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In this answer on another Physics StackExchange thread, the self-interaction potential terms of a classical field theory ($ \phi^n $ terms with $ n>2 $) are said to correspond to higher-order allowed vertices in Feynman diagrams.

I think this is interesting, because a priori, is there any reason to believe that the potential term of a field theory should be a polynomial at all, let alone end after just a few terms? Of course, the existence of particles at all suggests that we should restrict ourselves to $ \phi^2 $ terms, but any potential will look like that in the vicinity of a minimum anyway, so I'm not sure that that is an argument.

If we allow for (analytic over $\mathbb{R}$) potentials, we could Taylor expand and get an infinite family of possible self-interacting vertices for our Feynman diagrams. Seems like a problem, but if the coefficients of the expansion drop off suitable quickly, I feel like that should restrict the contributions from diagrams including those vertices.

So my specific questions would be:

  • Why do we assume that field potentials stop after at most a couple of terms? Are there technical theoretical reasons, or is it just because we don't need to based on experimental results?
  • Is there any way for us to experimentally probe the field potentials in a more detailed manner? Do we need to, and why/why not?

For context, I'm a third year undergraduate, so please excuse me if these questions are obvious/show a misunderstanding of the theory :)

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  • $\begingroup$ For a theory to be renormalizable the potential has to be truncated at fourth order. Odd powers make the theory not bounded from below, so they are not physically meaningful $\endgroup$ Mar 12, 2020 at 11:44

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Weinberg's view (in The Quantum Theory of Fields) is particularly illuminating for your questions. As an undergrad, it wouldn't be the best idea to spend time deciphering Weinberg's words but basically it goes like this. You start with creation and annihilation operators. Then, there is a theorem that says all operators can be expressed as a sum of product of creation and annihilation operators.

To construct the interaction Hamiltonian, a basic requirement is that it has to be Hermitian. Then if the interaction Hamiltonian can be expressed as a space integral of a Hamiltonian density, another requirement is that the Hamiltonian density at spacelike separations have to commute (for the S-matrix to be Lorentz invariant, also known as the causality condition). There is a third requirement from the cluster decomposition principle but let's not worry about that here.

Now we need a sum of product of creation and annihilation operators that simultaneously satisfy these conditions. The easiest way to do it is to construct a field with simple commutation relations from the creation and annihilation operators and then express everything else in terms of the field. That is the motivation for the construction of quantum fields.

Thus, the Hamiltonian as a sum of product of creation and annihilation operators can now be expressed as a polynomial of the field. Based on the condition that the Hamiltonian has to be bounded below (a physical theory must have a ground state) and the symmetry of the theory, one can decide which terms are allowed. A condition on renormalizability puts further restrictions on which terms one can use.

All things considered, if a theory is renormalizable, there are only a finite number of terms in the interaction.

In a sense, all the experiments in particles and high energy physics are attempts at "probing" the structure of a theory. But the "field potentials" as you mentioned are operators, we cannot probe its structure directly and can only measure a very limited number of observables.

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  • $\begingroup$ That's very interesting, thank you! Would you agree that starting from creation and annihilation operators is imposing that there be a well-defined "number of particles" for the theory? The answer I linked to in my OP states that particles are not well defined for the higher order theories I suggested. Do we know what those theories look like, and are they interesting (do they have any physical applications)? $\endgroup$
    – Movpasd
    Mar 12, 2020 at 17:20
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    $\begingroup$ @Movpasd Not really. What we construct above are operators, whereas the question about particle content is only applicable for states. One can construct asymptotic states with definite particle content, e.g. two particles so far way from each other that they are essentially non-interacting (complications do arise for theory with massless particles because in that case particles are always interacting). Now if those two particles get close to each other and start interacting, then there is no well-defined number of particles anymore. $\endgroup$
    – JF132
    Mar 12, 2020 at 18:30

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