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Why is $\frac{\hbar}{mc}$ a good estimate of the range of the four forces, where $m$ is the mass of the carrier particle of the force? Inputting the pion mass gives $1.4\ \mathrm{fm}$ for the hadronic force. I have one book that tries to justify the formula by the uncertainty principle and another book that says that the argument is bogus. So does this follow from the uncertainty principle?

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If you have access to it the correct argument is given in Zee, Quantum Field Theory in a Nutshell, chapter 1.4. Just a quick sketch here: What you do is calculate the energy of the field (in your case the pion field) in the presence of two sources, which are treated for the moment as static. You drop the self-energy contributions and what is left is the interaction energy. You do some integral and get the Yukawa potential (en.wikipedia.org/wiki/Yukawa_potential), which goes as $\exp(-mcr/\hbar)/r$. You do the exact same thing in electromagnetism to get Coulomb's law - a good exercise. – Michael Brown Jan 22 at 0:34
Thanks ill accept this as answer. – fisk Jan 22 at 7:59

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