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I'm trying to prove that the conformal weight of the bosonic vertex operator [itex]:e^{ik\cdot X}:[/itex] is [itex]\left(\frac{\alpha'k^2}{4},\frac{\alpha'k^2}{4}\right)[/itex].

I've done some algebra but I think I am making some mistake with a factor of 2 somewhere because I get a 1/2 instead of a 1/4. My attempt is detailed in the attachment.

Can someone please tell me what is wrong here? Its probably a dumb thing, but I don't see it.

Thanks!

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# Conformal weights of the vertex operator

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