Helicity operator in spinor-helicity variables

Question

How do I prove that the helicity operator is $$ H = \frac{1}{2} (\tilde{\lambda}_\dot{\alpha} \frac{\partial}{\partial \tilde{\lambda}_\dot{\alpha}} - \lambda_\alpha \frac{\partial}{\partial \lambda_\alpha}) $$ knowing that in general this operator is defined as $$ H = \frac{\vec{S} \cdot \vec{p}}{|\vec{p}|} ~?$$

Answer 1

To go from one formulation of the helicity operator, the one involving the spinors $(\lambda,\tilde{\lambda})$ and the typical one from QFT, as far as I can tell, is not possible, or at the very least not straight forward. Let me try to show you however where the $H$ you call the helicity operator comes from.

Given that you wrote down the helicity operator in terms of the spinor helicity formalism, I'll skip some basics and go right to the derivation. Recall that for a lightlike vector we can write its momentum as $p_{a\dot{dot}} = \lambda_a\tilde{\lambda}_{\dot{a}}$. Since on shell, the scalar product of two lightlike momenta are $2p\cdot p' = \langle\lambda,\lambda'\rangle[\tilde{\lambda},\tilde{\lambda}']$. Recall that under little group scaling, the spinors behave as $(\lambda,\tilde{\lambda})\rightarrow (t\tilde{\lambda},t^{-1}\tilde{\lambda})$ as $t^{-2h_i}$ for massless particles with helicity $h$ (I denoted the helicity as $h_i$ since this is true for every particle in a given amplitude, i.e. $\mathcal{A}(p_1,\cdots p_n) = t^{-2h_i}\mathcal{A}(p_1,\cdot p_i,\cdots p_n)$ for the $i$th particle in the amplitude). To see this, consider a massless particle moving in the $n$-direction. Then, a rotation by an angle $\theta$ around the $n$-axis acts on the spinors as \begin{equation} (\lambda,\tilde{\lambda})\rightarrow (e^{-i\theta/2}\lambda,e^{i\theta/2}\tilde{\lambda}) \end{equation} Thus, $\lambda$ and $\tilde{\lambda}$ carry $1/2$ or $-1/2$ units of angular momentum around the $n$-axis. Thus, the wave function of the particle gets transformed as $\psi\rightarrow e^{ih\theta}\psi$ under the rotation around $n$-axis, and so it obeys the auxiliary condition \begin{equation} \left(\lambda^a\frac{\partial}{\partial\lambda^a} - \tilde{\lambda}^{\dot{a}}\frac{\partial}{\partial\tilde{\lambda}^{\dot{a}}}\right)\psi(\lambda,\tilde{\lambda}) = -2h\psi(\lambda,\tilde{\lambda}). \end{equation} This argument can be extended to the amplitude by replacing $\psi\rightarrow\mathcal{A}$ and every spinor gets and index $i$.

This is the context in which you usually see or define or maybe some would say derive the helicity operator, but perhaps you could even start from its matrix definition \begin{equation} h = \frac{1}{2|p|} \begin{bmatrix} \sigma\cdot\hat{p} & 0\\ 0 &\sigma\cdot\hat{p} \end{bmatrix} \end{equation} and then plug-in the momenta in terms of spinors using the relation $p_{a\dot{b}}\equiv p_\mu(\sigma^\mu)_{a\dot{b}}$ to simplify the calculation.

Finally, everything discussed came from the following two papers. The first are lecture notes on twistors and string theory Lectures on Twistor Strings and Perturbative Yang-Mills Theory, which originated from the original paper on the subject by Witten (a ghastly 97 pages) Perturbative Gauge Theory as a String Theory in Twistor Space (you'll want pages 2-8).