Towards a kinetic theory for fermions with quantum coherence
  A new density matrix and corresponding quantum kinetic equations are introduced for fermions undergoing coherent evolution either in time (coherent particle production) or in space (quantum reflection). A central element in our derivation is finding new spectral solutions for the 2-point Green's functions written in the Wigner representation, that are carrying the information of the quantum coherence. Physically observable density matrix is then defined from the bare singular 2-point function by convoluting it with the extrenous information about the state of the system. The formalism is shown to reproduce familiar results from the Dirac equation approach, like Klein problem and nonlocal reflection from a mass wall. The notion of the particle number in the presence of quantum coherence is shown to be particularily transparent in the current picture. We extend the formalism to the case of mixing fields and show how the usual flavour mixing and oscillation of neutrinos emerges again from a singular shell structure. Finally, we show how the formalism can be extended to include decohering interactions. 
Matti Herranen, Kimmo Kainulainen and Pyry Matti Rahkila