# How to put the terms in correct order when deriving amplitudes in QED from the S-Matrix?

I am trying to derive the amplitude for the Møller scattering process $$e^-+e^-\rightarrow e^-+e^-$$ from "first principles", that is, by working out the calculation with the relevant S-matrix terms instead of applying directly the Feynman rules. Basically, I want to see that I obtain the same results as if I applied the Feynman rules or check that these are correct.

Through Feynman rules, I know that the amplitude for the t-channel of the process can be written as

$$i\mathcal{M}_t=(-ie)^2\bar{u}(p_3)\gamma^\mu u(p_1)\frac{-i}{t}\bar{u}(p_4)\gamma_\mu u(p_2)$$

In order to know the correct order for the terms here, I follow the fermionic lines in the opposite direction of the charge flow. This way I obtain a Dirac number $$\bar{u}(p_3)\gamma^\mu u(p_1)$$ multiplied by another Dirac number $$\bar{u}(p_4)\gamma_\mu u(p_2)$$. This makes sense because I could have chosen to write any of the two fermionic lines first, but since they just give a Dirac number each, the order does not matter.

However, when I now try to get the same result by calculating the S-matrix element $$$$and taking out the unnecessary constants and delta function of 4-momentum conservation, I need to apply the corresponding fields that annihilate the initial state and create the final state.

My understanding is that this should be done by normal ordering the terms of the fields that contribute to the process, which in fact gives me the terms that create on the left and the ones that annihilate on the right, in the order as $$$$ suggests.

In this case, following this argument, I obtain that the terms with Dirac structure (contributing fields and gamma matrices) should appear in the following order:

$$\bar{\psi}^-_3 \gamma^\mu \bar{\psi}^-_4 \bar{\psi}^+_1 \gamma_\mu \bar{\psi}^+_2$$

where fields with a - (negative frequencies) create and the ones with + (positive frequencies) annihilate.

After going to momentum space I have

$$\bar{u}(p_3) \gamma^\mu \bar{u}(p_4) u(p_1) \gamma_\mu u(p_2)$$

Doing the integrations I then obtain finally as the amplitude

$$i\mathcal{M}_t=(-ie)^2\bar{u}(p_3) \gamma^\mu \bar{u}(p_4) u(p_1) \gamma_\mu u(p_2)$$

The whole thing is a Dirac number, since we can contract first $$\bar{u}(p_4) u(p_1)$$, getting a number, and then it is just the $$\bar{u}(p_3)$$ multiplied with a square matrix multiplied with $$u(p_2)$$.

Nevertheless, the result is clearly not the same as the one that I obtain from the Feynman rules since they do not commute.

I am relatively sure that I need to order the fields differently from the beginning in order to obtain the same result, but since the S-matrix calculation is supposed to be independent from what I do with the Feynman rules (in fact, they should be proven from here), I do not understand how I am supposed to get to the same result without looking at the diagram and the direction of fermionic lines.

Another thing that I am not sure about is how to treat the gamma matrices in the mixture of terms when I am normal ordering fermionic fields... To be honest, in my procedure I have simply left them at the same space, only moving around the fields.

I know that normal ordering is used to determine if different scattering channels need to have a relative minus sign for the total amplitude or not, but I think that I am not applying it correctly in this context.

I hope I expressed myself clearly enough! Any help would greatly appreciated, thank you!

• It might help to keep track of all the spinor indices when doing this calculation – Triatticus Jun 10 at 19:09