conformal, Weyl transformations, apparent discrepancies and confusions

Question

Because of the apparent discrepancy of how some CFT and GR books define conformal transformation unlike in string theory area, I wanted to get rid of all the confusion from McGreevy's lecture notes:

http://ocw.mit.edu/courses/physics/8-821-string-theory-fall-2008/lecture-notes/lecture10.pdf

I have understood and happy with the definition of conformal group and conformal transformation now, but I have the following two questions:

1. In page 2, where he defined conformal invariance, did he mean invariance under the diffeomorphism part only (which gives the conformal group)? The reason I think so is that he changed the metric which according to his definition of conformal transformation don't change. So, the reason I am asking is that I thought conformal invariance will be a suitable name for a theory, invariant under conformal transformation.

2. Just to make sure: the transformation in page 2: $\delta g_{\mu\nu}=2\lambda g_{\mu\nu}$ and $\delta g_{\mu\nu}=\delta\Omega(x)\delta g_{\mu\nu}$, these are results of diffeomorphism ..right? and not Weyl in any way. In other words, what are these transformations?

All my questions are for general dimension and general space time (classically of course).

Answer 1

It's somewhat unclear how you "understood" and are "happy" about the definition of the conformal transformations because your questions, while referring to page 2 and other things, are nothing else than misunderstandings about the definition of a conformal transformation which is explained on page 1, not 2.

John wrote his equation 1 which states that conformal transformations "are" diffeomorphisms that only change the metric by a local scalar coefficient - by a Weyl rescaling. However, the invariance of a theory under these diffeomorphisms is a trivial property. If one is allowed to change the terms coupled to the "metric", a diffeomorphism-transformed theory has a different action in general, and it is always possible to rewrite the original action in the conformally transformed coordinates.

But what's physically nontrivial is the condition that the action, in its original form, is actually invariant under the operations - that's what we mean by the theory's being conformally invariant. If a theory is conformally invariant, we don't allow any "change of the coefficients" in the integral of the Lagrangian density. This condition of conformal invariance, as he shows, is equivalent to the invariance of the theory under the "Weyl rescaling" only: we just completely eliminate the diffeomorphisms from the picture.

So:

1. Again, the invariance under the combined "diffeomorphism" and "Weyl rescaling" (the latter changes the form of the action) is a tautology. Obviously, by conformal invariance, we don't mean a tautology, so by conformal invariance, we mean the invariance of the action under the diffeomorphism separately, without changing the form of the metric in the action. Because the invariance under the "combo" is tautological, conformal invariance is equivalent to the invariance under the "Weyl rescaling part" of the transformation only.

2. No, on page 2, the transformations are exactly what the equations say: point-dependent transformations of the metric tensor itself i.e. a Weyl rescaling. There is no diffeomorphism at this stage. The invariance under those Weyl transformations is equivalent to the invariance under some diffeomorphisms (with the modification of the metric erased), as explained in the previous point.

The formalism may depend on classical physics but your comment that it is "classically only of course" is incorrect, too. All those facts about conformal transformations are completely valid quantum mechanically as well - and indeed, this background is primarily mean to understand some quantum theories.