Summary: Given a mechanical system whose phase space \({\mathfrak{M}}^n\) 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 \(\mathscr{K} : {\mathfrak{M}} \to \mathbb{C}\mathbb{P} \big ({\mathscr M} \big)\) built in terms of \((J, H)\), with \(\mathscr{M} = H^0 \Big ({\mathfrak{M}}, L^2 \mathscr{O} \big (T^{\ast (n, 0)} ({\mathfrak{M}}) \otimes E \big) \Big)\), 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 \(\pm\) holomorphic maps of finite-dimensional compact Kählerian manifolds) to describe anti-holomorphic maps \({\mathscr K} : {\mathfrak{M}} \to \mathbb{C}\mathbb{P} ({\mathscr M})\) 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 \(\gamma = H \big (\sigma_0, \sigma_0 \big) \in AW(\Omega)\) (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 \(\gamma (z) = \gamma_a (z) = \big (\operatorname{Im} (z_n) - |z^\prime |^2 \big)^a, a > - 1\). (iii) We study the boundary behavior of a coherent state map \({\mathscr K} : \Omega \to \mathbb{C}\mathbb{P} \big [ L^2 H (\Omega_n, \gamma_a) \big ]\).
For the entire collection see [Zbl 07882604].