  Let $P$ be a Delzant polytope. We show that the quantization of the
corresponding toric manifold $X_{P}$ in toric K\"ahler polarizations and in the
toric real polarization are related by analytic continuation of Hamiltonian
flows evaluated at time $t = \sqrt{-1} s$. We relate the quantization of
$X_{P}$ in two different toric K\"ahler polarizations by taking the
time-$\sqrt{-1} s$ Hamiltonian "flow" of strongly convex functions on the
moment polytope $P$. By taking $s$ to infinity, we obtain the quantization of
$X_{P}$ in the (singular) real toric polarization.
  Recall that $X_{P}$ has an open dense subset which is biholomorphic to
$({\mathbb{C}}^{*})^{n}$. The quantization of $X_{P}$ in a toric K\"ahler
polarization can also be described by applying the complexified Hamiltonian
flow of the Abreu--Guillemin symplectic potential $g$, at time $t=\sqrt{-1}$,
to an appropriate finite-dimensional subspace of quantum states in the
quantization of $T^{*}{\mathbb{T}}^{n}$ in the vertical polarization. By taking
other imaginary times, $t= k \sqrt{-1}, k\in {\mathbb{R}}$, we describe toric
K\"ahler metrics with cone singularities along the toric divisors in $X_{P}$.
  For convex Hamiltonian functions and sufficiently negative imaginary part of
the complex time, we obtain degenerate K\"ahler structures which are negative
definite in some regions of $X_{P}$. We show that the pointwise and $L^2$-norms
of quantum states are asymptotically vanishing on negative-definite regions.
