# Conformal transformation vs diffeomorphisms

I am reading Di Francesco's "Conformal Field Theory" and in page 95 he defines a conformal transformation as a mapping $$x \mapsto x'$$ such that the metric is invariant up to scale:

$$g'_{\mu \nu}(x') = \Lambda(x) g_{\mu \nu} (x).$$

On the other hand we know from GR that under any coordinate transformation the metric changes as

$$g_{\mu \nu} (x) \mapsto g'_{\mu \nu}(x') = g_{\alpha \beta} (x) \frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} .$$

I feel like there is a notation problem (inconsistency) in these formulas, or maybe I am mixing active and passive coordinate transformations. For instance, if we consider a simple rotation (which is of course a conformal transformation with no rescaling, i.e. $$\Lambda(x)=1$$) then from the first formula we see that $$g'_{\mu \nu}(x') = g_{\mu \nu} (x)$$, whereas from the second formula we get something more complicated. Where is the flaw?

In the "String theory" lecture notes by David Tong the same definition of conformal transformation is given. Then he says:

A transformation of the form (4.1) has a diferent interpretation depending on whether we are considering a fixed background metric $$g_{\mu \nu}$$, or a dynamical background metric. When the metric is dynamical, the transformation is a diffeomorphism; this is a gauge symmetry. When the background is fixed, the transformation should be thought of as an honest, physical symmetry, taking the point $$x$$ to point $$x'$$. This is now a global symmetry with the corresponding conserved currents.

I think it has to do with my question, but I don't fully understand it...

• Related: physics.stackexchange.com/q/38138/2451 , physics.stackexchange.com/q/226464/2451 and links therein. Mar 28, 2019 at 17:48
• I feel my question is not fully answered there. Mar 28, 2019 at 17:54
• The actual definition of a conformal transformation is a mess. By my count, the usual references use at least 5 totally different definitions interchangeably, so you're right to be annoyed. I thought the top answer to the second question Qmechanic linked was good. Mar 28, 2019 at 17:56

OK I think I know what is going on. It's all about primes. Consider an active spacetime transformation:

$$x^{\mu} \mapsto x'^{\mu}(x) \, ,$$

$$g_{\mu \nu} (x) \mapsto g'_{\mu \nu} (x') = g_{\alpha \beta} (x) \frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} \, .$$

(the transformation of the metric tensor follows from the fact that it is a rank 2 tensor). With this notation both Di Francesco and David Tong are wrong (as far as I understand). The GR book by Zee on the other hand writes it properly. First of all consider an isometry. This is an spacetime transformation as before that leaves the metric invariant, meaning

$$g'_{\mu \nu} (x') = g_{\alpha \beta} (x) \frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} = g_{\mu \nu} (x') \, .$$

(watch the primes). On the other hand a conformal transformation is a transformation that satisfies a weaker condition: it leaves the metric invariant up to scale, meaning

$$g'_{\mu \nu} (x') = g_{\alpha \beta} (x) \frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} = \Omega^2(x')g_{\mu \nu} (x') \, .$$

Now there should be no inconsistency. Di Francesco's definition was wrong (according to this convention/notation/understanding) because it compared the metric before and after the transformation at different points, and you have to compare them at the same point.

• Did you check what equation do you get by imposing invariance under a small transformation? I mean, if you use your definition do you get the covariantized Killing equation $\nabla_{(a}v_{b)} = \nabla^cv_c\,g_{ab}$? Mar 28, 2019 at 23:10
• Yes, I get $\nabla_{\mu} v_{\nu} + \nabla_{\nu} v_{\mu} = \frac{2}{d} (\nabla \cdot v ) g_{\mu \nu}$. Mar 28, 2019 at 23:32
• Instead if you use the other one there are some pieces missing I imagine? This is very weird... Mar 28, 2019 at 23:38
• Yes, there are missing pieces and it is logical. Try it. From Di Francesco's definition, for a change of coordinates to be a conformal transformation you should have something like $\frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} = \Omega^2(x) \delta^{\alpha}_{\mu} \delta^{\beta}_{\nu}$, which doesn't seem like a conformal transformation in general. Take a look at Zee. Mar 28, 2019 at 23:40
• Damn it, I thought the second block-quoted expression for diffeomorphism in your question to be from Tong which is what I was defending (while thinking that you had quoted it from Tong because I had misread). Yes, now I have no disagreements. Thanks for the responses and for providing a nice addition to my PSE bookmarks ;)
– ACat
Nov 17, 2020 at 13:12

I'm a mathematician, not a physicist, so I learned all of these ideas with different notation, but I think I understand what might be confusing you.

Conformal transformations are indeed a special kind of diffeomorphism, and a rotation (say in the plane with the usual metric) is indeed conformal, so the two formulas you listed had better agree in this case.

But in fact, if your manifold is $$\mathbb{R}^2$$, your metric is the usual one ($$g_{\mu\nu}$$ is the identity matrix at every $$x$$), and your coordinate change is a rotation, the second formula you listed will show you that the metric looks unchanged in the new coordinates. (This is not a coincidence: preserving this metric is exactly the property that makes rotations special in the first place!) That is, there is no conflict between the two formulas here, it's just that seeing it involves a bit of computation.

Working this out is a very good exercise and I don't think you'd gain much from me typing it all out here. A hint that might help you get oriented is that, since rotations are linear in the coordinate system we've chosen, the Jacobian matrix at every point is the same as the matrix for the rotation itself.

• You are right, but I think that is not saying anything. Notice that for a simple metric like Euclidean metric, which doesn't depend on position, the formula for the transformation of the metric $g'_{\mu \nu}(x') = \frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} g_{\alpha \beta} (x)$ and the equation for an isometry $g'_{\mu \nu}(x') = g_{\mu \nu}(x')$ look the same. Mar 28, 2019 at 18:42
• Your answer indeed gives one of the five definitions of the phrase “conformal transformation”. But it isn’t the one that CFT is actually about. Your definition makes conformal invariance just a subset of diffeomorphism invariance. Mar 28, 2019 at 18:43
• This is definitely our (i.e. physicists’) fault. We stole a word from math and used it to mean several different things, and the notation conventionally used is so ambiguous one can never tell between them. Mar 28, 2019 at 18:44
• @knzhou: That's unfortunate! But it does seem like the definition I had in mind writing this is the one that the original question used in the first formula. Can you explain the discrepancy more clearly? I certainly don't want to contribute further to the confusion, so this answer ought to be edited if it's currently doing that. Mar 28, 2019 at 19:28
• @MBolin: As I said in the last comment there may be a difference in the definitions that I'm not aware of, but it's worth pointing out that I think what you just said is not strictly true. There are certainly even linear transformations on the plane that are not conformal (and therefore not isometries). An example is multiplying the $x$ coordinate by 2, that is, $\begin{pmatrix}2&0\\0&1\end{pmatrix}$. Mar 28, 2019 at 19:31

I deleted my previous answer because I was very confused when I wrote it. I realized that I just wanted to stress MBolin's answer.

From a differential geometric point of view, an active transformation of space(time) is a diffeomorphism $$\phi:M\rightarrow M$$. Given a curve $$\gamma$$ with initial velocity $$X=\dot{\gamma}(0)$$ at $$\gamma(0)$$, we can define the new transformed curve $$\phi\circ\gamma$$ which has velocity $$\phi_{*,\gamma(0)}X$$ at the transformed initial point $$\phi(\gamma(0))$$. This is the push-forward of the vector $$X$$. The conformal condition is then just that the angles between two vectors and their push-forwards coincide. In other words, that there is a function $$\Lambda:M\rightarrow\mathbb{R}$$ such that for any two vectors $$X$$ and $$Y$$ at $$p$$ we have $$g_{\phi(p)}(\phi_{*,p}X,\phi_{*,p}Y)=\Lambda(p) g_p(X,Y).$$ The left hand side can be thought of as the pull-back of the metric so this condition is at times written as $$\phi^*g=\Lambda g$$, but this is just a more condensed version of the statement above .

In a coordinate system $$x$$ defined on a chart containing both $$p$$ and its transformed point $$\phi(p)$$, we can write the equation above as $$\Lambda g_{\mu\nu}(x)=\frac{\partial x^\alpha\circ\phi}{\partial x^\mu}\frac{\partial x^\beta\circ\phi}{\partial x^\nu}g_{\alpha\beta}(x)\circ\phi.$$ In here I have used the physicists notation of writing $$g_{\mu\nu}(x)$$ for the composition $$g_{\mu\nu}\circ x$$. In here my convention is that $$g_{\mu\nu}(x(p))=g_p\left(\left(\frac{\partial}{\partial x^\mu}\right)_p,\left(\frac{\partial}{\partial x^\nu}\right)_p\right)$$

There is a parallel (although local) discussion on can make from the passive point of view. In here one is instead replacing the coordinates $$x$$ by a new system $$x':=x\circ\phi^{-1}$$ (thinking of active vs. passive rotations clarifies why this is the correct coordinates to choose). Then the trick for relating these two is to note that $$\frac{\partial x^\alpha\circ\phi}{\partial x^\mu}=\partial_\mu(x^\alpha\circ\phi\circ x^{-1})\circ x=\partial_\mu(x^\alpha\circ {x'}^{-1})\circ x'\circ\phi=\frac{\partial x^\alpha}{\partial x'^\mu}\circ\phi.$$ Then the conformal condition becomes $$\Lambda g_{\mu\nu}(x)=\left(\frac{\partial x^\alpha}{\partial x'^\mu}\frac{\partial x^\beta}{\partial x'^\nu}g_{\alpha\beta}(x)\right)\circ\phi.$$ We recognize that the term in parenthesis is just the metric in the new coordinates $$g'_{\mu\nu}(x')$$. Then we have $$g'_{\mu\nu}(x')\circ\phi=g'_{\mu\nu}\circ x'\circ\phi=g'_{\mu\nu}\circ x\equiv g'_{\mu\nu}(x)$$. We thus recover the statement as in Zee's book that the conformal condition is $$g'_{\mu\nu}(x)=\Lambda g_{\mu\nu}(x).$$