# Wick Theorem: Performing contractions in the right order

The first line is one of four terms that one gets after applying Wick theorem to the time-ordered product of these field operators and as far as i understand it is just a short-hand notation for which operators have to be contracted in this case.

I dont unterstand how to get the right order of the contractions as shown in the second line, given the first line. First guess was naively performing the contractions shown in the first line as they appear from left to right, starting with $a_{\textbf{p´}}^-$ contracted with $A^{\nu}(y)$ and then $b_{\textbf{q´}}^-$ contracted with $\overline{\psi}(y)$ and so on... which is obviously not the right way to do it.

Of course only from physical reasoning the ordering of the contractions in the second line makes perfect sense (for example: the Fermion propagator should stand between the gamma matrices). But i am looking for rigorous rules how to do the ordering in a strict way.

Has anyone an idea how to explain them in an easy way? • This site supports MathJax (LaTeX) notation for mathematical formulae and it's generally preferable to use that instead of embedded images. However, I'm not fully sure how to typeset Wick contractions, so that might require some additional help. – Emilio Pisanty Jan 6 '18 at 23:29
• @EmilioPisanty: how would you do the connecting lines above and below the expression? – flippiefanus Jan 7 '18 at 3:05
• @flippiefanus As I said, I don't know how to typeset the Wick contractions in the image; if I did, I would have edited it directly. In standard LaTeX there appears to be a simplewick package that can do this, but I suspect it won't be available on MathJax. This needs attention from someone who actually works on QFT to be typeset correctly. – Emilio Pisanty Jan 7 '18 at 3:15

## 1 Answer

I doubt that there is a typo in your second line, i.e. the order of two gamma matrices should be reversed. You can check it by observing that the expression is for a Compton scattering process.

Anyway, you can start from the first line to the form of second line by simply placing the contracted fields together (because they are C numbers from now on) with the gamma matrices in the correct order. Note that when a Grassmann number passes another, a change of sign is needed. That is why there is an overall -1 in the front.

• I dont see why the gamma matrices in the second line should be reversed. – Johnny90 Jan 8 '18 at 23:14
• On the one hand i know that a contraction of two fields should just be a C number but on the other hand that would mean i can simply pick a any order for writing down the contractions. But that can obviously not be true otherwise i could for example write the contraction with $\overline{\psi}(x)$ and $\psi (y)$ at the beginning and thus have, after conctracting all the other contractions afterwards, the fermion propagator at the beginning of the expression for the amplitude and not between expressions for external lines as it should be the case by looking at the corresponding feynman diagram – Johnny90 Jan 8 '18 at 23:29
• @Johnny90 The contraction between the two fermion fields is also just a complex number. – probably_someone Jun 15 '18 at 13:13