String Theory and Tachyons
A bigger problem than massless particles for string theory in the 1970's was the tachyon, a particle predicted by bosonic string theory that travels faster than the speed of light. Under a consistent bosonic string theory, the mathematical formulas demanded that tachyons exist, but the presence of tachyons in a theory represents a fundamental instability in the theory.
Solutions that contain tachyons will always decay into another, lower energy solution — possibly in a never-ending cycle. For this reason, physicists don’t believe that tachyons really exist, even if a theory initially looks like it contains such particles.
Strictly speaking, Einstein’s theory of relativity doesn’t absolutely forbid an object from traveling faster than the speed of light. What it says is that it would require an infinite amount of energy for an object to accelerate to the speed of light.
Therefore, in a sense, the tachyon would still be consistent with relativity, because it would always be moving faster than the speed of light (and wouldn’t ever have to accelerate to that speed).
Mathematically, when calculating a tachyon’s mass and energy using relativity, it would contain imaginary numbers. (An imaginary number is the square root of a negative number.)
This was exactly how string theory equations predicted the tachyon: They were consistent only if particles with imaginary mass existed. But what is imaginary mass? What is an imaginary energy? These physical impossibilities give rise to the problems with tachyons.
The presence of tachyons is in no way unique to bosonic string theory. For example, the Standard Model contains a certain vacuum in which the Higgs boson is actually a type of tachyon as well. In this case, the theory isn’t inconsistent; it just means that the solution that was applied wasn’t a stable solution.
It’s like trying to place a ball at the top of a hill — any slight movement will cause the ball to roll into a nearby valley. Similarly, this tachyon solution decays into a stable solution without the tachyons.
Unfortunately, in the case of bosonic string theory, there was no clear way to figure out what happened during the decay, or even if the solution ended up in a stable solution after decaying into a lower energy state.
With all of these problems, physicists don’t view these tachyons as actual particles that exist, but rather as mathematical artifacts that fall out of the theory as a sign of certain types of inherent instabilities. Any solution that contains tachyons quickly decays due to these instabilities.
Some physicists (and science fiction authors) have explored notions of how to treat tachyons as actual particles. But for now, just know that tachyons were one of the things that made physicists decide, at the time, that bosonic string theory was a failure.