Free theory time-ordered correlation functions with derivatives of fields

[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?

 

[eljose79]

I would like to clarify that the correlation function provided in the forum post is a hypothetical example and does not represent any specific physical phenomenon. However, in general, the number of Feynman diagrams that exist for a given correlation function depends on the number of distinguishable field points in the correlation function.

In this case, there are six distinguishable field points, but only four of them (x', x'', y', y'') can be connected by the derivative operator. This means that there are four distinguishable field points and therefore, four different Feynman diagrams that can be drawn for this correlation function.

It is important to note that the number of Feynman diagrams is not always equal to the number of distinguishable field points. In some cases, multiple distinguishable field points can be connected by the same operator, leading to a smaller number of Feynman diagrams.

Furthermore, the specific form and structure of the correlation function can also affect the number of Feynman diagrams. For example, if the correlation function contains terms that are symmetric or antisymmetric under exchange of field points, the number of Feynman diagrams may be reduced.

In summary, the number of different Feynman diagrams that exist for a given correlation function depends on the number of distinguishable field points and the specific form of the correlation function.