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?


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, $c$),


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

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.