Constrained Hamiltonian approach to the phase space of the coupled cluster method

The coupled cluster method (CCM) introduces a linked-cluster parametrization for the many-body wave function. The method can be formulated through an action principle, which introduces a symplectic structure into the CCM phase space. We study the phase spaces of the normal and extended coupled cluster methods (NCCM and ECCM, respectively) with a special emphasis on the ensuing characteristic non-Hermiticity of the action and the average-value functionals. By introducing the complex conjugates of the usual CCM cluster amplitudes the phase space is enlarged into a genuine complex manifold. The extra degrees of freedom are then eliminated by the Dirac bracket method, leaving as independent coordinates a minimal set of complex conjugate (eventually, real) coordinate functions. The constraint functions performing this elimination are shown to be of second class. The induced symplectic structure in the physical submanifold (or the constraint surface) is derived, and the physical shell itself found to be a Kahler manifold. Furthermore, the significance of the CCM star product in generating the average-value functionals of compound operators is emphasized. It is proven that the star product can also be defined on the constraint surface in both the NCCM and ECCM cases. The results suggest that Hermiticity-preserving truncation schemes based on the on-shell star product may become feasible.