(This question is related to: A pedagogical exposition of the hadron physics?)

I'm a mathematician who has been trying to learn quantum field theory for a while. I've gone through large parts of quite a few books, and there's a conceptual issue that's been bothering me for a while.

It's common to see statements like "the proton is composed of two up quarks and a down quark," or even equations like $\pi^0=\frac{1}{\sqrt{2}}(u\bar u-d\bar d)$. The reading I've done suggests that one builds an effective field theory containing fields corresponding to all the hadrons, but that the exact relationship between the hadron fields and the quark fields is just poorly understood, so, sad as it might be, there's no way to say what the hadron fields have to do with the quark fields.

But all the talk about hadrons being "made of" particular combinations of quarks and antiquarks makes it seem like this can't be the whole story. In particular, one finds these composite particles by looking for copies of the trivial representation in a tensor product of $SU(3)$ representations, so it seems like a "proton field" should somehow be related to some corresponding product of the up and down quark fields. How does this work? When someone says that a proton is made of two up quarks and a down quark, what are they saying about the relationship between the proton field and the up and down quark fields?

(This question is related to: A pedagogical exposition of the hadron physics?)

I'm a mathematician who has been trying to learn quantum field theory for a while. I've gone through large parts of quite a few books, and there's a conceptual issue that's been bothering me for a while.

It's common to see statements like "the proton is composed of two up quarks and a down quark," or even equations like $\pi^0=\frac{1}{\sqrt{2}}(u\bar u-d\bar d)$. The reading I've done suggests that one builds an effective field theory containing fields corresponding to all the hadrons, but that the exact relationship between the hadron fields and the quark fields is just poorly understood, so, sad as it might be, there's no way to say what the hadron fields have to do with the quark fields.

But all the talk about hadrons being "made of" particular combinations of quarks and antiquarks makes it seem like this can't be the whole story. In particular, one finds these composite particles by looking for copies of the trivial representation in a tensor product of $SU(3)$ representations, so it seems like a "proton field" should somehow be related to some corresponding product of the up and down quark fields. How does this work? When someone says that a proton is made of two up quarks and a down quark, what are they saying about the relationship between the proton field and the up and down quark fields?