On the Geometry of Coherent State Maps

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 \({\mathscr K} : \Omega \to \mathbb {C}\mathbb {P} \big [ L^2 H (\Omega _n \, , \, \gamma _a ) \big ]\) .