  Starting from the recently proposed dynamical exchange-correlation field
framework, the equation of motion of the diagonal part of the many-electron
Green function is derived, from which the spectral function can be obtained.
The resulting equation of motion takes the form of the continuity equation of
charge and current densities in electrodynamics with a source. An unknown
quantity in this equation is the current density, corresponding to the kinetic
energy. A procedure \`a la Kohn-Sham scheme is then proposed, in which the
difference between the kinetic potential of the interacting system and the
non-interacting Kohn-Sham system is shifted into the exchange-correlation
field. The task of finding a good approximation for the exchange-correlation
field should be greatly simplified since only the diagonal part is needed. A
formal solution to the continuity equation provides an explicit expression for
calculating the spectral function, given an approximate exchange-correlation
field.
