constant between the waves and the fluctuating medium can be treated in a lmost th e sam e way as in field theory (t heo ry of renormalization). These analogies are s hown to be du e to the fact that basic equ ations ex ist in the s tatistical th eo ry which correspond closely to th e fundamenta l equations in field th eory; i.e., to th e commu tation relations and the H e isenberg equa tion of motion. A probability d e nsity fun c· tion of waves is introduced here which co rresponds to the probability amplitude function in quantum mechanics. In this case, the boundary conditions at infinity for thi s probability den sity fun c tion are found to be ex pressed in th e sa me form as the vacuum boundary con dition s in field th eory. Th e theory of the statis ti cal Green's functions and their relationships to the ex pectation values of physical variab les is also extensively developed , usi ng auxiliary external so urces of the wave and of the flu c tuat-ing medium. It is found that th e re ex ists a one-to-one correspondence between the formalism of Green's functions prese nted here and that used in field theory. The above co rrespo nde nce may be important for th e further development of the s tatisti ca l theory of wave s, just as the advanced tec hniques of field theory have greatly influen ced the development of thermodynamics. The detai ls of the theory were recently publis hed [Furutsu, 19651.