Complex Extensions of Canonical Transformations and Quantum Mechanics

Abstract

Canonical transformations have played a fundamental role in the understanding and solution of problems of classical mechanics; however, in quantum mechanics, the main applications have concerned groups of point transformations, which involve the coordinates. This chapter discusses the three-body problem showing that potential and kinetic energy can be obtained from Gaussian expressions in the coordinates and momenta. The latter can be visualized as the bounded representations of complex extensions of canonical transformations. The permutations of the particles required to satisfy the Pauli principle are point transformations. The matrix elements to be evaluated are bounded representations of complex canonical transformations. The three-body system provides the simplest example for the clustering problem. Clustered states are defined by introducing Gaussian internal states for sets of nucleons. These states might be characterized by choosing the identity representation of a unitary group acting on the internal coordinates and momenta.