If we want to describe the Universe as simply and completely as possible, it takes 26 dimensionless constants to get us there. This is quite a small number, but not necessarily as small as we’d like. In an ideal world, at least from the point of view of most physicists, we’d like to think that these constants arise from somewhere physically meaningful, but no current theory predicts them.
Errr, that table is not showing the fundamental physical constants, even in 1986 (as Siegel reported in the article). To wit, none of them are dimensionless, and a lot of them can be derived from each other (e.g. the Josephson quotient is just twice e/h, which is also on the table).
@PdotdQ and @dga471, how do GUT theories or string theories change the number of fundamental constants? Do they improve on the Standard Model, reducing it from 26?
I once came across a proposal called the Framed Standard Model that explained the mass hierarchy nicely, reducing the number of parameters by a decent amount. I don’t know if it is still viable, though, but it is the only elegant explanation for why we have three generations of matter that I have seen.
The dream is that the “true physical theory” will give a prediction for these numbers.
In string theory, the forces are envisioned to be higher dimensional extensions to the usual 4-spacetime. Compactifying these higher dimensions to 4-spacetime produces a solution for the fundamental constants. The problem is that there are many ways to compactify higher dimensional spaces to the usual 4-spacetime, and each one generally produces a different solution for the fundamental constants. For 10 dimensional string theory, there are ~10^500 ways to do this. This is known as the string landscape.
It is still a subject of research whether we can find our universe within the string landscape. Note that while the number of solutions of physical constants (the jargon in the field is vacua) is dauntingly large, it is not infinite, which is a good thing.
No, these fundamental dimensionless physical constants are all independent of each other.
Fourneutrino mixing parameters for the particle physicists under the sky, Six constants for the theorists in their halls of of tome, Fifteen for particle masses doomed to collide, One cosmological constant for Stephen Hawking on his wheeled throne In the Land of Physics where the Constants lie. One cosmological constant to rule them all, one cosmological constant to find them, one cosmological constant to bring them all, and in the mathematics bind them, In the Land of Physics where the Constants lie.