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

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 tuating 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.