# Are fields ignored in conservation of 4-momentum problems?

When we encounter particle-collision problems that call for invoking the conservation of four-momentum, are we tacitly assuming a field-free idealization (or at least negligible potential energy)?

For example, say particles 1 and 2 collide elastically. Then the conservation of four-momentum says:
$$\mathbf{P}_{1,i} + \mathbf{P}_{2,i} = \mathbf{P}_{1,f}+ \mathbf{P}_{2,f}$$ (where ##i## means initial and ##f## means final).

But in reality, there's potential energy associated with the (changing) relative positions of the particles, isn't there? So to express the full picture, would we add ##\mathbf{P}_{\textrm{field},i}## to the left side and ##\mathbf{P}_{\textrm{field},f}## to the right side?

robphy
Homework Helper
Gold Member
We do something similar in nonrelativistic physics.

ChrisVer
Gold Member
But in reality, there's potential energy associated with the (changing) relative positions of the particles, isn't there?
Well, you'd better ask yourself what would happen if you consider the momentum of the particles 1,2,3,4 pretty "far-away" , that is final corresponding to $t \rightarrow \infty$ and initial to $t \rightarrow - \infty$ (or you can see infinity as 'very large').
As long as no new particles as asymptotic states are produced by the interaction of 1,2 to 3,4 the momenta of initial and final should be equal by conservation of energy/momentum.... no matter what happened inbetween, since anything that happens inbetween is going to conserve the momentum..

Khashishi
$e^- e^+ \rightarrow P(^1S_0) \rightarrow \gamma \gamma$
again you can use $p_{e-} + p_{e+} = p_{\gamma} + p_{\gamma}$... as if you forget what happened at the intermediate step.