01-214 Carlos Villegas-Blas

The Bargmann Transform and Canonical Transformations (413K, Postscript) Jun 11, 01

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Abstract. This paper is about a relationship between the kernel of the Bargmann transform (BT) and the corresponding canonical transformation. We study this fact for a BT (introduced by L. Thomas and S. Wassell) when the configuration space is the 2-sphere $\EST$ and for a BT that we introduce for $\ESTR$. It is shown that the kernel of the BT is a power series of a function which is a generating function of the corresponding canonical transformation (CT) (a classical analog of the BT). We show in each case that our CT is a composition of other two CT involving the complex quadric in ${\bf C}^3$ or ${\bf C}^4$. We also study the quantizations of those other two CT by dealing with spaces of holomorphic functions on the mentioned quadric. We show reproducing kernels for those spaces. We also relate some of the quantizations with work of V. Bargmann$^{20}$ and V. Guillemin$^{21}$. Since powers of the generating functions are coherent states for $L^2(\EST)$ or $L^2(\ESTR)$, we finally show that the studied BT are actually coherent states transforms.

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