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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
in string theory or in QFT context ? –  igael Jun 4 at 5:01

3 Answers 3

The higgs mechanism contains this paradox that mass is created as a result of mass. the coupling constant (to the higgs field) is mass itself. But it is not gravitational mass. Obviously mass is a dualist entity. How the one mass (higgs mass) becomes the other (gravitational mass) is still a secret of nature.

May be we have to write mass as 4-vector comprising of gravitational mass, higgs mass, weak charge and electric charge. We have to devise four 4x4 matrices to extract the required entity out of these 4 components of the vector for our calculations. And we have to devise four 4x4 tensors for converting the energy-momentum tensor into a hamiltonian 4-vector of 4x4 matrices as components.

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Complex mass means gravitational mass + i.lambda.higgs mass. The weak coupling constants are all proportional to (higgs) mass with Higgs vacuum value (=246 GeV) as the proportionality constant.

Thus mass necessarily becomes a complex number.

The real part produces attractive gravity forces and the imaginary part produces repulsive "weak" forces. The factor lambda regulates the relative strength and range of the weak force.

The weak force possibly acquires a long-range in company of the gravity force.

Mass must thus be written as m.exp(phi(x)).

Phase phi being function of x (with contravariant-indexes 0,1,2,3), this phase automatically creates/necessitates new gauge fields and intermediary bosons.

This prevents the singularity in the black hole.

The equivalence principle has not been tested in high density matter (nucleus, white dwarfs, neutron stars, black holes etc.) and below the atto-m ranges.

A complete description of causality is not possible without the inclusion of repulsive forces (and their geodesics) in General Relativity.

A complex mass leads to a complex energy momentum tensor and a complex metric. The complex metric can explain anti-particle trajectories (rotated or reversed geodesics).

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care to provide a reference to read more about this interpretation? –  lurscher Mar 27 at 16:01

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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