Dimensionless physics: Planck constant as an element of Minkowski metric

TL;DR

This paper explores the idea that the Planck constant may be interpreted as a parameter of the Minkowski metric, suggesting it has a length dimension and proposing implications for quantum gravity and vacuum states.

Contribution

It extends Diakonov's quantum gravity theory by proposing the Planck constant as a Minkowski metric parameter with length dimension, affecting invariance properties and vacuum relations.

Findings

Planck constant may have length dimension as a Minkowski metric parameter

Diffeomorphism invariant quantities are dimensionless under this framework

Different Minkowski vacua could have distinct Planck constant values and related thermal laws

Abstract

Diakonov theory of quantum gravity, in which tetrads emerge as the bilinear combinations of the fermionis fields,\cite{Diakonov2011} suggests that in general relativity the metric may have dimension 2, i.e. $[g_{\mu\nu}]=1/[L]^2$. Several other approaches to quantum gravity, including the model of superplastic vacuum and $BF$-theories of gravity support this suggesuion. 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 $\Lambda$, scalar curvature $R$, scalar field $\Phi$, 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 nonzero dimension -- the dimension of length [L]. 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: $\hbar_1/T_1= \hbar_2/T_2$.