Many-particle and semiclassical limit transitions for nonrelativistic bosons in a quantized electromagnetic field

Using the method of the analytic germ, we obtain a system of equations for the amplitudes of one-particle phase densities of a system of several species of classical particles with electromagnetic interaction. The corresponding equations result from an extremely complicated limit transition in the theory of bosons interacting with a quantized electromagnetic field rather than in the classical equations for $N$ particles in a magnetic field. This transition implies a double limit: first, the limit of large numbers of particles and photons and, second, the semiclassical limit. Moreover, in the first of these limits under some additional assumptions, we obtain the equations that are the steady-state conditions for an action functional considered in a recent paper by Faddeev and Niemi.