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  • particles with real-mass have time-like kinematics ($ds^2 > 0$).
  • particles with zero-mass have light-like kinematics ($ds^2 = 0$).
  • particles with imaginary-mass have space-like kinematics ($ds^2 < 0$) (tachyons).

So the question is pretty simple:

What would be the kinematics of a particle with both non-zero real and imaginary parts?

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Maybe such a particle is decaying or being born? –  Vladimir Kalitvianski Oct 26 '11 at 18:54
This is something people doing PT quantum mechanics study. Complex classical mechanics is a field which is about 5 years old. –  Ron Maimon Oct 30 '11 at 19:57
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1 Answer

I think the question has no meaningful answer, at least in our universe. If you look at $$E^2 - p^2 = m^2$$ then if $m$ is complex with non-zero real and imaginary components, then $m^2$ is also complex with non-zero real and imaginary components and therefore either $E$ or $p$ (or both) must also be complex with non-zero real and imaginary components. I don't think there is any meaningful description of the kinematics of a particle with complex energy or momentum.

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thanks for the answer. Yeah i've thought this as well, since in the lorentz transform expression, fixing the $\beta=\frac{v}{c}$ factor and $E$ to be real implies that $m$ must be either real or immaginary, but since in twistor geometry one might want to study complexified poincare geometries (where the above asumptions about $\beta$ and $E$ being real do not necessarily hold anymore), i wondered if in a twistor description a complex mass would have a meaningful kinematics –  lurscher Oct 31 '11 at 3:41
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