  We present a general analytic method for evaluating the generally
time-dependent pointer states of a subsystem, which are defined by their
capability not to entangle with the states of another subsystem. In this way,
we show how in practice the global state of the system and the environment may
evolve into a diagonal state as a result of the natural evolution of the total
composite system. We explore the conditions under which the pointer states of
the system become independent of time; so that a preferred basis of measurement
can be realized. As we show, these conditions include the so-called quantum
limit of decoherence and the so-called quantum measurement limit; as well as
some other specific conditions which are discussed in the paper. We relate the
mathematical conditions for having time-independent pointer states to some
classes of possible symmetries in the Hamiltonian of the total composite
system. Indeed, our theory would serve as a generalization of the existing
theory for determination of the preferred basis of measurement. By exploiting
this new theory we can obtain those regimes of the parameter space for a given
total Hamiltonian defining our system-environment model for which a preferred
basis of measurement can be realized. Moreover, we can predict the
corresponding preferred basis of measurement for each regime. We can also
obtain the time-dependent pointer states of the system and the environment in
most of the other regimes where the pointer states of the system are
time-dependent and a preferred basis of measurement cannot be realized at all.
This ability to obtain time-dependent pointer states is specifically important
in decoherence studies; as pointer states correspond to those initial
conditions for the state of the system and the environment for which we can
have longer decoherence times.
