  Our purpose is to show the existence of a Calabi-Yau structure on the
punctured cotangent bundle $T^{*}_{0}(P^2\mathbb{O})$ of the Cayley projective
plane $P^{2}\mathbb{O}$ and to construct a Bargmann type transformation from a
space of holomorphic functions on $T^{*}_{0}(P^2\mathbb{O})$ to $L_{2}$-space
on $P^{2}\mathbb{O}$. The space of holomorphic functions corresponds to the
Fock space in the case of the original Bargmann transformation. A K\"ahler
structure on $T^{*}_{0}(P^{2}\mathbb{O})$ was shown by identifying it with a
quadrics in the complex space $\mathbb{C}^{27}\backslash\{0\}$ and the natural
symplectic form of the cotangent bundle $T^{*}_{0}(P^2\mathbb{O})$ is expressed
as a K\"ahler form. Our method to construct the transformation is the pairing
of polarizations, one is the natural Lagrangian foliation given by the
projection map ${\bf q}:T^{*}_{0}(P^2\mathbb{O})\longrightarrow
P^{2}\mathbb{O}$ and the polarization given by the K\"ahler structure.
  The transformation gives a quantization of the geodesic flow in terms of one
parameter group of elliptic Fourier integral operators whose canonical
relations are defined by the graph of the geodesic flow action at each time. It
turn out that for the Cayley projective plane the results are not same with
other cases of the original Bargmann transformation for Euclidean space,
spheres and other projective spaces.
