CPTM symmetry, closed time paths and cosmological constant problem in the formalism of extended manifold

The problem of the cosmological constant is considered in the formalism of an extended space-time consisting of the extended classical solution of Einstein equations. The different regions of the extended manifold are proposed to be related by the charge, parity, time and mass (CPTM) reversal symmetry applied with respect to the metric fields of the manifolds. There are interactions between the points of the extended manifold provided by scalar fields present separately in the different patches of the extended solution. The value of the constant is obtained equal to zero at the classical level due the mutual contribution of the fields in the vacuum energy, it’s non-zero value is due the quantum interactions between the fields. There are few possible scenario for the actions of the fields are discussed. Each from the obtained variants is similar to the closed time path approach of non-equilibrium condensed matter physics and among these possibilities for the closed paths, there is a variant of the action equivalent to the formalism of Keldysh. Accordingly, we consider and shortly discuss the application of the proposed formalism to the problem of smallness of the cosmological constant and singularities problem.