Discrete Z4 Symmetry in the Lattice Theory of Gravity: Symmetric and Asymmetric Phases

The lattice theory of gravity at ultrahigh temperatures is discussed. The global Z 4 transformation of Dirac fermionic and bosonic fields (tetrad and holonomy elements) is determined, which are used to construct the lattice action. This Z 4 transformation is the “root” of the Z 2 or PT transformation I have previously proposed. At the highest temperatures, the action in question is invariant under the Z 4 transformation, and the corresponding symmetry is not broken. As the temperature decreases, the Z 4 symmetry is broken to the Z 2 symmetry, and the order parameter is the Z 2 -symmetric contribution to the action, which is transformed in the long-wavelength limit into the Hilbert–Einstein action. The Z 4 transformation mixes particles and antiparticles, so that the difference between particles and antiparticles in the Z 4 -symmetric phase disappears (in terms of Dirac fields).