CPT symmetry in the mirror universe

A model of a two-sheeted universe in the quantum theory of gravity is proposed, based on the definition of 3D invariant and gauge-invariant proper time of the universe. A uniform time in a closed universe is introduced in the class of equivalent reference systems defined by the spectrum of a Hermitian 3D operator on the space of Dirac bi-spinors, the equality to zero of which is equivalent to a system of gravitational constraints. Based on the analogy with the Lorentz-invariant quantum field theory in Minkowski space and the definition of discrete C-, P- and T-transformations separately, the principle of CPT symmetry in a two-sheeted universe is formulated. A covariant quantum theory of matter gauge fields consistent with the charge conjugation operation in a two-sheeted universe is proposed. The CPT symmetric state of a homogeneous isotropic two-sheeted model of the universe is constructed by an iterative method in which the no-boundary Hartle--Hawking wave function is used as a zeroth approximation.