Brazil is experimenting an increase of scientific production. Today, I am going to explain this brazilian paper http://arxiv.org/abs/0711.4276v2 concerning the number of fundamental constants.
The Okun cube of fundamental constant, firstly introduced by Gamov, Ivanenko and Landau, has raised some questions about what we should consider as fundamental “unit” we already had but now with more intensity. I mentioned the trialogue of fundamental constants between Veneziano, Duff and Okun himself more than a decade ago. Veneziano argued that 2 fundamental constants were well enought to fix everything. However, it is not the “accepted” and “more popular” approach in these days, but the brazilian paper about defends such a claim!
What do they claim? They basically argue that what we need is a convention for space and time measurements and nothing else. Specifically, they say that every physical observable 
with 
can be expressed as follows:
(1) 
and where 
are pure dimensionless numbers, while 
denote “basic units” of space and time. We could argue that these two last “fundamental units” of “space and time” were “quanta” of “space” and “time”, the mythical “choraons” and “chronons” some speculative theories of Quantum Gravity seem to suggest, but it would be another different story not related to this post!
After introducing the above statement, they discuss 2 procedures to measure with clocks and rulers, what they call 
-protocol and 
-protocol. They begin assuming some quantity in the CGS system (note that the idea is completely general and they could use the MKSA or any other traditional system of units):
(2)
![\[ \boxed{\mathcal{D}_i= \Delta_i T^{\alpha_i}L^{\beta_i}M^{\gamma_i}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-04adf0fc84616f7202e5a3fbf89ad21b_l3.png "Rendered by QuickLaTeX.com")
where 
are dimensionless constants. And then, the 2 protocols are defined:
1st. G-protocol. Multiply the above equation (2) by 
and identify with 
or 
. Rewriting all the physical quantities and laws in terms of this protocol in terms of instead 
we gain some bonuses:
i) The unit M from CGS “vanishes” or is “erased” from physical observables.
ii) G disappear from every physical law.
iii) Masses being measured in 
imply that from Newton’s gravitational law 
we deduce that
![\[ \boxed{g=-m^{(G)}/d^2}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-97b150b839284b63474bb9304f91cc91_l3.png "Rendered by QuickLaTeX.com")
where 
are units with physical dimension 
. 
, the gravitational constant, is some kind of conversion factor between mass and “volume acceleration” 
. This G-protocol applied to the Planck constant provides
![\[ \boxed{h^G=hG}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-b56862df1b08b2d5e3f50c4a87f46329_l3.png "Rendered by QuickLaTeX.com")
and it has dimensions of 
.
2nd. h-protocol. From equation (2), if we divide by 
and we identity 
with 
we get the so-called h-protocol. The consequences are:
i) M units disappear from physical laws and quantities, as before.
ii) h is erased and vanishes from every equation, law and quantity.
iii) Masses are measured in units of 
, e.g., from the Compton equation we get in the h-protocol
![\[ \Delta \lambda= \dfrac{1}{m^{(h)}c}\left(1-\cos\theta\right)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-59a8efd1e09c7019c6f143545558ce9f_l3.png "Rendered by QuickLaTeX.com")
and where 
are units of mass in the h-protocol with dimensions 
. Therefore, h is the conversion factor between inverse areolar velocity and mass 
. In this protocol the inverse of the Compton length measures “inertia”, and indeed this fact fits with some recent proposals to determine a definition of kg independent from the old MKSA pattern (the famous iridium “thing”, which is know now not to have a 1 kg mass). Moreover, we also get that
![\[ G^{(h)}=Gh\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-96fc87d67d9fb04ff086cbb4d3a487cc_l3.png "Rendered by QuickLaTeX.com")
and
![\[ G^{(h)}=h^{(G)}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-c3bfa538f3ab8703b30f5922dabe1a30_l3.png "Rendered by QuickLaTeX.com")
The two protocols can be summarized in a nice table
They also derive the mysterious relations between charge and mass that we saw in the previous post about Pavšič units, i.e., they also derive
![\[ e^{(G)}=2\cdot 10^{21}m_e^{(G)}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-a43a20e33c18b3b261c83d65ec24db39_l3.png "Rendered by QuickLaTeX.com")
and it is equivalent to 
. Somehow, and electron is more electrical/capacitive than gravitational/elastic!
Finally, in their conclusions, they remark that two constants, 
instead three 
seems to be well enough for physical theories, and it squashes or squeezes the Gamov-Ivanenko-Landau-Okun (GILO) cube to a nice plane. I include the two final figure for completion, but I urge you to read their whole paper to get a global view before you look at them.
Are 2 fundamental constants enough? Are Veneziano (from a completely different viewpoint) and these brazilian physicists right? Time will tell, but I find interesting these thoughts!
See you soon in another wonderful post about Physmatics and systems of units!
