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I have problem with deriving $$T(z)X^{\mu}(w,\bar{w})\sim \frac{1}{z-w}\partial X^{\mu}(w,\bar{w}) + \cdots .$$

We know that $$\langle X^{\mu}(z)\partial X^{\nu}(w)\rangle =\frac{1}{4}\eta^{\nu\nu}\frac{1}{z-w}.$$ On the other hand, $$T(z)=-2:\partial X(z)\cdot \partial X(z):$$ Then we have: $$T(z)X^{\mu}(w)= -2:\partial X(z)\cdot \partial X(z): X^{\mu}(w).$$ Now what we can do? I don't know how to use Wick's theorem here. I would appreciate if someone could help me.

Qmechanic
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Mahtab
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1 Answers1

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For three operators $X,Y,Z$, we have: $$XY=:XY:+\langle XY\rangle$$ and $$XYZ=:XYZ:+X\langle YZ\rangle +Y\langle ZX\rangle +Z\langle XY\rangle .$$ Then we have: $$:XY:Z=(XY-\langle XY\rangle )Z=XYZ-\langle XY\rangle Z$$ $$=:XYZ:+X\langle YZ\rangle +Y\langle XZ\rangle .$$ In particular, $$: \partial X\cdot \partial X:X^{\mu}=:\partial X^{\nu}\partial X_{\nu}:X^{\mu}$$ $$=:\partial X^{\nu}\partial X_{\nu}X^{\mu}:+\partial X^{\nu}\langle \partial X_{\nu}X^{\mu}\rangle +\partial X^{\nu}\langle \partial X_{\nu}X^{\mu}\rangle$$ $$=:\partial X^{\nu}\partial X_{\nu}X^{\mu}:+2\partial X^{\nu}\langle \partial X_{\nu}X^{\mu}\rangle \sim 2\partial X^{\nu}\langle \partial X_{\nu}X^{\mu}\rangle .$$ Therefore, $$T(z)X^{\mu}(w)\sim -4\partial X^{\nu}(z)\langle \partial X_{\nu}(z)X^{\mu}(w)\rangle .$$

Mahtab
  • 952