  It is well-known that the pointer basis of a quantum system satisfies the
condition to diagonalize the interaction Hamiltonian between the subsystems. We
show that this condition can be translated into the form $\delta\Lambda=0,$
where $\Lambda$, so-called the action, is the time integrated interaction
energy: it is found out naturally in the phase of state vectors due to diagonal
interaction. The careful treatment of a two states system demonstrates that the
states of the total system branch into the states with different values of the
action. Mathematically the pointer states are selected out by the saddle point
condition on the phase $\Lambda$. This study helps us to understand the precise
mechanism and the general dynamics of decoherence.
