Planck Constants in the Symmetry Breaking Quantum Gravity

Abstract

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 (Formula presented.) in the emergent general relativity is dimensionless. Several other approaches to quantum gravity, including the model of superplastic vacuum and (Formula presented.) 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 (Formula presented.), scalar curvature R, scalar field (Formula presented.), wave function (Formula presented.), etc. The composite fermion approach to quantum gravity suggests that the Planck constant ℏ can be the parameter of the Minkowski metric. Here, we extend this suggestion by introducing two Planck constants, bar ℏ and slash (Formula presented.), which are the parameters of the correspondingly time component and space component of the Minkowski metric, (Formula presented.). The parameters bar ℏ and slash (Formula presented.) are invariant only under (Formula presented.) transformations, and, thus, they are not diffeomorphism invariant. As a result they have non-zero dimensions—the dimension of time for ℏ and dimension of length for (Formula presented.). 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.