On the geometry of coherent state maps

Given a mechanical system whose phase space $\mathfrak{M}$ is equipped with a complex structure J, and a Hermitian line bundle $(E, H) \to \mathfrak{M}$, a coherent state map is an anti-holomorphic embedding $\mathscr{K} \to \mathbb{CP} (\mathfrk{M})$ built in terms of $(J, H)$, with $\mathfrak{M} = H^0 \Big( \mathfrak{M}, L^2 \mathscr{O} \big( {T^\ast}^{(n,0)} (\mathfrak{M}) \otimes E \big) \Big)$, such that for any pair of classical states $z, \zeta \in \mathfrak{M}$ the number $\langle \mathscr{K} (z), \mathscr{K}(\zeta ) \rangle$ is the transition probability amplitude from the coherent state $\mathscr{K} (z)$ to $\mathscr{K}(\zeta )$. We examine three related questions, as follows: (i) We generalize Lichnerowicz’s theorem (on ± holomorphic maps of finite-dimensional compact Kählerian manifolds) to describe anti-holomorphic maps $\mathscr{K} : \mathfrak{M} \to \mathbb{CP} ( \mathfrak{M})$ as harmonic maps that are absolute minima within their homotopy classes. (ii) If the phase space is a domain $\mathfrak{M} = \Omega \subset \mathbb{C}^n$ and $E \to \Omega$ is a trivial Hermitian line bundle such that $\gamma = H(\sigma_0, \sigma_0) \in AW(\Omega)$ (i.e., $\gamma$ is an admissible weight), we discuss the use of $K_\gamma (z, \zeta )$ [the $\gamma$-weighted Bergman kernel of $\Omega$] vis-a-vis to the calculation of the transition probability amplitudes, focusing on the case where $\Omega = \Omega_n$ is the Siegel domain and $\gamma (z) = \gamma_a (z) = \big( Im (z_n) - |z^\prime|^2 \big)^a$, $a > -1$. (iii) We study the boundary behavior of a coherent state map $\mathscr{K}: \Omega \to \mathbb{CP}[L^2 H(\Omega_n , \gamma_a )]$.