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

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月1日

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  • S. N. Vergeles

摘要

The lattice theory of gravity at ultrahigh temperatures is discussed. The global Z4 transformation of Dirac fermionic and bosonic fields (tetrad and holonomy elements) is determined, which are used to construct the lattice action. This Z4 transformation is the “root” of the Z2 or PT transformation I have previously proposed. At the highest temperatures, the action in question is invariant under the Z4 transformation, and the corresponding symmetry is not broken. As the temperature decreases, the Z4 symmetry is broken to the Z2 symmetry, and the order parameter is the Z2-symmetric contribution to the action, which is transformed in the long-wavelength limit into the Hilbert–Einstein action. The Z4 transformation mixes particles and antiparticles, so that the difference between particles and antiparticles in the Z4-symmetric phase disappears (in terms of Dirac fields).