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

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.

Erratum: a previous version of this article erroneously claimed that Noether’s second theorem is related to Ward identities that guarantee gauge invariance at the quantum level. This is not the case, to our knowledge. Indeed, the Ward identity is a statement about averaging over field configurations, which necessarily depends on the behaviour of the path integral measure, a quantity that Noether never concerned herself with! Interestingly, there is a connection between the second theorem and large residual gauge symmetries, as pointed out in https://arxiv.org/abs/1510.07038.

Spinor Helicity Formalism – Twistors to the Rescue

Dang! Didn’t get my teeth into enough supersymmetry today. I’m standing at the gateway though, so I’ll be able to tell you much more tomorrow. For now, let’s backtrack a bit and take a look at spinor helicity formalism.

First things first, I need to remind you that on-shell matter particles (specifically spin $\frac{1}{2}$ fermions) are represented as spinors in quantum field theory. For a massless fermion, the spinor encodes the momentum and helicity of the particle. We introduce the so called spinor helicity notation

$\displaystyle [p| = +\textrm{ve helicity particle with momentum }p \\ \langle p| = -\textrm{ve helicity particle with momentum }p$

Their hermitian conjugate spinors give the corresponding antiparticles, as you’d expect if you’re familiar with QFT. One can thus naturally define the inner product of a particle and antiparticle state by contracting their corresponding spinors. We see these contractions a lot in the Parke-Taylor formula, for example.

Now it turns out that every null future pointing vector can be represented in terms of it’s corresponding spinor helicity as according to the identification

$\displaystyle p \leftrightarrow |p]\langle p|\qquad (A)$

This can be made formal easily using the Weyl equation that the spinor states must satisfy. But what exactly is the use of this?

Well we saw that writing down scattering amplitudes in the spinor helicity formalism was particularly easy, since we could keep the amplitudes manifestly “on-shell” throughout the process. However I did sweep under the carpet a little algebraic manipulation. I can be more explicit about that now. The only difficult steps in the simplification I omitted are due to momentum conservation.

Usually momentum conservation for an $n$ particle process takes the form

$\displaystyle \sum_{i=1}^n p_i = 0$

This is easy to implement in the usual Feynman diagram formalism, because it is linear! But in the spinor helicity world, we see that this formula becomes quadratic on account of the identification (A) above. This quadratic relation is somewhat troublesome to deal with, and requires annoying identity manipulation to impose.

But what if we want to have our cake and eat it? Imagine a world where we could have the spinor helicity simplifications and yet keep the simplicity of linear momentum conservation. Fortunately for us, such a world exists. We can get there by means of an abstract tool called the dual momentum twistor. There’s not enough time to tell you about that now, but watch out for its appearance in a later post.

So tomorrow will be some supersymmetry then, and maybe a short aside on calculating graviton scattering amplitudes easily. Easily, you say? Well it’s a doddle… at tree level… with a small number of particles…

My thanks to Andi Brandhuber for an enlightening discussion on this point.