1.4.2 Euclidean gravitational path integral

Conformal transformation

A conformal transformation is a spacetime transformation that is a specific combination of a general coordinate transformation and a Weyl transformation that leaves the metric intact.

The general coordinate transformation part (can be called conformal coordinate transformation) $x\to x’$ of a conformal transformation should be of the form such that \begin{equation} \label{general-def-ct} g_{\rho\sigma}^{\prime} ( x^{\prime} ) \frac{\partial x^{\prime\rho}} {\partial x^{\mu}} \frac{\partial x^{\prime\sigma}} {\partial x^{\nu}}= g_{\mu\nu} ( x )\quad\text{such that}\quad \frac{\partial x^{\prime\rho}} {\partial x^{\mu}} \frac{\partial x^{\prime\sigma}} {\partial x^{\nu}}=\Lambda( x ) \end{equation}

The Weyl transformation part should precisely reverse the above. \begin{equation} g_{\mu\nu}\rightarrow e^{-2\omega (x)}g_{\mu\nu}\quad\text{such that}\quad e^{-2\omega (x)}=\Lambda( x ) \end{equation}

In the first step, the line element $ds^2$ remains the same, but it gains a local factor of $\Lambda (x)$ in the second step. Together, $ds^2\to \Lambda (x) ds^2$, but the angles remain invariant. These 2 steps are often combined and written obscurely as \begin{equation}\label{CFTdefinition} g_{\rho\sigma}^{\prime} ( x^{\prime} ) \frac{\partial x^{\prime\rho}} {\partial x^{\mu}} \frac{\partial x^{\prime\sigma}} {\partial x^{\nu}}= \Lambda( x ) g_{\mu\nu} ( x ) \end{equation}

The CFT is on a non-dynamical manifold. Usually, this manifold is the Minkowski spacetime.

Rant

Almost everywhere, people specify that conformal transformation is a type of ``coordinate transformation”¹. But that is false. Not all spacetime transformations are coordinate transformations. Every physical theory (not just general relativity) has diffeomorphism invariance (general coordinate invariance), and you can check the beginning of the General relativity chapter in my \href{http://ksr.onl/FP}{FP notes} if you want to know why every physical theory from Newtonian mechanics has diffeomorphism invariance.

If conformal transformation were just a subset of coordinate transformations, then conformal symmetry would be present in every theory, which is nonsense. A Weyl transformation changes the distances because it directly changes the metric without changing the coordinates. A coordinate transformation can never change distances, as $ds^2$ is invariant under coordinate transformations. So, you can never scale distances with them. This is the definition of coordinate transformations for centuries. It is an abuse of terminology to come now and say that conformal transformations are a subset of coordinate transformations and change the centuries-old definition.

Weyl symmetry implies conformal symmetry, but the converse is not true. Since if you take a CFT and promote the metric into a dynamical variable, then we might not have Weyl symmetry because only for the specific fixed metric of our CFT we know from the definition of CFT that it is invariant under Weyl transformations. Which might not be true for other metrics.

If we take the subset of conformal transformations where the Weyl part of the transformation is trivial, i.e., $\Lambda(x)=1$, then the corresponding coordinate transformations will give us a subset of diffeomorphisms that locally scale the metric. So, conformal transformations contain a subset of diffeomorphisms. But there are coordinate transformations such as Cartesian to spherical coordinates that don’t locally scale the metric, and these coordinate transformations can’t be part of conformal transformations. But every CFT has the full diffeomorphism invariance, not just this subset.

Conformal Geometry vs Projective Geometry

The geometry that is defined by demanding conformal invariance is called Conformal Geometry ^[1] and on it, lengths are not well defined, but angles are well defined as they are conformally invariant. There is a slightly similar geometry called Projective Geometry, which also has local scaling symmetry. But for a plane, instead of angle preserving, here we have \href{https://en.wikipedia.org/wiki/Cross-ratio}{cross-ratios} preserving geometry since cross-ratios are projective invariant quantities defined on a projective line. Also, Projective Geometry has a preferred origin, unlike the conformal case. Unfortunately, in $D>2$, projective invariant quantities are complicated.

At this point, there is a confusion. If lengths are not defined for conformal geometry, then why do we have lengths in CFT correlation functions, etc., instead of only having angles? The reason is that physicists don’t use conformal geometry \& instead use ordinary Riemannian geometry, and just like we can fix some coordinate system (manifest diffeomorphism invariance is lost), we have fixed some arbitrary (local) length scale for that CFT, losing the manifest conformal symmetry. Distances are well defined in our case \& if we scale all distances locally, it’s like changing the gauge \& the theory doesn’t change. The conformal symmetry is still there even if it is no longer manifest.

If we properly use conformal geometry, correlation functions like $\langle V_{\alpha_1}(z_1,\bar{z}1) V{\alpha_2}(z_2,\bar{z}_2) \rangle$ are not defined as they are length dependent but in $2D$ the ratios of correlation functions that depend only on cross-ratios are well defined. Because in $2D$, cross-ratios are conformal invariant under global conformal transformations (M"{o}bius transformations) \& are determined by angles.

Non-conformal QFTs cannot be defined on conformal geometries because they introduce a length scale from their masses and break the conformal symmetry of that conformal geometry, making it an ordinary manifold. That is, for a non-conformal QFT to live on a conformal manifold, the manifold needs extra structure to measure distances, just like general relativity can’t be defined in topological spaces and needs extra structure.

Mass \& causal structure

Angles in the Lorentzian signature contain the causal structure. So, the causal structure is conformal invariant. Also, if we have a massive particle, its trajectory is a timelike geodesic whose length is the proper time interval from the endpoints \& this length is not conformal invariant. Null geodesics always have zero length between any 2 points, so their length is not well defined. Null geodesics are conformal invariant which is why mass is not allowed for CFTs.

Infinitesimal conformal coordinate transformations: We need to find the subset of coordinate transformations that locally scale the metric. First, let’s consider the infinitesimal case.

\begin{equation} x’^\rho=x^\rho+\epsilon^\rho(x)+\mathcal{O}(\epsilon^2) \, , \end{equation}

From general-def-ct, after defining $K(x) = \Lambda(x) - 1$, it follows that

\begin{equation} \label{inf-constraint-K} \partial_{\mu}\epsilon_\nu + \partial_{\nu}\epsilon_\mu = K(x) \eta_{\mu\nu} \, . \end{equation}

By tracing on both sides, we get $K(x) = \frac{2 \partial^{\mu}\epsilon_\mu} {d}$. Plugging it back gives

\begin{equation} \label{inf-cft-e-condition} \partial_{\mu}\epsilon_\nu + \partial_{\nu}\epsilon_\mu = \frac{2}{d} (\partial\cdot\epsilon) \eta_{\mu\nu} \, . \end{equation}

Now applying $\partial^{\nu}$ on both sides gives \begin{equation} \label{2dCFTinfty} \partial_{\mu} (\partial\cdot\epsilon) + \square\epsilon_\mu = \frac{2}{d} \partial_{\mu} (\partial\cdot\epsilon) \, . \end{equation}

We then take $\partial_{\nu}$ to both sides to get \begin{equation} \label{inf-cft-e-condition-2} \partial_{\mu}\partial_{\nu} (\partial\cdot\epsilon) + \square \partial_{\nu}\epsilon_\mu = \frac{2}{d} \partial_\nu\partial_{\mu} (\partial\cdot\epsilon) \, . \end{equation}

Interchanging $\mu \leftrightarrow \nu$, adding back to inf-cft-e-condition-2 and using inf-cft-e-condition, we get

\begin{equation} \label{e-useful-relation-1} ( \eta_{\mu\nu}\square + (d-2) \partial_{\mu}\partial_{\nu} ) (\partial\cdot\epsilon) = 0 \, . \end{equation}

If we apply trace, we get \begin{equation} \label{e-useful-relation-2} (d-1)\square (\partial\cdot\epsilon) = 0 \, \implies \square (\partial\cdot\epsilon) = 0 \text{ for } d\neq 1. \end{equation}

We can also apply $\partial_{\rho}$ to inf-cft-e-condition and then permute the indices and add/subtract them to get the below.

\begin{equation} \label{inf-cft-e-relation-3} 2\partial_{\mu} \partial_{\nu} \epsilon_\rho = \frac{2}{d} ( - \eta_{\mu\nu} \partial_\rho + \eta_{\rho\mu} \partial_{\nu} + \eta_{\nu\rho} \partial_{\mu} ) (\partial\cdot\epsilon) \, . \end{equation}

From e-useful-relation-2, it is obvious that $\epsilon$ being quadratic is sufficient for $\square (\partial\cdot\epsilon) = 0$, but it is not necessary since there are examples like $\epsilon^\mu=(\frac{t^4}{2},2tx^3,0,0)$. But we can also use the other conditions to show that $\epsilon^\mu$ must be quadratic as in e-ansatz. Plugging $\square (\partial\cdot\epsilon) = 0$ in e-useful-relation-1 tells us that $\partial\cdot\epsilon$ is linear. Again, quadratic is not necessary for $\partial\cdot\epsilon$ to be linear. But plugging $\partial\cdot\epsilon$ in inf-cft-e-relation-3 tells us $\epsilon^\mu$ must be quadratic.

\begin{equation} \label{e-ansatz} \epsilon_\mu = a_\mu + b_{\mu\nu} x^\nu + c_{\mu\nu\rho}x^\nu x^\rho \, , \end{equation} where we demand $ c_{\mu\nu\rho} = c_{\mu\rho\nu} $ as the anti-symmetric part doesn’t contribute. $a_\mu$ corresponds to infinitesimal translations. From 2dCFTinfty, we get a constraint that the symmetric part of $b_{\mu\nu}$ must be proportional to the metric. The symmetric part corresponds to dilation, and the anti-symmetric part corresponds to rotations.

\begin{equation} b_{\mu\nu} = \alpha\eta_{\mu\nu} + m_{\mu\nu} \,. \end{equation}

If we insert $c_{\mu\nu\rho}$ in inf-cft-e-relation-3, it tells us that $c_{\mu\nu\rho}$ is completely fixed by $f_\mu=\frac{1}{d}{c^\rho}{\rho\mu}$ such that $ c{\mu\nu\rho} = \eta_{\rho\mu}f_\nu + \eta_{\mu\nu}f_\rho - \eta_{\nu\rho}f_\mu \, $. These $f_\mu$ give the so called infinitesimal special conformal transformations

\begin{equation} \label{inf-SCT} x’^\mu = x^\mu + 2(x\cdot f)x^\mu -(x\cdot x)f^\mu \, , \end{equation}

Generators: For all these transformations, we can find generators using the definition

\begin{equation} \label{def-generator} iG_a\Phi = \frac {\delta x^\mu} {\delta \omega^a} \partial_{\mu} \Phi \,. \end{equation}

Generator of special conformal transformations

\begin{equation} iK_\nu = \frac {\delta (2(x_\rho f^\rho)x^\mu -(x\cdot x)f^\mu)} {\delta f^\nu} \partial_{\mu}=2 x_\nu x^\mu\partial_{\mu} - (x\cdot x)\partial_\nu \end{equation}

\begin{table}[htbp] \centering \begin{tabular}{l|l|l} \toprule & Transformations & Generators
\midrule translation & $x’^\mu=x^\mu+a^\mu$ & $P_\mu=-i \partial_\mu$
dilatation & $x’^\mu= \alpha x^\mu$ & $D=-ix^\mu \partial_\mu $
rotation & $x’^\mu ={M^\mu}\nu x^\nu $ & $L{\mu\nu}= i( x_\mu\partial_\nu - x_\nu\partial_\mu ) $
SCT & $x’^{\mu}= \frac {x^\mu-(x\cdot x)f^\mu} { 1-2(f\cdot x) + (f\cdot f)(x\cdot x)} $ & $K_\mu=-i( 2 x_\mu x^\nu\partial_{\nu} - (x\cdot x)\partial_\mu ) $
\bottomrule \end{tabular} \caption{Finite conformal transformations and corresponding generators} \end{table}

Finite conformal transformation: We can exponentiate the generators to obtain finite transformations.

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The first time I saw the correct definition was in ^[2] in my undergrad final year, and it resolved my longstanding confusion about this definition. ↩