Planck Constants in the Symmetry Breaking Quantum Gravity

We consider the theory of quantum gravity in which gravity emerges as a result of the symmetry-breaking transition in the quantum vacuum. The gravitational tetrads, which play the role of the order parameter in this transition, are represented by the bilinear combinations of the fermionic fields. In this quantum gravity scenario the interval ds in the emergent general relativity is dimensionless. Several other approaches to quantum gravity, including the model of superplastic vacuum and BF theories of gravity support this suggestion. The important consequence of such metric dimension is that all the diffeomorphism invariant quantities are dimensionless for any dimension of spacetime. These include the action S, cosmological constant Lambda, scalar curvature R, scalar field Phi, wave function psi, etc. The composite fermion approach to quantum gravity suggests that the Planck constant (h) over bar can be the parameter of the Minkowski metric. Here, we extend this suggestion by introducing two Planck constants, bar (h) over bar and slash h, which are the parameters of the correspondingly time component and space component of the Minkowski metric, g(Mink)(mu nu) = diag(h2, /h2, /h2, /h2). The parameters bar (h) over bar and slash /h are invariant only under SO(3) transformations, and, thus, they are not diffeomorphism invariant. As a result they have non-zero dimensions-the dimension of time for h and dimension of length for /h. Then, according to the Weinberg criterion, these parameters are not fundamental and may vary. In particular, they may depend on the Hubble parameter in the expanding Universe. They also change sign at the topological domain walls resulting from the symmetry breaking.