What Gets Conserved at Vertices in Feynman Diagrams?

The simple answer is – everything! If there’s a symmetry in your theory then the associated Noether charge must be conserved at a Feynman vertex. A simple and elegant rule, and one of the great strengths of Feynman’s method.

Even better, it’s not hard to see why all charges are conserved at vertices. Remember, every vertex corresponds to an interaction term in the Lagrangian. These are automatically constructed to be Lorentz invariant so angular momentum and spin had better be conserved. Translation invariance is built in by virtue of the Lagrangian spacetime integral so momentum is conserved too.

Internal symmetries work in much the same way. Color or electric charge must be conserved at each vertex because the symmetry transformation exactly guarantees that contributions from interaction terms cancel transformations of the kinetic terms. If you ain’t convinced go and check this in any Feynman diagram!

But watch out, there’s a subtlety! Suppose we’re interested in scalar QED for instance. One diagram for pair creation and annihilation looks like

scalarqed

Naively you might be concerned that angular momentum and momentum can’t possibly be conserved. After all, don’t photons have spin and mass squared equal to zero? The resolution of this apparent paradox is provided by the realization that the virtual photon is off-shell. This is a theorist’s way of saying that it doesn’t obey equations of motion. Therefore the usual restrictions from symmetries do not apply to the virtual photon! Thinking another way, the photon is a manifestation of a quantum fluctuation.

There’s a coda to this story. Rather magically, charge still does have a meaning for the off-shell photon. This is courtesy of Noether’s second theorem, which says that local symmetries have charges that remained conserved off-shell. This critical insight is related to the powerful Ward identities which ensure that observables are gauge invariant.

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