I want to clarify the relations between a few different sets of operators in a conformal field theory, namely primaries, descendants and operators that transform with an overall Jacobian factor under a conformal transformation. So let us consider the the following four sets of operators:(adsbygoogle = window.adsbygoogle || []).push({});

Primaries:Defined by the fact that they are annihilated (commute with) the lowering operators of the theory.

Descendants:Obtained by acting on primaries with raising operators.

Set 3:Operators that scale as [itex]\mathcal{O'}(x')=\lambda^{-\Delta/d}\mathcal{O}(x)[/itex] under a scaling (dilatation).

Set 4:Operators that transform as [itex]\mathcal{O'}(x')=|\partial x'/\partial x|^{-\Delta/d}\mathcal{O}(x)[/itex] under any conformal transformation.

What are the relation between the above sets? I think all descendants are also part of set 3 (but not the converse, since there could probably also exist linear combinations of descendants that are part of set 3). All primaries should also be part of set 4, but I am not sure about the converse.

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# A Primaries, descendents and transformation properties in CFT

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