- #1
spaghetti3451
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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?
$$\langle 0 | T \{ \phi(x_{1}) \phi(x_{2}) \phi(x) \partial^{\mu}\phi(x) \partial_{\mu}\phi(x) \phi(y) \partial^{\nu}\phi(y) \partial_{\nu}\phi(y) \} | 0 \rangle.$$
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?