Are Feynman rules unique?

Question

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.

Answer 1

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 $N$.