# A Free theory time-ordered correlation functions with derivatives of fields

1. Nov 18, 2016

### spaghetti3451

Consider the following time-ordered correlation function:

$$\langle 0 | T \{ \phi(x_{1}) \phi(x_{2}) \phi(x) \partial^{\mu}\phi(x) \partial_{\mu}\phi(x) \phi(y) \partial^{\nu}\phi(y) \partial_{\nu}\phi(y) \} | 0 \rangle.$$

The derivatives can be taken out the correlation function to give

$$\partial^{\mu'}\partial_{\mu''}\partial^{\nu'}\partial_{\nu''}\langle 0 | T \{ \phi(x_{1}) \phi(x_{2}) \phi(x''') \phi(x') \phi(x'') \phi(y''') \phi(y') \phi(y'') \} | 0 \rangle.$$

There are six distinguishable field points in the correlation function:

$$\phi(x_{1})\qquad\phi(x_{2})\qquad\phi(x''')\qquad\phi(x')\qquad\phi(y''')\qquad\phi(y').$$

$\phi(x')$ and $\phi(x'')$ are not distinguishable because the derivative operator acts on both of these fields. Same goes for $\phi(y')$ and $\phi(y'')$.

How many different Feyman diagrams exist for this time-ordered correlation function?

2. Nov 23, 2016