# Dimensional analysis, fermions masses, and universal constants

I was wondering if it was possible to have a theory one day from which we will be able to derive the numerical value of the speed of light or Planck's constant. After a quick Google search the answer to that question was negative. The reason given then was that the numerical values of universal constants depend on what system of units we are using.

I do not understand what is the relation between the negative answer and the fact that any dimensionful quantity will have different numerical values in different system of units. The answer implies that as if the predictions of any theory must be dimensionless. Looks like a postulate to me more than an answer. Could someone provide an intuitive explanation why this is the case?

I am even more confused because when people talk about the origin of mass in the standard model, they are trying to understand the pattern of fermions masses. Is not that considered as trying to build a theory beyond standard model from which we can calculate the fermions masses? similarly the numerical values of the masses depend on the units chosen. Is not that similar to the case with universal constants?

So why it cannot be done for the speed of light, $c$, say but it can be done for masses.

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I don't know whether it's a postulate or answer but what's important is that it's obviously true: values of dimensionful quantities can't be calculated from the first principle. Why? Proof by contradiction follows. Calculate that the value of the speed of light (similarly for any dimensionful quantity) is 299,792,458 m/s. But there exists another nation that uses a different definition of a "meter" so the speed of light in that nation will be 314,159,265.35 m/s or any other real number. So the first number couldn't be better than the other; no numerical value is more right than others. – Luboš Motl Jan 25 '12 at 12:09
@LubošMotl But when we calculate the spectrum of the hydrogen atom, quantum mechanics gives us the energies in whatever units we are using. Alien civilization who discovered QM will be able to calculate the same spectrum. Their spectrum will have different values of energies because they have different units. So in principle, theoreis can predict numerical values for dimensionful quantities after all. Why not for the speed of light? – Revo Jan 25 '12 at 13:22
Dear @Revo, your reasoning seems totally irrational to me. You were originally asking whether one can calculate the numerical value of Planck's constant or the speed of light in certain units. The answer is No and a trivial proof is in my comment. Whether one may calculate the hydrogen binding energy is a totally different question. One may show it is $(e^2/4\pi\epsilon_0)^2 m_e/2\hbar^2$, so it is not independent from the other constants that appear in the formula. But again one may only calculate 13.6 eV numerically if we know the numerical values of all the constants in chosen units, too. – Luboš Motl Jan 25 '12 at 14:29
To return to your sentences: "So in principle, theoreis can predict numerical values for dimensionful quantities after all. Why not for the speed of light?" - They're just incorrect. Indeed, what holds for the numerical value of Planck's constant, the speed of light also holds for the binding energy of the hydrogen atom or any dimensionful quantity in the Universe, too. But what holds is the opposite of what you claim: one clearly can't calculate the numerical values of any of these dimensionful quantities out of nothing because all of them depend on units which are social conventions. – Luboš Motl Jan 25 '12 at 14:33
possible duplicate of Can Planck's constant be derived from Maxwell's equations? – Qmechanic Jan 25 '12 at 14:43

Suppose you have some super-unified theory that explains life, the universe and everything. Ask the question: does this theory contain any constants with dimensions i.e. units of mass, length and/or time?

If the theory does contain such constants then the theory isn't a "first principle" because it contains constant(s) that someone had to put it.

If the theory doesn't contain any such constants then it can't predict constants that have dimensions. This follows from simple dimensional analysis.

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This is not a complete answer.
The absolute values of masses are far less important than the ratio between masses.

After reading the post of Lubos Could the Koide formula be real? I found that the relation between the masses of the charged leptons, the electron, the muon, and the tau lepton, with the help of the eureqa(now formulize) program, obey the simple formula:

$N = e^{( 0.69651574 - 0.0065921498 * mass )}$, for N = {0,1,2}

Dividing each mass by 0.0065921498 (it is a change of units) the formula is simpler:
$N = e^{( 0.69651574 - mass )}$, for N = {0,1,2}
( with correlation 1.000000 , R-squared 1.000000 )
(the data:
N "lepton mass"

0 1776.84
1 105.65836
2 0.5109989
)
This relation is interesting by its simplicity but not really relevant, in my POV, because using only three real valued points no significant relation can be extracted.

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