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I am reading the 7th chapter of Di Francesco's CFT book. It builds, for example in section 7.3, a null state |x> which is orthogonal to the whole Verma Module. The author asserts that the field x representing the state has the property that any correlation function containing the field x vanishes, which the author does not provide the reader with a justification.

Now I am confused by the assertion. Why is the assertion true? Can anyone give me an explanation? Thanks.

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The two-point functions of the null state $N(z,\bar z)$ with an operator $O(z,\bar z)$ is proportional to the inner product of the corresponding states $\langle n| o\rangle $, so it vanishes. The three-point and higher-point functions $NOP\cdots $ vanish as well because the products $OP\cdots $ may be expanded, using OPEs, in terms of "single operators", and that is how the correlator is reduced to the two-point function.

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  • $\begingroup$ Thanks Luboš Motl. I was just struggling over why the two point functions vanish. $\endgroup$ – Li Xinghe Jan 27 '14 at 17:13

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