In QFT how do you write down the most general interactions?

Question

This past year I took a QFT class and I now feel comfortable solving scattering problems, but I am still a bit perplexed by how physicists write down a Lagrangian in the first place. In particular, given some particle menu, is there a method for writing down the most general Lagrangian interaction terms?

It would be very helpful if someone could provide a concrete example, say if you had 1 fermion field and 1 vector boson field, what are all of the possible interaction terms (including non-renormalizable terms)?

Answer 1

In physics there is no general criterion on how to write down suitable Lagrangians, rather than a posteriori check on the equations of motions: all the Lagrangians generating the same dynamics are equally correct. For example, as an exercise, you may try to write down all the possible Lagrangians giving you back $F_j = m \ddot{x}_j$.

This said, to directly answer your question: in QFT the tendency is to write all the possible terms that are invariant under the Poincaré group and satisfy the conservation laws you want to be preserved. All the possible couplings of any order in the fields are allowed and there is plenty of papers in journals introducing this or that kind of interaction. Eventually, you will have to check the renormalisability of the theory, but there is no general method to cook fields up rather than explicitly adding terms and checking a posteriori what kind of dynamics they produce (provided all the invariances of the theory to be fulfilled).

Answer 2

If physics isn't an issue, you can add arbitrarily many terms. Once the physics comes in though, you will encounter a few restrictions :

• As said by Gennaro, it is assumed that the Poincaré symmetry applies.
• Higher derivative terms (second derivatives and above) are generally bad news. They can cause vacuum instability (energies can be arbitrarily negative, making the vacuum radiate away infinitely) and, in the worst cases (such as infinitely many such terms), non-locality of the solutions. There are plenty of threads on stackexchange regarding the evils of higher derivatives, if you wish.
• Some interaction terms can also cause vacuum instability. A $\varphi^3$ interaction term also has no lowest energy.
• If any gauge symmetry is present, it will further constrain the form of the gauge field, as it will otherwise break gauge symmetry (that is part of the reason the Higgs mechanism was developed, as mass terms broke gauge symmetry for the weak field)

A domain where generic interactions are somewhat used is effective field theories. To avoid dealing with horrors like quantum chromodynamics, chiral perturbation theory writes down the most general lagrangian that obeys some approximate symmetry of the system (usually SU(2) or SU(2)xSU(2)), something of the form

$\mathcal{L}_{eff} = \sum_{(k,l)} \mathcal{L}_{(k,l)}$

With $\mathcal{L}_{(k,l)} \approx (\partial \varphi)^k \phi^l$

Usually most terms are dropped, but if you want an example of a Lagrangian in a very general form, you might want to look into it.