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:
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):
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
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
and it has dimensions of .
2nd. h-protocol. From equation (2), if we divide by and we identigy 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
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 famours iridium “thing”, which is know now not to have a 1 kg mass). Moreover, we also get that
The two protocols can be summarized in a nice table
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 system of units!
Example 1. Compton effect.
Let us define as “a” a photon of frequency . Then, it hits an electron “b” at rest, changing its frequency into , we denote “c” this new photon, and the electron then moves after the collision in certain direction with respect to the line of observation. We define that direction with .
We use momenergy conservation:
We multiply this equation by to deduce that
Using that the photon momenergy squared is zero, we obtain:
Remembering the definitions and and inserting the values of the momenta into the respective equations, we get
It is generally defined the so-callen electron Compon’s wavelength as:
Remark: There are some current discussions and speculative ideas trying to use the Compton effect as a tool to define the kilogram in an invariant and precise way.
Example 2. Inverse Compton effect.
Imagine an electron moving “to the left” denoted by “a”, it hits a photon “b” chaging its frequency into another photon “c” and the electron changes its direction of motion, being the velocity and the angle with respect to the direction of motion .
The momenergy reads
Using the same conservation of momenergy than above
Supposing that , and then
This inverse Compton effect is important of importance in Astronomy. PHotons of the microwave background radiation (CMB), with a very low energy of the order of , are struck by very energetic electrons (rest energy mc²=511 keV). For typical values of , the second term in the denominator dominates, giving
Therefore, the inverse Compton effect can increase the energy of a photon in a spectacular way. If we don’t plut we would get from the equation:
If we suppose that the incident electron arrives with certain angle and it is scattered an angle . Then, we would obtain the general inverse Compton formula:
In the case of , i.e., , and then
In conclusion, there is an energy transfer proportional to . There are some interesting “maximal boosts”, depending on the final energy (frequency). For instance, if , then provides:
a) In the radio branch: , a maximal boost . It corresponds to a wavelength about 300nm (in the UV band).
b) In the optical branch: , a maximal boost . It corresponds to photons in the Gamma ray band of the electromagnetic spectrum.
Example 3. Bremsstrahlung.
An electron (a) with rest mass arrives from the left with velocity and it hits a nucleus (b) at rest with mass . After the collision, the cluster “c” moves with speed , and a photon is emitted (d) to the left. That photon is considered “a radiation” due to the recoil of the nucleus.
The equations of momenergy are now:
In clusters of galaxies, typical temperatures of provide a kinetic energy of proton and electron at clusters about . Relativistic kinetic energy is and it yields for hydrogen nuclei (i.e., protons ). If , then we have . Then, the electron kinetic energy is almost completely turned into radiation (bremsstrahlung). In particular, bremsstrahlung is a X-ray radiation with .