Let me Know What you Think of This Toy Theory

[KyleSingh]

Hello all, My name is Kyle. I am a physics student at Columbia. I do a lot of science for the public in the New York area and one of my projects involves helping out a group of high schoolers advance in there physics knowledge and self study. Currently I am teaching them the basics of QFT and they took the initiative as a group and sent me this toy theory they were working on. The problem is, the whole computation looks wrong but I can't put my finger on why. Do you guys have any ideas? Perhaps I am just doubting them because they are high schoolers x) Thanks for the help! I have attached it here as a PDF.

 

[PeterDonis]

Hi Kyle,

The computation is wrong because it assumes that you can just declare by fiat that there are three kinds of "particles" in the theory. But you can't. The number of kinds of "particles" in the theory is determined once you write the Lagrangian. In the case of this theory, with the standard massive scalar field Lagrangian, there is only one kind of "particle", a scalar (spin zero) particle with the given mass. An "in" or "out" state in a scattering process can contain multiple particles of this kind, but they will all be identical particles of the same kind; you can't have three different kinds.

(Actually, it's even more complicated than that, because not all states of quantum fields even have a useful "particle" interpretation to begin with. But that's a further level of complexity that I don't think needs to be gone into at this point.)

 

[KyleSingh]

Thank you so much for the response! I figured that this was the case but couldn't articulate why they could not just posit three different particles.

 

[PeterDonis]

I figured that this was the case but couldn't articulate why they could not just posit three different particles.

There is another issue as well: interactions between the particles don't just appear out of nowhere. They have to be in the Lagrangian. The Lagrangian in the paper has no interaction terms. An example of a Lagrangian for a single scalar particle with an interaction term is this:

$$
L = - \frac{1}{2} \partial^\mu \partial_\mu \phi - m^2 \phi^2 - \lambda \phi^4
$$

The $\lambda \phi^4$ term is the interaction term: basically, it says that a pair of these $\phi$ particles can scatter off of each other.

A theory of three different scalar particles would have three different fields in it; you could call them $\phi_\alpha$, $\phi_\beta$, and $\phi_\gamma$. The total Lagrangian would include a kinetic $\partial^\mu \partial_\mu$ term and a mass $m^2$ term for each field, plus interaction terms that could be between the same field, such as a $\lambda$ term for each field, or could be between different fields, such as a term like $g \phi_\alpha^2 \phi_\beta^2$, which would describe scattering of an $\alpha$ and $\beta$ particle off of each other. Other interactions are also possible. But they all have to appear in the Lagrangian; you can't just conjure them up out of nowhere.

(Btw, the paper also seems to think that there can be separate derivative operators for each of the fictitious particles, as $\partial_\alpha$, $\partial_\beta$, etc. This is another error. The derivative operator has a spacetime index; it describes variations of fields in spacetime. It is the same operator regardless of which field, which kind of "particle", it is applied to.)