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 Pis'ma v Zh. Èksper. Teoret. Fiz., 2009, Volume 90, Issue 11, Pages 793–799 (Mi jetpl596)

GRAVITY, ASTROPHYSICS

$\hbar$ as parameter of Minkowski metric in effective theory

G. E. Volovikab

a Low Temperature Laboratory, Helsinki University of Technology, FIN-02015 HUT, Finland
b Landau Institute for Theoretical Physics RAS

Abstract: With the proper choice of the dimensionality of the metric components and matter field variables, the action for all fields becomes dimensionless. Such quantities as the vacuum speed of light $c$, the Planck constant $\hbar$, the electric charge $e$, the particle mass $m$, the Newton constant $G$ never enter equations written in the covariant form, i.e., via the metric $g^{\mu\nu}$. The speed of light $c$ and the Planck constant $\hbar$ are parameters of a particular two-parametric family of solutions of general relativity equations describing the flat isotropic Minkowski vacuum in effective theory emerging at low energy: $g^{\mu\nu}_\mathrm{Minkowski}=\mathrm{diag}(-\hbar^2,(\hbar c)^2,(\hbar c)^2,(\hbar c)^2)$. They parametrize the equilibrium quantum vacuum state. The physical quantities which enter the covariant equations are dimensionless quantities and quantities which have dimension of rest energy $M$ or its power. Dimensionless quantities include the running coupling ‘constants’ $\alpha_i$; the geometric $\theta$-parameters which enter topological terms in action; and geometric charges coming from the group theory, such as angular momentum quantum number $j$, weak charge, electric charge $q$, hypercharge, baryonic and leptonic charges, number of atoms $N$, etc. Dimensionful parameters are mass matrices with dimension of $M$; gravitational coupling $K$ with $[K]=[M]^2$; cosmological constant with dimension $M^4$; etc. In effective theory, the interval $s$ has the dimension of $1/M$; it characterizes dynamics of particles in quantum vacuum rather than space-time geometry. The action is dimensionless reflecting equivalence between action and the phase of a wave function in quantum mechanics. We discuss the effective action, and the measured physical quantities including parameters of metrology triangle.

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English version:
Journal of Experimental and Theoretical Physics Letters, 2009, 90:11, 697–704

Bibliographic databases:

PACS: 03.65.-w, 04.20.-q, 05.20.Jr
Revised: 09.11.2009
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Citation: G. E. Volovik, “$\hbar$ as parameter of Minkowski metric in effective theory”, Pis'ma v Zh. Èksper. Teoret. Fiz., 90:11 (2009), 793–799; JETP Letters, 90:11 (2009), 697–704

Citation in format AMSBIB
\Bibitem{Vol09} \by G.~E.~Volovik \paper $\hbar$ as parameter of Minkowski metric in effective theory \jour Pis'ma v Zh. \Eksper. Teoret. Fiz. \yr 2009 \vol 90 \issue 11 \pages 793--799 \mathnet{http://mi.mathnet.ru/jetpl596} \transl \jour JETP Letters \yr 2009 \vol 90 \issue 11 \pages 697--704 \crossref{https://doi.org/10.1134/S0021364009230027} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000275104100002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77949418957} `

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2. Uzan J.-Ph., Living Reviews in Relativity, 14 (2011), 2
3. Brodsky S.J., Hoyer P., Phys Rev D, 83:4 (2011), 045026
4. Acosta D., Fernandez De Cordoba P., Isidro J.M., Santander J.L.G., Int. J. Geom. Methods Mod. Phys., 9:5 (2012), 1250048
5. De Raedt H., Katsnelson M.I., Michielsen K., Ann. Phys., 347 (2014), 45–73
6. De Raedt H., Katsnelson M.I., Donker H.C., Michielsen K., Ann. Phys., 359 (2015), 166–186
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