
VOLUME 90 (2009)  ISSUE 11 
PAGE 793

as parameter of Minkowski metric in effective theory
G. E. Volovik
Low Temperature Laboratory, Helsinki University of Technology, FIN02015 HUT, Finland Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia
PACS: 03.65.w, 04.20.q, 05.20.Jr
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 , 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^{μν}. The speed of light c and the Planck constant
are parameters of a particular twoparametric family of solutions of
general relativity equations describing the flat isotropic Minkowski vacuum
in effective theory emerging at low energy:
.
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'
α_{i}; the geometric θparameters 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 spacetime 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.

