Canonical transforms. II. Complex radial transforms

Continuing the line of development of Paper I [J. Math. Phys. 15, 1295 (1974)], we enlarge the concept of canonical transformations in quantum mechanics in two directions: first, by allowing the definition of a canonical transformation to be made through the preservation of an so(2,1) algebra, rather than the usual Heisenberg algebra, and providing the bridge between the classical and quantum mechanical descriptions, and, second, through the complexification of the transformation group. In this paper we study specifically the transformations which can be interpreted as the radial part of n‐dimensional complex linear transformations in Paper I. We show that we can build Hilbert spaces of analytic functions with a scalar product defined through integration over half the complex plane of a variable which has the meaning of a complex radius. A unitary mapping to the ordinary Hilbert space ${\mathcal{L}}_{r^{\text{n−1}}}^2\text{(0,∞)}$ is provided with a kernel involving a Bessel function. Special cases of this are shown to be the Barut‐Girardello, one‐dimensional Bargmann and Hankel transforms. The transform kernels provide a series of representations of a subsemigroup of $\text{SL(2,}\mathbb{C})$ and allow the construction of coherent states for the harmonic oscillator with an extra centrifugal force. We present a hyperdifferential operator realization of these transforms which yields new Baker‐Campbell‐Hausdorff and special function relations.