  The quantum electromagnetic dielectric tensor for a multi species plasma is
re-derived from the gauge invariant Wigner-Maxwell system and presented under a
form very similar to the classical one. The resulting expression is then
applied to a quantum kinetic theory of the electromagnetic filamentation
instability. Comparison is made with the quantum fluid theory including a Bohm
pressure term, and with the cold classical plasma result. A number of
analytical expressions are derived for the cutoff wave vector, the largest
growth rate and the most unstable wave vector.
