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I am reading a paper by J. Polchinski, called "What is string theory", hep-th/9411028. In eq. 1.1.9, and the line before it, the author seems to have used:

$$\langle \partial_z \partial_{\bar{z}} X(z,\bar{z}) X(z',\bar{z}')\rangle = \partial_z \partial_{\bar{z}} \langle X(z,\bar{z}) X(z',\bar{z}')\rangle ,$$

where the symbol $\langle F(X)\rangle$ denotes the expected value of $F(X)$ with respect to the "measure" $\exp(-S) [\mathrm dX]$. In other words,

$$\langle F(X)\rangle = \int [\mathrm dX] \exp(-S) F(X)$$

(it is a path integral). My question is, why can one pull out the partial derivatives $\partial_z$ and $\partial_{\bar{z}}$ outside the path integral? Or did I misunderstand something? Can someone please give an argument for this, or explain what the author meant perhaps (in case I misunderstood)?

UPDATE: I understood better what is going on after I have read section 7-2 in the book "Quantum Mechanics and Path Integrals" by Feynman and Hibbs.

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The complex world sheet (WS) coordinate $z\in\mathbb{C}$ contains a WS time and a WS space coordinate. It is important to realize that the time derivatives inside the Boltzmann factor in the path integral should respect the underlying time slicing procedure. See e.g. this & this Phys.SE answers, and my Phys.SE answer here. Thus it is implicitly understood that the time-differentiations should be taken after the time ordering (although the notation sometimes unfortunately suggests the opposite order).

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  • $\begingroup$ Ok, I thank you Qmechanic. It will take me some time to absorb this well. I will carefully read the 3 posts you have linked to. $\endgroup$ – Malkoun May 26 '16 at 15:12

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