Correlation function of time derivative of fields

Question

Suppose you have calculated a two point function for a field $\phi$, and the result is some function of the positions (it can be a generic function, not necessary a function of the distance $x_1-x_2$): \begin{equation} \langle\phi(x_1)\phi(x_2) \rangle=f(x_1,x_2) \end{equation} How do I compute the correlator of the time derivative of the fields? Namely \begin{equation} \langle\dot{\phi}(x_1)\phi(x_2) \rangle \end{equation} and \begin{equation} \langle\dot{\phi}(x_1)\dot{\phi}(x_2) \rangle \end{equation} I can imagine that I have somehow to derive $f$. However the left hand side contains two fields, so I don't know whether the Leibniz rule is involved or not.

Answer 1

Consider first a correlator which is not time ordered $$G_n(x_1,\cdots,x_n)=\langle \Omega|\phi(x_1)\cdots \phi(x_n)|\Omega\rangle\tag{1}.$$

This is a function of the $n$ points. Therefore it can be differentiated with respect to the various coordinates $x_i^\mu$. Say you take the derivative with respect to $x_i^\mu$, then the derivative does not affect any of the $\phi(x_j)$ where $j\neq i$ and also won't affect the states which do not depend on the points at all:

$$\dfrac{\partial}{\partial x_i^\mu}G(x_1,\cdots,x_n)=\langle \Omega|\phi(x_1)\cdots \phi(x_{i-1})\dfrac{\partial \phi}{\partial x_i^\mu}(x_i)\phi(x_{i+1})\cdots \phi(x_n)|\Omega\rangle.\tag{2}$$

If the correlator is time ordered $$G_n(x_1,\cdots,x_n)=\langle \Omega |T\{\phi(x_1)\cdots\phi(x_n)\}|\Omega\rangle\tag{3}$$

then care must be taken because $T\{\}$ hides step functions in the time coordinates of the various points and so there is extra dependence on the various $x_i^0$ coordinates. For example $$G_2(x_1,x_2)=\theta(x_1^0-x_2^0)\langle \Omega|\phi(x_1)\phi(x_2)|\Omega\rangle+\theta(x_2^0-x_1^0)\langle\Omega|\phi(x_2)\phi(x_1)|\Omega\rangle\tag{4}.$$

In that case if you take derivatives with respect to $x_i^a$, where $a=1,2,3$ does not involve time, then the derivative will pass through the step functions, and hence pass through the $T\{\}$ and just hit $\phi(x_i)$. But if some derivative involves $x_i^0$ then you must use the product rule for derivatives and in particular differentiate the step function.