# Correlation function of time derivative of fields

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 $$x_1-x_2$$): $$$$\langle\phi(x_1)\phi(x_2) \rangle=f(x_1,x_2)$$$$ How do I compute the correlator of the time derivative of the fields? Namely $$$$\langle\dot{\phi}(x_1)\phi(x_2) \rangle$$$$ and $$$$\langle\dot{\phi}(x_1)\dot{\phi}(x_2) \rangle$$$$ I can imagine that I have somehow to derive $$f$$. However the left hand side contains two fields, so I don't know whether the Leibniz rule is involved or not.

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.