# Identity in CFT

I heard and read couple of times reference to a certain identity in conformal field theory (maybe specific to two dimensions). The identity relates the trace of stress-energy tensor to the beta functions of all the operators in the simplest way: $$T^{\mu}_{\mu} = \sum_{i}\beta({\cal{O_i}}){\cal{O_i}}$$ It's an operator identity and the sum runs over all operators (primary and descendants).

It makes senses since vanishing beta functions and "tracelessness" are two marks of conformal symmetry. If it exists, I would find it really elegant (who cares?) and I would appreciate to read the demonstration, it strangely isn't part of the textbook material on the subject.

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The trace of the stress-energy tensor is the generator of the $x^\alpha\to \lambda x^\alpha$ scaling transformations. Such a generator is an operator so it's enough to know how it acts on all the states or, equivalently, operators. Under such a scaling, an operator changes by the appropriate multiple of its beta-function (times the same operator), so if you sum this transformation over all operators, you get the operator that correctly transforms everything, and the identity above therefore holds. I didn't write this as answer because I expect the answer to use the formulae. –  Luboš Motl May 7 '13 at 10:18