Ju n 20 08 Third - quantized functional Green ’ s functions for the Hamiltonian constraint in Chang – Soo variables

In this paper we compute the main ingredients necessary to construct the Hamiltonian part of the propagator for the third quantized theory stemming arising from the quantum constraints of general rela-tivity in the Chang–Soo variables. This necessitates the use of momentum space methods on the infinite dimensional functional space of the configuration variables. It is found that a naive inversion of all nine degrees of freedom in the momentum space conjugate to the Chang– Soo variables produces a divergent result. As a necessary condition for a finite state of quantum gravity in the full theory, a contraint is imposed which separates the unphysical from the physical degrees of freedom, making the result convergent leading to the possibility of a finite state. It is envisioned that the requirement to produce con-vergent functional Green's functions is relevant to the issue of reality conditions in quantum gravity.