lie algebra - Generator of the Special Conformal Transformation

In this thread Integrating the generator of the infinitesimal special conformal transformation, the generator of the 'flow' of the transformation is written as

$$G_b = 2(b \cdot x)x - x^2 b,$$ where $b$ parametrises the SCT. Now, we know that under an infinitesimal SCT, the coordinates transform like so $$x'^{\mu} = x^{\mu} + 2(b \cdot x)x - x^2b.$$ So, in the same vein that, under a translation $x'^{\mu} = x^{\mu} + a^{\mu}$ we don't speak of $a^{\mu}$ being a generator but rather a parameter, what merits the calling of $2(b \cdot x)x - x^2b$ a generator?

Answer

General remarks on flows and their generators.

Let an $\epsilon$-parameter flow $\Phi(\epsilon):\mathbb R^d\to \mathbb R^d$ be given. Let the flow be defined on some $\epsilon$-neighborhood containing $0$. Provided the flow is sufficiently smooth, we can expand the flow in the parameter $\epsilon$;

$$\begin{align} \Phi(\epsilon) = \Phi(0) + \epsilon\Phi^{(1)} + O(\epsilon^2). \end{align}$$ If the flow "starts at the identity," namely if $\Phi(0) = \mathrm{id}$, where $\mathrm{id}$ is the identity on $\mathbb R^d$, then $\Phi(0) = \mathrm{id}$. In addition, in anticipation of the terminology to come, we define $G = \Phi^{(1)}$, namely $G$ is just the function that when multiplied by $\epsilon$ gives the first order change caused by the flow. So we have $$\begin{align} \Phi(\epsilon) = \mathrm{id} + \epsilon G + O(\epsilon^2). \end{align}$$ At this point, one usually defines $G$ as the generator (or infinitesimal generator depending on who you talk to ) of the flow, but of course we gain no insight into why it's called that. To see why this first order coefficient is called a generator, consider some point $x_0\in \mathbb R^d$, and suppose that we apply the flow to $x_0$, then as $\epsilon$ increases from $0$ to some $\epsilon >0$, the point $x_0$ travels along a curve, $\gamma$ defined by $$\begin{align} \gamma(\epsilon) = \Phi(\epsilon)(x_0). \end{align}$$ Now consider taking the derivative of $\gamma$ with respect to $\epsilon$ at $\epsilon = 0$; $$\begin{align} \dot\gamma(0) = \frac{d}{d\epsilon}\Phi(\epsilon)(x_0)\Big|_{\epsilon=0} = \frac{d}{d\epsilon} \big(x_0 + \epsilon G(x_0) + O(\epsilon^2)\big)\Big|_{\epsilon=0} = G(x_0) \end{align}$$ but recall also that $\gamma(0) = x_0$, so we find that $$\begin{align} \dot \gamma(0) = G(\gamma(0)). \end{align}$$ Since the derivative $\dot\gamma(0)$ is just the tangent vector to $\gamma$ at zero, this equation means that the tangent vector of $\gamma$ at zero agrees with $G$, the generator (a vector field) at $\gamma(0)$. Actually, we can say something much stronger than this. Notice that we can compute the derivative of $\gamma$ at any parameter value $\epsilon$, not just at $\epsilon = 0$, as follows: $$\begin{align} \dot \gamma(\epsilon) &= \frac{d}{dt} \gamma(\epsilon+t)\Big|_{t=0} \\ &= \frac{d}{dt} \Phi(\epsilon + t)(x_0) \Big|_{t=0} \\ &= \frac{d}{dt} \Phi(t)(\Phi(\epsilon)(x_0)) \Big|_{t=0} \\ &= \frac{d}{dt} \Big(\Phi(\epsilon)(x_0) + t G(\Phi(\epsilon)(x_0)) + O(t^2)\Big) \Big|_{t=0} \\ &= G(\Phi(\epsilon)(x_0)), \end{align}$$ where we have used the property $\Phi(t+s) = \Phi(t)\circ \Phi(s)$ of flows. But $\Phi(\epsilon)(x_0)$ is precisely $\gamma(\epsilon)$! So we get $$\begin{align} \dot\gamma(\epsilon) = G(\gamma(\epsilon)). \tag{$\star$} \end{align}$$ In other words

If $\gamma$ is a curve generated by a flow, then $G$, the infinitesimal generator of the flow is tangent to $\gamma$ everywhere along $\gamma$.

This is a very powerful fact, because it tells us that if we are given the infinitesimal generator of a flow (which is a vector field), then we can reconstruct the entire flow by solving $(\star)$, a first order system or ordinary differential equations!

This also explains the terminology "infinitesimal generator" when referring to $G$. It is "infinitesimal" because it tells us how the flow behaves to first order, which is a good approximation when $\epsilon$ is small, and it is a "generator" in the sense that the flow can be reconstructed from it by solving the system $(\star)$.

Example. Special conformal transformation

Recall that in the other physics.SE post you refer to in your question:

Integrating the generator of the infinitesimal special conformal transformation

we saw that the special conformal flow is given by

$$\begin{align} \Phi_b(\epsilon)(x) = \frac{x - x^2 (\epsilon b)}{1-2x\cdot (\epsilon b) + x^2(\epsilon b)^2}. \end{align}$$ (although there we used $t$ instead of $\epsilon$ as the flow parameter. If we Taylor expand this around $\epsilon = 0$, then we find that $$\begin{align} \Phi_b(\epsilon)(x) = x + \epsilon(2(x\cdot b)x-x^2b) + O(\epsilon^2), \end{align}$$ so we can immediately identify the infinitesimal generator of this flow as $$\begin{align} G_b(x) = 2(x\cdot b)x-x^2b. \end{align}$$

How does the SCT act on fields?

Consider a scalar field $\phi:\mathbb R^d\to\mathbb R^d$. Suppose that $\phi$ is such that the action of a conformal transformation $f$ on $\phi$ is

$$\begin{align} \phi_f(x) = \phi(f^{-1} x) \tag{$\star\star$}. \end{align}$$ Then we can ask the following question:

What happens to $\phi$ when an infinitesimal special conformal transformation is acted in it?

In other words, we are asking what $\phi_f$ is to first order in $\epsilon$ when $f = \Phi_b(\epsilon)$. Well, let's calculate:

$$\begin{align} \phi_{\Phi_b(\epsilon)}(x) &= \phi(\Phi_b(-\epsilon)(x)) \\ &= \phi(x - \epsilon G_b(x) + O(\epsilon^2)) \\ &= \phi(x) - \epsilon (G_b)(x)\cdot\partial_\mu\phi(x) + O(\epsilon^2) \\ &= \Big(\mathrm{id} - \epsilon G_b(x)\cdot \partial+ O(\epsilon)^2\Big)\phi(x) \end{align}$$ which shows that for a given $b$, the differential operator $$\begin{align} -G_b(x) \cdot \partial \end{align}$$ is the infinitesimal generator of the action of a special conformal transformation on such scalar fields as opposed to on points in $\mathbb R^d$. You'll see that this agrees with eq.4.18 in Di Francesco since in that book, as is often conventional, he strips off the $b$-dependence and adds a factor of $i$ when defining the infinitesimal generator, namely he defines the infinitesimal generators $K_\mu$ such that $$\begin{align} ib^\mu K_\mu = - G_b^\mu(x) \cdot \partial. \end{align}$$ Ok so this object is just the infinitesimal generator for the action of SCTs on spinless fields that obey $(\star\star)$ (or as Di Francesco writes it, fields for which $\mathcal F(\Phi) = \Phi$), namely there is no-nontrivial target space transformation that happens when such a field transforms.

However, if the field has spin, then there will by definition be a non-trivial target space transformation in which case the differential operator that represents SCTs on such fields picks up the following extra terms:

$$\begin{align} \kappa_\mu +2 x_\mu \tilde\Delta - x^\nu S_{\mu\nu}. \end{align}$$