# 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?

• The answer doesn't argue that such a particle doesn't exist. The question is about an electron absorbing a photon, and absorbing is commonly understood to mean that the thing that absorbs something is still there afterward. One cannot exclude a reaction $e+\gamma \to ???$ with a conservation argument, only the $e+\gamma\to e$. Dec 25 '15 at 15:30
• @ACuriousMind: Thank you for that clarification. (I hadn't been aware of the conventional meaning for "absorb".) My question about $e+\gamma\rightarrow ???$ does, however, remain. Dec 25 '15 at 15:31
• Just inspect the standard model. The only vertex it has with an electron and a photon is the 3-vertex with two electrons/positrons and the photon, you just don't get any other particle. Of course, you can let the electron interact after that with $W$ or $Z$ bosons and try to change it into a muon or something, but this then has nothing to do with the photon. Dec 25 '15 at 15:33
• @ACuriousMind: Thanks for that too. I'm not sufficiently familiar with the standard model to know whether the existence/non-existence of various vertices is due entirely to the existence/non-existence of certain group representations, so I'm still not sure I know the answer to my question. (I hope it's clear that this is not meant as a complaint, and that I'm very grateful for your comments.) Dec 25 '15 at 15:37
• I wonder if the photon wave bounces or if it is emited back by the electron
– user46925
Dec 25 '15 at 19:38