Abstract
For local conformal field theories, it is shown how to construct an expression for the energy-momentum tensor in terms of a Wilsonian effective Lagrangian. Tracelessness implies a single, unintegrated equation which enforces both the Exact Renormalization Group equation and its partner encoding invariance under special conformal transformations.
Highlights
A interesting feature of this equation is that, in general, is the scaling dimension a priori unknown but, so too, is the action
Within the context of approaches that are intrinsically local, are there any options to improve upon the ERG? A clue comes from the fact that (1.6) is only a statement of scale invariance; it does not automatically incorporate full conformal invariance
Whilst it is true that for many theories of interest scale invariance enhances to conformal invariance, this is not a general property of all solutions of (1.6)
Summary
As emphasised in [4], if one is to consider various representations of the conformal algebra, one must be prepared to consider associated representations of the energy-momentum tensor. The right-hand side of (2.19b) is readily seen to be independent of a and so this ambiguity in Hτλ has no effect on the energymomentum tensor Having discussed these generalities, let us return to our explicit solution (2.13) and take the trace: Tαα = T αα + ∂λ Fαλα − Fααλ + Wλαα. Let us return to our explicit solution (2.13) and take the trace: Tαα = T αα + ∂λ Fαλα − Fααλ + Wλαα Comparing this with (2.3c) gives a consistency condition: There is an additional, interesting subtlety in d = 2: it is possible to have a theory for which the action satisfies both dilatation and special conformal invariance but, the quantum theory is not conformal!
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