  We consider the imaginary time flow of a quadratic hyperbolic Hamiltonian on
the symplectic plane, apply it to the Schr\"odinger polarization and study the
corresponding evolution of polarized sections. The flow is periodic in
imaginary time and the evolution of polarized sections has interesting
features. On the time intervals for which the polarization is real or K\"ahler,
the half--form corrected time evolution of polarized sections is given by
unitary operators which turn out to be equivalent to the classical
Segal-Bargmann transforms (which are usually associated to the quadratic
elliptic Hamiltonian $H=\frac12 p^2$ and to the heat operator). At the right
endpoint of these intervals, the evolution of polarized sections is given by
the Fourier transform from the Schr\"odinger to the momentum representation. In
the complementary intervals of imaginary time, the polarizations are
anti--K\"ahler and the Hilbert space of polarized sections collapses to
${\mathcal H}= \{0\}$.
  Hyperbolic quadratic Hamiltonians thus give rise to a new factorization of
the Segal-Bargmann transform, which is very different from the usual one, where
one first applies a bounded contraction operator (the heat kernel operator),
mapping $L^2$--states to real analytic functions with unique analytic
continuation, and then one applies analytic continuation. In the factorization
induced by an hyperbolic complexifier, both factors are unbounded operators but
their composition is, in the K\"ahler or real sectors, unitary.
  In another paper [KMNT], we explore the application of the above family of
unitary transforms to the definition of new holomorphic fractional Fourier
transforms.
