  We formulate a complex action theory which includes operators of coordinate
and momentum $\hat{q}$ and $\hat{p}$ being replaced with non-hermitian
operators $\hat{q}_{new}$ and $\hat{p}_{new}$, and their eigenstates ${}_m
<_{new} q |$ and ${}_m <_{new} p |$ with complex eigenvalues $q$ and $p$.
Introducing a philosophy of keeping the analyticity in path integration
variables, we define a modified set of complex conjugate, real and imaginary
parts, hermitian conjugates and bras, and explicitly construct $\hat{q}_{new}$,
$\hat{p}_{new}$, ${}_m <_{new} q |$ and ${}_m <_{new} p |$ by formally
squeezing coherent states. We also pose a theorem on the relation between
functions on the phase space and the corresponding operators. Only in our
formalism can we describe a complex action theory or a real action theory with
complex saddle points in the tunneling effect etc. in terms of bras and kets in
the functional integral. Furthermore, in a system with a non-hermitian
diagonalizable bounded Hamiltonian, we show that the mechanism to obtain a
hermitian Hamiltonian after a long time development proposed in our letter
works also in the complex coordinate formalism. If the hermitian Hamiltonian is
given in a local form, a conserved probability current density can be
constructed with two kinds of wave functions.
