We would like both 4-momentum and angular momentum to be conserved in a field theory such as QED.(adsbygoogle = window.adsbygoogle || []).push({});

My understanding is that in the case of 4-momentum, there are two different descriptions of the same theory. (1) We can impose conservation of 4-momentum p at each vertex, which requires that virtual particles be off-shell. (An electron can't emit an on-shell photon while conserving p.) This automatically guarantees that you can't compute a nonzero amplitude for any process that violates conservation of p. (2) There are also descriptions in which p fluctuates according to the uncertainty principle, but by the time you calculate amplitudes, it emerges that the amplitude for any p-nonconserving process cancels out.

I'm confused about how this works when it comes to conservation of angular momentum. The analog to approach #1 above would be that we impose conservation of spin at each vertex. Just as we were forced to allow particles to be off shell, I guess we're forced to allow them to have the "wrong" spin. For example, the static repulsion between two electrons arises from interference between the no-exchange diagram and the one-particle-exchange diagram. If the internal state of the emitting or absorbing particle changes, then you can't have this interference. Therefore the virtual photon has to have spin 0.

Is this analogy valid? I can see how there's at least some logical link between being off-shell and having "wrong" spin properties, since off-shell photons don't propagate at c, and therefore Lorentz invariance forbids you from constraining them to have transverse polarization. But that's about a component of the spin vector, which isn't a built-in property of the particle, whereas the analogy with #1 requires the photon to have a spin vector with the wrong *magnitude*.

If virtual photons can have spins 0 and 1, is there any way to rule out other spins? Why not spin 2, or spin 1/2? Do we rule them out based on knowledge of the properties of the classical near fields, e.g., knowing that the static field of a point charge has zero angular momentum density everywhere?

Is there an analog of #2, where we would not impose conservation of spin at each vertex, but amplitudes for nonconservation of angular momentum would end up canceling out?

Is it valid to talk about conservation of spin at a vertex, or should it be spin plus orbital angular momentum? Classically, interactions at a point automatically conserve angular momentum if they conserve momentum. However, a vertex in a Feynman diagram doesn't necessarily represent anything localized, and if we're taking approach #2 for momentum, then there's also no guarantee that momentum is conserved.

An example that's confusing the heck out of me is on p. 24 of the Feynman Lectures on Gravitation (not the same book as the undergrad Feynman Lectures). He's discussing crazy ideas for describing gravity using ordinary QFT, and here he's specifically talking about describing it using the exchange of virtual neutrinos. (I know, it's crazy.) He gives the argument I gave above about how you can't have static attraction if the emitting or absorbing body changes its internal state, and says that this rules out a description of gravity using the exchange of a spin-1/2 particle. (He then goes on and tries to salvage the model by making it a three-body interaction.) The argument he *doesn't* invoke, which would seem much simpler to me, is that we expect force carriers to be bosons, not fermions, because otherwise we'd be violating the expectation that at any three-leg vertex, we should have an even number of fermions -- otherwise the [itex]\Delta S[/itex] at the vertex would have to be a half-integer, and therefore nonzero. So if he's taking the analog of approach #1 by imposing spin conservation at each vertex, then it automatically fails. But if he's taking the analog of approach #2 by allowing nonconservation of spin at a vertex, then his argument about a change of internal state seems invalid, because he could just decree it to stay the same.

The trouble I'm having figuring out this kind of thing from field theory books is that the spins are all sort of hidden behind a bunch of gamma matrices, and I've long since lost whatever fluency I had 20 years ago in interpreting those. There also seems to be a tendency to explain this kind of thing by writing down Lagrangians and reading off their properties, but I'm not fluent in that either at this point.

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# Conservation of angular momentum in field theory: imposed, or emergent?

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