Quantum kinetic equation in the closed-time-path formalism
A systematic derivation of the quantum kinetic equation is presented in the framework of a closed-time-path formalism.  Introducing a probe, the expectation value of the number operator is calculated as a functional of the probing source.  Then, solving for the source by inverting the relation, the removal of the source leads to the quantum kinetic equation as the equation of motion for the number, which gives a generalization of the Boltzmann equation including memory.  The inversion formula is used in the course of the derivation.  The calculation is presented up to third order in interaction, and the effect of initial correlations is also considered.