Discrete Z₄ Symmetry in Quantum Gravity

We consider the discrete (Formula presented.), which takes place in the scenario of quantum gravity where the gravitational tetrads emerge as the order parameter—the vacuum expectation value of the bilinear combination of fermionic operators. Under this symmetry operation, (Formula presented.), the emerging tetrads are multiplied by the imaginary unit, (Formula presented.). The existence of such symmetry and the spontaneous breaking of this symmetry are also supported by the consideration of the symmetry breaking scheme in the topological superfluid ³He-B. The order parameter in ³He-B is also the bilinear combination of the fermionic operators. This order parameter is the analog of the tetrad field, but it has complex values. The (Formula presented.) -symmetry operation changes the phase of the complex order parameter by (Formula presented.), which corresponds to the (Formula presented.) discrete symmetry in quantum gravity. We also considered the alternative scenario of the breaking of this (Formula presented.) symmetry, in which the (Formula presented.) -operation changes sign of the scalar curvature, (Formula presented.), and thus the Einstein–Hilbert action violates the (Formula presented.) -symmetry. In the alternative scenario of symmetry breaking, the gravitational coupling (Formula presented.) plays the role of the order parameter, which changes sign under (Formula presented.) -transformation.