By user260822


2015-12-29 05:37:17 8 Comments

I come up with this conclusion after reading some books and review articles on conformal field theory (CFT).

CFT is a subset of FT such that the action is invariant under conformal transformation of the fields and coordinate but leave the metric unchanged.

Is this correct?

Let me explain further and take the $\phi^4$ theory in $4$-dim as an example (just discuss classical invariance, I know that loops break the invariance). In $4$-dim consider a scalar field with conformal weight $\Delta=1$ such that \begin{align} x \to x' = \lambda x,\\ \phi'(x')=\phi'(\lambda x) = \lambda^{-1}\phi(x). \end{align} Then the action is unchanged \begin{align} S'& = \int d^4 x' \sqrt{g}\left\{\frac{1}{2}g^{\mu\nu}\partial'_{\mu}\phi'(x')\partial'_{\mu}\phi'(x')-\phi'^4(x')\right\}\\ &= \int d^4 x \lambda^4 \sqrt{g}\left\{\frac{1}{2}g^{\mu\nu}\lambda^{-4}\partial_{\mu}\phi(x)\partial_{\mu}\phi(x)-\lambda^{-4}\phi^4(x)\right\} \\ &= \int d^4 x \sqrt{g} \left\{\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi(x)\partial_{\mu}\phi(x)-\phi^4(x)\right\}. \end{align} Note that I did not use $g'^{\mu\nu} = \lambda^2g^{\mu\nu}$, all metrics are unprimed. In this example we see that conformal invariance is realized without changing the metric. I was confused at the beginning since all textbooks and articles derive the conformal group and representations by considering the change of the metric.

If we use $g'_{\mu\nu}x'^{\mu}x'^{\nu} = g_{\mu\nu}x^{\mu}x^{\nu}$, the physical distance does not change at all and we are just choosing a new coordinate chart. My interpretation is, as what we meant by a physical scaling or transformation, we really change the distance between two points. Another reasoning is, the metric in CFTs are just background (not being integrated in the path integral) thus we do not change them. If we consider a theory including the metric as a dynamical field (we path integrate it and perhaps quantize it), the actions has to be invariant including the transformation of the metric.

Is the above correct? Please give me some comments and point out the wrong concepts if there is any. Thank you very much.

If you have time, could you please take a look at my other question.

2 comments

@MannyC 2018-12-13 23:59:54

Imposing that an action should be conformally invariant has a subtle but important difference with respect to imposing that it should be diffeomorphism invariant. Let me try to emphasize the differences between Diffeomorphisms, Conformal transformations and Weyl transformations. I will also clarify how to impose the invariance of the action, namely I'll argue whether the fields or the coordinates must be changed.

Diffeomorphisms

The action on the coordinates is given by a general differentiable function with differentiable inverse: $x^\mu \to {x'}^\mu = {x'}^\mu(x)$. The action on fields with spin 0, 1 and 2 is the following $$ \begin{align} \phi(x) &\to \phi'(x') = \phi(x)\,, \\ \partial_\mu\phi(x) &\to \partial'_\mu\phi'(x') = \frac{\partial x^\nu}{\partial{x'}^\mu}\partial_\nu\phi(x)\,, \\ g_{\mu\nu}(x) &\to g'_{\mu\nu}(x') = \frac{\partial x^\alpha}{\partial {x'}^\mu}\frac{\partial x^\beta}{\partial {x'}^\nu}g_{\alpha\beta}(x)\,. \end{align} $$ These transformations are just changes of variables, every theory is invariant under them.

Weyl transformations

Weyl transformations do not act on the coordinates but simply rescale the fields $$ \begin{align} \phi(x) &\to \tilde{\phi}(x) = \Omega^{-\Delta}(x)\phi(x)\,, \\ \partial_\mu\phi(x) &\to \partial_\mu\tilde{\phi}(x) = \partial_\mu(\Omega^{-\Delta}(x)\phi(x))\,, \\ g_{\mu\nu}(x) &\to \tilde{g}_{\mu\nu}(x) = \Omega^{2}(x)g_{\mu\nu}(x)\,. \end{align} $$

Conformal transformations

Conformal transformations are coordinate transformations that preserve the metric up to a scale factor, that is $x^\mu \to {x'}^\mu$ and $$ \frac{\partial x^\alpha}{\partial {x'}^\mu}\frac{\partial x^\beta}{\partial {x'}^\nu} g_{\alpha\beta}(x)= \Omega^{-2}(x)g_{\mu\nu}(x)\,.\tag{1} $$ In OP's example ${x'}^\mu = \lambda x^\mu$ and $\Omega(x) = \lambda$. Now, this is the key point: we don't want to impose that the theory is invariant under this change of variables, this is always true because it is a particular case of diffeomorphism invariance. We want instead to compare theories with the same metric. That is, we always imagine a conformal transformation as a diffeomorphism satisfying (1) followed by a Weyl rescaling that cancels the $\Omega$ factor. Therefore the transformation laws are $$ \begin{align} \phi(x) &\to \tilde{\phi}'(x') = \Omega^{-\Delta}(x)\phi(x)\,, \\ \partial_\mu\phi(x) &\to \partial'_\mu\tilde{\phi}'(x') = \frac{\partial x^\nu}{\partial{x'}^\mu}\partial_\nu(\Omega^{-\Delta}(x)\phi(x))\,, \\ g_{\mu\nu}(x) &\to \tilde{g}'_{\mu\nu}(x') = g_{\mu\nu}(x)\,. \end{align} $$ With the notation "tilde prime" I denote the composition of the two last transformations.

Invariance of the action

Consider a generic Lagrangian that depends on the field $\phi$, its first derivative and the metric tensor. The action then reads $$ S = \int d^4x \, \mathcal{L}[g_{\mu\nu}(x),\phi(x),\partial_\mu\phi(x)]\,. $$ The transformation acts only on the fields and leaves $x$ unchanged since it is a dummy variable (integrated over). Proving diffeomorphism invariance amounts to requiring $$ \begin{align} S &= \int d^4x\, \mathcal{L}\left[g'_{\mu\nu}(x),\phi'(x),\partial'_\mu\phi'(x)\right] = \\ &=\int d^4x'\, \mathcal{L}\left[g'_{\mu\nu}(x'),\phi'(x'),\partial'_\mu\phi'(x')\right] = \\ &=\int d^4x\,|\Omega(x)|^4 \mathcal{L}\left[\Omega^{-2}(x)g_{\mu\nu}(x),\phi(x),\frac{\partial x^\nu}{\partial{x'}^\mu}\partial_\nu\phi(x)\right]\,, \end{align} $$ in the second step I just renamed $x$ to $x'$, it's a trivial operation. In the last step I used the transformation properties of the measure and of the fields (See Di Francesco - Mathieu - Sénéchal for example).

A Weyl rescaling requires simply $$ S = \int d^4x\, \mathcal{L}\left[\Omega^2(x)g_{\mu\nu}(x),\Omega^{-\Delta}(x)\phi(x),\partial_\mu(\Omega^{-\Delta}(x)\phi(x))\right]\,. $$ Finally, the requirement for conformal invariance can be obtained by composing the two transformations above $$ S = \int d^4x\, |\Omega(x)|^4\mathcal{L}\left[g_{\mu\nu}(x),\Omega^{-\Delta}(x)\phi(x),\frac{\partial x^\nu}{\partial{x'}^\mu}\partial_\nu(\Omega^{-\Delta}(x)\phi(x))\right]\,. $$ This is consistent with what OP says. Notice that since diffeomorphism invariance is always true, just proving Weyl invariance is enough. The converse is not obvious and it is object of current research. We could say that by imposing conformal invariance we are imposing Weyl invariance only by those $\Omega$'s that make $g_{\mu\nu}$ and $\Omega^2 g_{\mu\nu}$ diffeomorphic.

@knzhou 2018-12-22 16:11:01

This answer looks great to me, but if it really is this way, then why do books seem to go out of their way to hide it? Does, e.g. di Francesco ever say anything explicit about this?

@MBolin 2019-03-28 18:26:25

Sorry but I am confused about something. The transformation of the metric tensor under a general diffeomorphism comes up in GR by imposing that the line element $ds^2$ doesn't change under the change of coordinates. But this doesn't allow for instance a rescaling, where distances grow. So from this it seems that conformal transformations are not a subset of general diffeomorphisms, while in the definition you give of conformal transformations they are clearly a subset.

@MBolin 2019-03-28 19:03:22

Moreover, if you take your definition of a conformal transformation and consider a simple rotation (conformal transformation as far as I know), it is clear that you don't have $\frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} g_{\alpha \beta}(x) = \Omega^{-2}(x) g_{\mu \nu}(x)$.

@MannyC 2019-03-28 19:40:15

Diffs are changes of variables, a rescaling would be like measuring a distance in miles rather thank kilometers, so the physical line element is unchanged, while the metric does take an overall factor. Rotations are a subset of the conformal group for a conformally flat metric (i.e. a metric proportional to $\eta_{\mu\nu}$). Recall that for a rotation the Jacobian is an orthogonal matrix $O$ and thus $O^T\eta O=\eta$.

@MBolin 2019-03-28 22:15:42

I think your definition of conformal transformation is wrong. You should have $x'$ on the right hand side. See my self-answer here: physics.stackexchange.com/questions/469205/…

@MannyC 2019-03-28 23:07:02

Thanks for the comment. I have to agree that your definition seems more natural in terms of the analogy with isometries, but the one I wrote here is the standard one. It is equivalent to the statement "Conformal transformations are special Weyl transformations such that the transformed metric is diffeomorphic to the original metric" -- arxiv.org/abs/1702.07079

@P. C. Spaniel 2019-04-04 04:54:18

I have a question about conformal transformations being a diffeomorphism followed by a Weyl transformation. In Francesco's CFT (and many others) the typical representation of the generator of scale transformations is $D=-i(x^{\mu}\partial_{\mu}+\Delta)$ meaning that when we transform the fields we get $\phi'(x')=\lambda^{-\Delta}\phi(x)$ without the need of any Weyl transformation to compensate anything.

@knzhou 2019-04-04 08:45:40

@P.C.Spaniel No, if you just do a diffeomorphism then you get $\phi'(x') = \phi(x)$ for a scalar $\phi$, essentially by definition. You can't get the $\lambda^{-\Delta}$ factor without also doing the Weyl transformation. But di Francesco (and many others) do a very poor job of making this clear.

@MannyC 2019-04-04 17:14:26

Elaborating on that, the reason why you see that textbooks do rescale by that $\lambda$ factor when doing a scale transformation is because you always have to compare theories with the same metric. So you have to do a Weyl transformation to bring back the metric to the original $g_{\mu\nu}$.

@P. C. Spaniel 2019-04-04 17:39:22

okey... Everything you are saying really makes sense. Is there a book about cft or string theory saying anything about this? or is it an obscure piece of information that no book says but everyone silently knows? I'd like to read more about it. Thanks for the answers!

@knzhou 2019-04-04 23:37:48

@P.C.Spaniel David Tong's notes on string theory partially make the distinction. He spends a paragraph distinguishing conformal transformations with a "dynamical metric" (diffeomorphism alone) and a "background metric" (the notion used here, which is the one people actually care about). These notes don't use the full language used here, though, and they also are occasionally a little bit sloppy.

@L. Spodyneiko 2015-12-29 09:03:25

I think in your case it is better to see conformal transformation this way:

  1. You don't change coordinates
  2. Field with conformal dimension $\Delta$ transforms as $\widetilde \Phi = \lambda^{- \Delta} \Phi$
  3. Field $\phi$ has $\Delta=1$ and metrics $g_{\mu\nu}$ has $\Delta = 2$.

So parts of the action change just as you wrote, but for different reasons. Volume form change beacause $\sqrt{\det \widetilde g} = \sqrt{\det \widetilde g_{\mu\nu}}= \lambda^{4} \sqrt {\det g}$. Derivative $\partial_\mu$ doesn't change, however term $g^{\mu\nu}$ changes as $\widetilde g^{\mu\nu}=\lambda^{-2}g^{\mu\nu}$. And the $\phi$ scales the same way.

This gives the same result as you wrote in your question, but here it is more clear that we do actual physical transformation, not just change of coordinates.

In your case, you make conformal transformation as I just described it along with coordinate transformation $x \rightarrow \lambda x$. All together the fields transform as you wrote $dx^\mu \rightarrow \lambda dx^\mu, g_{\mu\nu}\rightarrow g_{\mu\nu}$ e.t.c., except that metric actually changes $\widetilde{ ds}^2 = \lambda^2 g_{\mu\nu} dx^\mu dx^\nu=\lambda^2 ds^2$.

@user260822 2015-12-29 09:51:06

Great! This is another side of my question. It seems that we can think of it like either changing the metric or coordinate but never both. If I regard it as a change of metric and the field contents, then how is it differ from Weyl symmetry? Thank you very much.

@L. Spodyneiko 2015-12-29 17:05:13

Weyl transformation is a local change of metric and here we've been discussing the global one.

@user260822 2015-12-30 06:29:19

Conformal can be also local. Do you mean that they are the same? difference between conformal and weyl

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