From the introduction: “Polar decompositions provide a general method to construct isometries between Hilbert spaces. By performing a polar decomposition of some extremely simple operator, we obtain a radial part which is a heat semigroup, and a phase part which is an isometrical integral transform. The coherent states arise naturally as the kernels of the isometry. We illustrate this point by several examples: the Bargmann transform, Hall’s coherent state transforms, and the Weyl correspondence. All these are connected with coherent states”.