Dimensionless Physics: Planck Constant as an Element of the Minkowski Metric

Abstract

Diakonov theory of quantum gravity, in which tetrads emerge as the bilinear combinations of the fermionic fields, suggests that in general relativity the metric may have dimension 2; i.e., [{{g}_{{mu nu }}}] = 1{text{/}}{{[L]}^{2}}. 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, interval s, cosmological constant Λ, scalar curvature R, scalar field Φ, etc. Here we are trying to further exploit the Diakonov idea, and consider the dimension of the Planck constant. The application of the Diakonov theory suggests that the Planck constant hbar is the parameter of the Minkowski metric. The Minkowski parameter hbar is invariant only under Lorentz transformations, and is not diffeomorphism invariant. As a result, the Planck constant hbar has the dimension of length. Whether this Planck constant length is related to the Planck length scale, is an open question. In principle there can be different Minkowski vacua with their own values of the parameter hbar . Then in the thermal contact between the two vacua their temperatures obey the analog of the Tolman law: {{hbar }_{1}}{text{/}}{{T}_{1}} = {{hbar }_{2}}{text{/}}{{T}_{2}}.