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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. I apologize for my confusion, and this post, particularly if I have confused others.

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.

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 think that all it is is just an abuse of notation, so to speak. So when the author wrote the right-hand side, he really meant the left-hand side, which can also be reinterpreted in terms of expected values of operators. I understood better after I read section 7-2 in the book "Quantum Mechanics and Path Integrals" by Feynman and Hibbs. I apologize for my confusion, and this post, particularly if I have confused others.

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. I apologize for my confusion, and this post, particularly if I have confused others.

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)?

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 think that all it is is just an abuse of notation, so to speak. So when the author wrote the right-hand side, he really meant the left-hand side, which can also be reinterpreted in terms of expected values of operators. I understood better after I read section 7-2 in the book "Quantum Mechanics and Path Integrals" by Feynman and Hibbs. I apologize for my confusion, and this post, particularly if I have confused others.