Why particles with certain properties can't exist This is inspired by a recent post on why a free electron can't absorb a photon, though my question below is about something considerably more general.
The argument in the accepted answer goes (in essence) like this:  
Taking $c=1$, and working in the rest frame of the electron, let $p\neq 0$ be the momentum (and hence the energy) of the photon and let $m\neq 0$ be the mass (and hence the energy) of the electron.  Then the combined particle must have momentum $p$ and energy $m+p$, which means that its mass $\hat{m}$ must satisfy $\hat{m}^2=(p+m)^2-p^2=m^2+2mp$.  
Now if we suppose that the combined particle has the mass of the original electron, we get $2mp=0$, which is a contradiction.  
This shows that the combined particle cannot be an electron.  It does not, by itself, show that the combined particle cannot exist.  This leads me to wonder how the argument can be completed, and to the more general question of how one shows that a given particle cannot exist.  (I expect this is a very naive question.)
It's been suggested (in comments on the referenced post) that the argument for non-existence can be completed by invoking additional conservation laws, such as conservation of lepton number.  But I don't see how this can possibly suffice, as we can always suppose that the lepton number of the combined particle is the sum of the lepton numbers of the electron and the photon, and likewise for any other quantity that needs to be conserved.
In the end, then, we have a particle with a prescribed mass and prescribed values for a bunch of  other conserved quantities.  
My question is:  What tells us that this particle can't exist?  
My guess is that this comes down to some exercise in representation theory, where the particle would have to correspond to some (provably non-existent) representation of the Poincare group.
Is this guess right, or is there either more or less to it than that?
 A: There are only two stable particles in the Universe, the electron and proton, excluding EM waves. All other particles are excited electron and proton states or from their interactions, like neutrinos. They, and the Up and Down quarks constituting protons, are precise arithmetic or exponential reciprocal multiples of Sommerfeld's alpha = 0.007297 fine structure constant correlating light speed to hydrogen ground state potential energy root. This energy density root factor is coincident in the nuclear, atomic and gravitational energy domains, relating them all to the hc = 1.986445684 x 10^-25 Jm electromagnetic impedance of space. Thus the reciprocal of the proton's radius and light year distance correlate closely since the light speed events within the particle and gravitational domains both relate to c. These correlations are enumerated on page 21 of A Nuclear-Gravitational Electrodynamic Framework at QuantumEnergySystems.com. Because of this energy density framework electrons can only absorb photon energies corresponding to their muon, pion and tau energy states. The quarks in the proton however offer a much wider range of photon absorption states.
