Is there any theory in physics that might support the existence of tachyons? According to Einstein, we all know that light is the fastest thing and it's impossible to beat it's speed. But isn't there a way to go around this? I read somewhere that tachyons gain speed per the energy they lose, is this valid? What are the conclusions we can draw from this?
 A: Let's see, we know that, Tachyons are hypothetical particles that always move faster than the speed of light. In a paper published in 1967 in Physics Review, Feinberg proposed that tachyonic particles could be quanta of a quantum field with negative squared mass.
Perhaps the most famous example of a tachyon is the Higgs boson of the Standard model of particle physics. In its uncondensed phase, the Higgs field has a negative mass squared, and is therefore a tachyon.
Most people think these cannot exist as they're inconsistent with laws of Physics, but still there are potentially consistent theories, that break break the Lorentz Invariance allow for Tachyons's existence.
Despite theoretical arguments against the existence of faster-than-light particles, experiments have been conducted to search for them, and papers are being published supporting the evidence. For example, this paper by Robert Ehrlich and others as well.
Still work is being done, with different people holding different views, but still theories and papers do come out, trying to unify and explain the existence of Tachyons based on our current model of Physics.
A: About the question in the title, when representations of the (universal covering group of the) Poincaré group are considered, one makes the spectral assumption that the joint spectrum of the generators of the translations (i.e. the components of the 4-momentum $p_\mu$) is contained in the closed future light-cone. If you relax this hypothesis and also consider other irreducible representations you will get a kinematic description of particles à la Wigner that can travel faster than the speed of light. So in this sense, tachyons can be derived from the standard mathematical framework of quantum mechanics, but this is usually not done as representations that do not satisfy to the spectral condition are usually deemed as unphysical. The same happens for the continuous-spin representations if one is willing to admit non trivial representations for the translation part of the stabiliser (little group) $S^1\ltimes\mathbb C$ for the massless case.
