No-Go Theorem for tachyons!

A physical theory cannot have any tachyon.
A propagator has at a nonzero value of q^2 a pole implies the existence of particle
moving with a velocity faster than light, a so-called tachyon.
The possible existence of such states was conjectured for a long time
since it was an allowed state in an irreducible representation of the Poincare group.
They were an imporant output of infinite-component wave equations
which were a popular subject of study in the sixties.
See for example this paper. They also exist in the best existing quantum field theory
so far, the quantum electrodynamics of electrons and photons (QED).
Textbook can be read online here.
There they appear at very large masses which lie far above the validity of this theory.
They are called Landau ghosts.
They also appear in attempts to quantize gravity by making it emerge from a conformally invariant action.
See the last chapter in the above textbook here..

At the end of Chapter 18 of the above book a no-tachyon theorem is stated as follows:
A particle moving faster then light is removed in nature by the system undergoing a phase transition
to a state of lower energy. In field theory this is usually a state with another vacuum expectation value of the fields.
This is the typical way how nature deals with all unstable physical systems:
For example, a timber-framed house will be unstable
if its small vibrations have some eigenfrequency with a negative square value.
Upon a small vibration, the amplitudes of such oscillations grow large exponentially fast in time.
As a consequence, the structure will collapse into a state of lower energy.
If there is again a negative square frequency, the intermediate state will collapse again. This process will go on until the stucture has settled as a stable ruin.
In the ruin, all frequencies have positive square values, and no tachyon is left.
Click on this link to read the last paragraph in Chapter 18!