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A descendant is not something that has derivatives in it. It is something that is a total derivative of something else $\mathcal{O}' = \partial_\mu \mathcal{O}$. More precisely, the definition of a primary operator is $$ K_\mu \mathcal{O}(x) = 0\,, $$ where $K_\mu$ is the generator of special conformal transformations. You can prove that you operator is a ...


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First, the one-particle state localized in the point, e.g. $\phi(x)|0\rangle$ is very singular. What behaves more nicely insteas is a smeared state $\int_V d^D x \eta(x) \phi(x)|0\rangle$. However if we look at this "particle at a point" we will see, $$ \phi(0)=\int \frac{d^{D-1}p}{(2\pi)^D\sqrt{2 E_p}} a^\dagger(\vec{p})|0\rangle $$ I.e. to produce ideally ...


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