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 Schrodinger'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 [itex]\psi[/itex], 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.

 

[Entropia]

Yes, conservation of momentum can be directly derived from quantum field theory, specifically in the context of QED. In QFT, momentum is treated as a conserved quantity due to the symmetries of the theory, such as translation invariance. This means that the laws of physics are the same regardless of where and when an experiment is conducted, leading to the conservation of momentum.

In QED, the Dirac equation, which describes the behavior of electrons, is a relativistic equation that takes into account the effects of special relativity. However, in the nonrelativistic limit, the Dirac equation reduces to Schrodinger's wave equation, which is a reflection of Newton's second law. This implies that conservation of momentum in QED is consistent with classical mechanics.

Furthermore, in QFT, momentum is not just conserved in the three-dimensional space, but also in the four-dimensional spacetime. This is known as conservation of 4-momentum, which takes into account the effects of special relativity and the fact that energy and momentum are closely related. So, in addition to conservation of classical momentum, QFT also predicts conservation of 4-momentum.

In summary, conservation of momentum, both classical and 4-momentum, can be directly derived from quantum field theory, specifically in the context of QED. This is due to the symmetries of the theory and the consistency with classical mechanics in the nonrelativistic limit.