I was wondering if the speed of light (c) can be derived from Planck's constant (h) or the opposite way. I know that they are both fundamental constants, but I'm sure they are connected somehow. Saying it on a different way: if Planck's constant would be different, would the speed of light change? Which would be the way on how to proceed for a hypothetical proof?
The name â€˜fundamental constantsâ€™ is misleading; G, h and c are not constants at all: in the Fundamental Planks units G=h=c=1. But you can vary parameters of the standard model instead.
It can be "derived" if one day in the future we will understand why the fine structure constant is dimensionless and why its value is close to 1/137. But asking such questions is being strongly discouraged by what is known as "mainstream physics". Once upon a time Princeton physicists (Dyson) were interested in these questions, but they are not for an ordinary folk to ask.
fine structure constant is part of parameters of dimensionless parameters of the Standard model. It is not very special - there are many other dimensionless constants.
Perhaps for you, and for others that are happy with the standard model, it is not very special. For Pauli it was, for Feynman it was, for Dyson it was. Perhaps it will be such for someone in the future - why not to keep such a possibility in mind as a working hypothesis that is waiting for a verification?