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.
Categories: math.DG