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I have always considered quantities like mass, charge, momentum etc to be completely real quantities as them being imaginary doesn't make much sense to me. But for tachyons to exist, they should have imaginary mass to be compatible with special relativity. Or their rest mass should be imaginary for their masses to be real. So is the mass of tachyons real or imaginary? And if it's imaginary, can there be imaginary charge, momentum etc?

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    $\begingroup$ "but for tachyons to exist..." but they do not. Period. $\endgroup$ – AccidentalFourierTransform Oct 24 '18 at 14:57
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    $\begingroup$ @AccidentalFourierTransform: Whether they exist or not is a matter of observation, and observations tomorrow could show that they exist. See physics.stackexchange.com/q/63297 . You may suspect that they shouldn't exist because it raises theoretical problems, but that doesn't mean they don't exist. $\endgroup$ – Ben Crowell Oct 24 '18 at 15:12
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    $\begingroup$ @BenCrowell Not really. Serious physics doesn't consider tachyons to be a thing (beyond an unstable solution, but those do not behave as "imaginary mass" particles in any meaningful sense whatsoever). Configurations with "negative mass squared" are the bread and butter of particle physics; they are very well understood, and one can even refer to them as a "tachyonic configurations". But those look nothing like the pop-sci notion of a tachyon. At the level of OP, "tachyons are not a thing" is the only possible answer. $\endgroup$ – AccidentalFourierTransform Oct 24 '18 at 15:16
  • $\begingroup$ @AccidentalFourierTransform: Serious physics doesn't consider tachyons to be a thing Counterexample: plenty of "serious physicists" published plenty of papers on tachyonic neutrinos in 2011. Your theoretical reasons for not believing in tachyons are perfectly reasonable, and I share them, but that doesn't mean that tachyons are not a possible topic of discussion. $\endgroup$ – Ben Crowell Oct 24 '18 at 20:36
  • $\begingroup$ @BenCrowell Literally everyone in the scientific community knew that faster-than-light neutrinos were nothing but an experimental error, from day one. Popular media did its thing, and took things out of context and released grandiloquent and unjustified statements, and cheap scientists jumped on the bandwagon. $\endgroup$ – AccidentalFourierTransform Oct 24 '18 at 20:52
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First let's clear up the possible confusion about rest mass and mass. All working physicists these days define mass to be the same thing as what used to be referred to as "rest mass" back in the 1950's. The modern convention has slowly filtered down into textbooks and is now a universal standard except in popularizations.

The definition of mass is $m^2=E^2-p^2$ (in units with $c=1$). Mass is not additive.

A tachyon has a spacelike world-line by definition. By symmetry, its energy-momentum is parallel to the tangent vector to its world-line. Therefore its energy-momentum is spacelike, and $m^2<0$.

And if it's imaginary, can there be imaginary charge, momentum etc?

We don't do general relativity on complex manifolds, we do it on real manifolds, so for compatibility with GR, the energy-momentum must have real components.

I'll leave it to others to discuss whether imaginary charge makes sense. This might depend on what you pick as your definition of charge, whether you're talking about classical or quantum physics, and what kind of distasteful outcomes you are or are not willing to accept.

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