  For a group of charged particles obeying quantum mechanics interacting with
an electromagnetic field, the charge and current density in a pure state of the
system are expressed with the many-body wave function of the state. Using these
as sources, the microscopic Maxwell equations can be written down for any given
pure state of a many-body system. By employing semi-classical radiation theory
with these sources, the microscopic Maxwell equations can be used to compute
the strong radiation fields produced by interacting charged quantal particles.
For a charged quantal particle, three radiation fields involve only the vector
potential $\mathbf{A}$. This is another example demonstrating the observability
of vector potential. Five radiation fields are perpendicular to the canonical
momentum of a single charged particle. For a group of charged particles, a new
type of radiation field is predicted to be perpendicular to
$\mathbf{A}(\mathbf{x}_{j},t)\times
\lbrack\nabla\times(\nabla_{j}\Psi^{\prime})]$, where $\Psi^{'}$ is the
many-body wave function. This kind of radiation does not exist for a single
charged particle. The macroscopic Maxwell equations are derived from the
corresponding microscopic equations for a pure state by the Russakoff-Robinson
procedure which requires only a spatial coarse graining. Because the sources of
fields are also the responses of a system to an external field, one also has to
give up the temporal coarse graining of the current density which is often
supposed to be critical in the kinetic approach of conductivity.
