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# Dimensionless physics: Planck constant as an element of Minkowski metric

Sep 2022

Several approaches to quantum gravity (including the model of superplasticvacuum; Diakonov tetrads emerging as the bilinear combinations of the fermionisfields; and $BF$-theories of gravity) suggest that in general relativity themetric may have dimension 2, i.e. $[g_{\mu\nu}]=1/[L]^2$. The importantconsequence of such metric dimension is that all the diffeomorphism invariantquantities are dimensionless for any dimension of spacetime. These include theaction $S$, interval $s$, Newton constant $G$, cosmological constant $\Lambda$,scalar curvature $R$, scalar field $\Phi$, etc. Another consequence of themetric dimension $1/[L]^2$ is that the Planck constant $\hbar$ is the elementof the Minkowski metric. The Minkowski parameter $\hbar$ is invariant onlyunder Lorentz transformations, and is not diffeomorphism invariant. As a resultthe 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 withtheir own values of the parameter $\hbar$. Then in the thermal contact betweenthe two vacua their temperatures obey the analog of the Tolman law:$\hbar_1/T_1= \hbar_2/T_2$.

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