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When one wants to include weak relativistic effects in classical equations, usually kinetic energy term is expanded into Taylor series about $p=0$. But the complete dispersion relation is $E=\sqrt{m^2+p^2}$, which has branch points at $p=\pm im$, thus its Taylor series diverges for any $p: |p|\ge m$.

Does such divergence have any physical significance? Does it mean that there're some completely new physical effects starting right at $p\approx m$, and trying to just include more complete dispersion relation will lead to nonsensical results? Maybe it is somehow related to possibility of pair production/annihilation?

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It just tells us that the original assumption allowing us to expand the expression into a Taylor series, i.e. that the momentum is small with respect to mass, is not satisfied anymore. In other words, if the expansion parameter of a series is not small, the result of the expansion is not meaningful anymore.

Physically speaking, this means that the approximation for low momentum (I have added the speed of light, $c$),

$$E=mc^2+\frac{p^2}{2m},$$

is not valid at high momentum. I would not attribute too much physical meaning to the value of $|p|$ for which it starts diverging.

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