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I am new here and I got a question that I never read in any textbook. Are Feynman rules actually unique or can one derive different rules, which are equivalent?

Of course, we have a freedom to choose a gauge. So my question concerns whether or not Feynman rules are unique after we have chosen such a gauge and if there is still the possibility to gain another Feynman rule, which is equivalent to the other one.

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  • $\begingroup$ There is some freedom you have certainly, since for example you can use different choices of gauge in QED which result in a different expression for the propagator but give the same physical answers. $\endgroup$
    – JamalS
    Mar 22, 2018 at 10:22
  • $\begingroup$ Okay, of course there is a gauge freedom we have. I forgot to mention that I am aware of this. But how is it besides the gauge that we can choose? After we have chosen a gauge, are the derived Feynman rules unique then? $\endgroup$
    – Laboon
    Mar 22, 2018 at 10:29
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    $\begingroup$ Can you try to clarify the question? $\endgroup$
    – innisfree
    Mar 22, 2018 at 10:35
  • $\begingroup$ Sure. :) After we derived a particular Feynman rule and have chosen a particular gauge: is that derived Feynman rule then unique or is it possible, that still another equivalent Feynman rule can be derived? I hope, I could clarify my question. (Maybe I also should edit my original post to prevent confusion.) $\endgroup$
    – Laboon
    Mar 22, 2018 at 10:39
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    $\begingroup$ Hi and welcome to PSE. There is no need to put the "edit" word when you edit. The question should be as concise as possible. $\endgroup$
    – Shing
    Mar 22, 2018 at 10:46

1 Answer 1

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There are lots of ways to rewrite Feynman rules! Here are just a few.

  • As has already been pointed out, different gauges have different Feynman rules.
  • The vertices depend on how you split the Lagrangian into free and interacting parts. For example, the propagator for a massive scalar field is usually $1/(p^2 + m^2)$, but one could also say the free part is the kinetic term alone, giving a propagator $1/p^2$. Then we gain a two point self-interaction from the mass term, which gives a factor of $m^2$. This point of view is useful when the Higgs mechanism is involved, because every 'free' particle really is massless.
  • Similarly, the allowed vertices depend on the basis we choose for our propagating particles. For example, quarks/neutrinos in the mass basis can change generations upon interacting with a $W$ boson, but can't in the flavor basis. Which you use is up to convention and context.
  • "Integrating out" some fields will give new Feynman rules from the effective interactions generated. For example, for scales much lower than the $W$-boson mass we can ignore the $W$ boson, at the cost of having a new, effective four-fermion interaction.
  • Feynman rules depend on how we group fields together; for example, a Dirac field is usually drawn as a plain line even though it contains four independent fields. You could alternatively have different lines for the Weyl spinors inside the Dirac field, as done here, and regard the Dirac mass as an interaction that swaps them. This is really the same as the third point, switching from the mass basis to the chirality basis.
  • The adjoint representation for gluons is "almost" the same as the fundamental times and antifundamental, as $$N \times \bar{N} = (N^2 - 1) + 1.$$ Therefore we can write Feynman rules for gluons as if they're made of a fictitious charged particle and antiparticle pair, even though the Lagrangian says no such thing. This is called t'Hooft double line notation and is especially useful for large $N$.
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  • $\begingroup$ I see, there are many different ways to construct the Feynman rules. And I was not aware of such a variety. So, many ways exist and all of them can make the Feynman rule look different, right? Let say, after we have chosen a particular way, the Feynman rule is unique in that sense? If yes, then I think my Question is solved. Thank you very much by now! :) $\endgroup$
    – Laboon
    Mar 22, 2018 at 12:28
  • $\begingroup$ @Laboon Yup, there are a huge variety of ways to use Feynman rules. That reflects the fact that Feynman diagrams are tools for physicists, not ground truth, and we can use a tool however we want. $\endgroup$
    – knzhou
    Mar 22, 2018 at 12:36

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