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From the "flatness problem" in Cosmology to the "strong CP problem" and "hierarchy problem" in the Model standard, a lot of problems in physics deal with Naturalness: a certain parameter in our theory is too small or too big and so we look for a better understanding of it. To resolve this "unnatural fine tuning", in Cosmology Guth proposed the Inflation, and for the Standard model we have the SUSY hypothesis.

If so, why there isn't a naturalness problem also for the velocity of light and all the fundamental constants that we know in nature? It seems to me that the only way to prevent a naturalness problem is to create a theory that determines itself with no free parameter, and this is impossible.

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  • $\begingroup$ Comment to the post (v2): Naturalness is linked to that dimensionless physical constants should be "of the order 1". E.g. the speed of light $c$ is not dimensionless. Which dimensionless physical constants are you thinking of? Related: physics.stackexchange.com/q/8373/2451 $\endgroup$ – Qmechanic Dec 11 '16 at 20:11
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As far as I know the "naturalness" problem comes from trying to understand some physical constants at lower energies from an assumed high energy model. So if I have some Lagrangian at low energies with a small physical constant , I could consider a possible ultra violet completion. In the high energy Lagrangian I could do a renormalization group flow to the lower energy Lagrangian. It turns out that in some theories constants that are small in the lower energy Lagrangian get multiplied by positive powers the cut off after the RG flow and so these constants depend sensitively on the cutoff. The question then becomes if we observe them to be small there must be some mechanism to cancel the ultraviolet corrections coming from the high energy Lagrangian. If you can't find such a mechanism then you have to tune these constants finely so that everything works out.

On the other hand there are constants that get multiplied by negative powers of the cutoff and so do not matter at low enough energies. So the issue is not simply about constants being " too small or too big". For example the mass of the electron is smaller than the Higgs particle but no one worries about the mass of the electron.

Other constants could be determined by a symmetry. So for example, the fact that there is a limit on the speed information can be transferred is a result of Lorentz symmetry. This speed is the speed of light in vacuum; it may be a big number but it is determined by a symmetry.

There is a separate question of whether one can calculate all physical constants in terms of mathematical constants. This is a separate issue but no unrelated. It is not unrelated because if all constants are determined by mathematical constants then it means that "naturalness problem" is ultimately a fake. We are just not smart enough or know enough to figure out the right theory.

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