# Electron matrix element in a most simple QFT problem, the e+ e- annihilation

In the beginning of my new QFT book there is this short chapter called Invitation: Pair Production in $e^{+}$ $e^{-}$ Annihilation. An electron and a positron collide and a couple muon & antimuon are the result. The first non-zero term of the transition amplitude comes from a process in which the electrons collide, a virtual photon results and then the virtual photon gives rise to the muons:

$$\langle\mu^{+}\mu^{-}|H_{I}|\gamma\rangle^{\mu}\langle\gamma|H_{I}|e^{+}e^{-}\rangle_{\mu}$$

This can be thought of as the scalar product of two vectors (hence the dumb index $\mu$). And the book says that the four elements $\langle\gamma|H_{I}|e^{+}e^{-}\rangle_{\mu}$ are $\propto (0,1,i,0)$

The electron flies parallel to the $x$ axis the positive direction and the positron goes against them. So the total angular momentum they carry is $\frac{1}{2}-\frac{-1}{2}$ in the $x$ direction (is it that why the second coordinate is $1$ ?). Now, my question:

What is that imaginary $i$ that appears in the third place? It is supposed to be a component in the $y$ axis (the collision happens in the $\rm XZ$ plane). And what is the first coordinate ($0$)?

I had undergraduate QM long ago. That $i$ reminds me of things largely forgotten. It might have something to do either with the polarization of the photon or the spin of the electron?

I hope I can survive through the QFT book by reviewing QM concepts as they appear.

• My two cents: that chapter is just an invitation, that is, it is meant to motivate what you are going to learn. You don't need to understant the details, but to "get a feeling" of what is to come. Try to read this chapter and get an overall picture of what you'll be doing in the following chapters: you are going to calculate transition amplitudes. When you learn how to do this in details, come back to this first chapter and you'll see that the details were actually not that important. Anyway, this is just my opinion. Keep on reading the book, and don't worry for $(0,1,i,0)$ yet... Jan 27, 2016 at 22:10
• @AccidentalFourierTransform Thanks. +1 That is certainly a strategy to avoid getting stuck without progressing through the book. However there is always a balance between avoiding getting stuck vs advancing without building an understanding of what you left behind. Too much of the later soon takes you into a position where you do not understand anything at all. Jan 28, 2016 at 1:13