By perfectly infinite range I mean a potential which is a constant, no matter the distance.

Would one just recover a single-particle behaviour?

Whatever any particle does, everything else has to mimic it...

  • if V is a constant then the field is zero – Wolphram jonny Nov 9 at 0:10
  • Did you mean a force which is constant, no matter the distance? – probably_someone Nov 12 at 10:51
  • Yeah I probably meant force... I’ll ask another question I guess. – SuperCiocia 2 days ago

As pointed out in a comment if the potential is a constant then you have no force and therefore you recover a free theory that is every particle motion is uniform and rectilinear. For example you can see this from the variation of the action or directly from the Euler-Lagrange equations or Newton's equations. Take for example the Euler Lagrange equations, you'd have in classical mechanics $$ \frac{d}{dt} \frac{\partial \mathcal{L}}{\partial \dot{q}} + \frac{\partial \mathcal{L}}{\partial {q}} = 0 $$

Since the Lagrangian $\mathcal{L}$ depends on the coordinates $q$ only through the potential wich is constant you have that the second term is $0$, so all that's left is the equation for the free theory:

$$ \frac{\partial \mathcal{L}}{\partial \dot{q}} = 0 $$

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