Representations of the Poincare group

[tom.stoer]

Hello,

in relativistic quantum field theories all particles are members of (unitary) representations of the Poincare group. For massive particles m² > 0 one gets the usual scalar / spinor / vector representations with spin J = 0, 1/2, 1, 3/2, ... and dim. rep. = 2J+1. For massless particles m² = 0 the spin is no longer defined in the usual sense (angular momentum in rest frame does not make sense) and is replaced by helicity. Again one finds the usual representations with dim. rep. = 1 and helicity = +1 and -1, respectively.

Question: what happens for m² < 0? how do the representations look like and why are they unphysical?

 

[DrDu]

These are the famous tachyons. As m is the energy in the rest frame, tachyons would have to move with speeds higher than c for their energy to be real valued.

 

[Norman]

You can try and go through a simple model where m^2<0 (for instance try quantizing the Klein-Gordon scalar fields with this assumption). All sorts of odd things occur. This was a final exam problem in my QFT class.

An interesting read of the simplest tachyon field is by http://prola.aps.org/abstract/PR/v159/i5/p1089_1. You might find it enlightening.

But to answer your question about why they are unphysical- the traditional argument is they violate causality.