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I often hear this explanation and there's something that bugs me every time. What does it mean? Does the constant's value change when we observe it at different length scales? Is it something that changes with the observer then? Is it related to the energy scale of the particle or to the energy scale of the device necessary to measure it?

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  • $\begingroup$ Related answer here. $\endgroup$ – knzhou Sep 23 '16 at 1:06
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I believe what you ask can be explained simply by noting that the coupling constants are not really constant, and they depend on the energy scale. One can change the energy scale and thus observe different values for the coupling constant.

For instance, consider Coulomb's law. In the mass scale, in the limit where $\vec{p}^{2} \gg m^{2}$, this law holds for an energy-dependant charge $$e_{\mu^{\prime}}^{2} = \frac{e_{\mu}^{2}}{1 - b e_{\mu}^{2} \ln \left(\frac{\mu^{\prime 2}}{\mu^{2}}\right)},$$ where $b$ is a constant (unimportant for the sake of this discussion), $\mu^{\prime}$ is a cutoff and $\mu$ is the energy scale. As the energy changes, so does the charge.

However, in the limit where $\vec{p}^{2} \ll m^{2}$, $e_{\mu^{\prime}}^{2} $ acquires a fixed value, $$\frac{e_{\mu^{\prime}}^{2}}{4\pi} = \frac{m}{\sqrt{c}},$$ and thus the constant stops running, so Coulomb's law holds for this constant value of the charge.

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  • $\begingroup$ Thank you for your answer. What do you mean by energy scale? Is it some caracteristic value? Could you explain why does electric charge change with the energy scale? Is it trully variable or is it something that happens because of the manipulations one does to renormalize? $\endgroup$ – P. C. Spaniel Sep 23 '16 at 17:14

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