Conservation of momentum in QFT

[jdstokes]

Can conservation of momentum be directly derived from quantum field theory (e.g. QED).

My feeling is this should be true since the Dirac equation reduces to Schrödinger's wave equation in the nonrelativistic limit which is a reflection of Newton's second law, thereby implying conservation of classical momentum.

What about conservation of 4-momentum?

 

[Fredrik]

Yes. You can actually see it without involving the fields. You just assume that there must exist operators that tell you how a Lorentz transformed observer would describe a state that you describe as $\psi$, and when you examine the mathematical properties of those operators, conservation of 4-momentum is one of the results. See chapter 2 in Weinberg's QFT book if you're interested.

You can also do it by using Noether's theorem. The Lagrangian is invariant under Poincaré group transformations. The Poincaré group is a 10-dimensional Lie group, so you will get 10 conserved quantities. 4 of them are the components of 4-momentum.