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I am trying to understand a calculation using Wick's theorem. Let $T(z)$ be the analytic part of a stress-energy tensor, and $\phi(z)$ a free boson field.

Now, $$T(z)\partial_{w}\phi(w)=-2\pi:\partial_{z}\phi(z)\partial_{z}\phi(z):\partial_{w}\phi(w).$$ Using Wick's theorem, we know that $$:\partial_{z}\phi(z)\partial_{z}\phi(z):\partial_{w}\phi(w)=:\partial_{z}\phi(z)\partial_{z}\phi(z)\partial_{w}\phi(w):+2\langle \partial_{z}\phi(z)\partial_{w}\phi(w)\rangle :\partial_{z}\phi(z):$$. Then why is this just equal to $$2\langle \partial_{z}\phi(z)\partial_{w}\phi(w)\rangle \partial_{z}\phi(z)?$$ as stated in many CFT books?

huyichen
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1 Answers1

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Normal-ordered non-singular terms in the OPE are still there in principle, but they are sometimes omitted (and therefore only implicitly implied) in the notation. This is because many important physical quantities only depend on the singular terms of the OPE.

By the way, speaking of implicitly implied things, be aware that many authors don't write the radial ordering symbol $\cal R$ explicitly.

Qmechanic
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