  In quantum theory its action is usually taken to be real, but we can consider
another theory whose action is complex. In addition, in the Feynman path
integral, the time integration is usually performed over the period between the
initial time $T_A$ and some specific time, say, the present time $t$. Besides
such a future-not-included theory, we can consider the future-included theory,
in which not only the past state $| A(T_A) \rangle$ at the initial time $T_A$
but also the future state $| B(T_B) \rangle$ at the final time $T_B$ is given
at first, and the time integration is performed over the whole period from the
past to the future. Thus quantum theory can be classified into four types,
according to whether its action is real or not, and whether the future is
included or not. We argue that, if a theory is described with a complex action,
then such a theory is suggested to be the future-included theory, rather than
the future-not-included theory. Otherwise persons living at different times
would see different histories of the universe.
