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I have these basic questions about the OPEs and path integrals. In this question, I am considering string theory on worlds sheet with conformal gauge.

Q1. When we write OPE between two operators say $X(z,\bar{z})X(0,0)$ does it really mean the insertion of the product in the path integral?

As a mathematical expression is this correct for the OPE?

$$X(z_1,\bar{z}_1)X(z_2,\bar{z}_2)= \int \,[..] X(z_1,\bar{z}_1)X(z_2,\bar{z}_2) \exp(-S)\tag{1}$$

Where $S$ is the Polyakov action in the conformal gauge.

Q2. Say $Q$ is some conserved charge and $X(z,\bar{z})$ is some operator when we write the commutation $[Q, X(z,\bar{z})]$ for some conserved charge $Q$, it becomes a contour integral of time-ordered product between conserved current associated to $Q$ and $X(z,\bar{z})$. Then for that time-ordered product, we use the OPE derived using the path integral. So, does this means the commutator $[Q, X(z,\bar{z})]$ is actually inside the path integral, even though we do not write it explicitly?

Qmechanic
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htr
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1 Answers1

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  1. OP's identification (1) is not correct: The path integral $\frac{1}{Z}\int\!{\cal D}X~F(X) e^{iS/\hbar}$ is equal to the correlation function $\langle F(X)\rangle$, not the operator $F(X)$ itself. Here the operator $F(X)$ could be an OPE.

  2. For the relation between commutator and OPE, see e.g. this Phys.SE post.

Qmechanic
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  • That is correct. But my question was about operator products as the operators approach each other. The question is motivated by the Polchinski equations 2.2.1, 2.2.2, and 2.2.3. – htr Jan 28 '23 at 05:21