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Speeds greater than light imply imaginary *energies*, not imaginary
time. (Although I like the concept of imaginary time, and I'll have to
think about what it might mean! You do need to keep in mind that time is
not a variable, but rather a coordinate. I think having a complex
coordinate would be equivalent to adding a new dimension to the universe.
New dimensions of time have been speculated upon, but it's not a
popular idea.)

For many decades, the fact that a faster-than-light object would have
imaginary energy was enough to "disprove" it in many people's minds.
After all, even though imaginary numbers are necessary in quantum
mechanics, any *observable* quantity must be real (there's a theorem
that proves this in quantum mechanics). When imaginary numbers are used in
engineering, it is simply a tool to make the math easier, and the real part
of the result is always used at the end. Because energies are observable,
they can't be complex.

But in 1967 a loophole was discovered. What if the mass of the particle
was also imaginary? (Mass is not directly observable, so that's okay by
quantum.) In that case, it turned out that the energy of the
faster-than-light particle would now be *real*! In fact, this
imaginary-mass particle couldn't go *less* than lightspeed -- it must
always go faster than light. As you may have guessed, this type of
particle was named a tachyon. Tachyons have never been detected, but no
one has proven they can't exist. So maybe faster-than-light particles are
possible after all...

Here is a speculative article on tachyons. If you're interested in why observable quantities can't be complex, I'm afraid I'll have to direct you to a quantum mechanics text.

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