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This relates to the derivation of equation (5.15) if Elvang and Huang's textbook. The idea is to transform the spinor helicity variables we are using, $(|i\rangle_{\dot{a}},[j|^a)$ to go into twistor space. The Fourier transformation is given as \begin{equation} |i\rangle^{\dot{a}}\rightarrow -i\frac{\partial}{\partial\langle\tilde{\mu}_i|_{\dot{a}}} \end{equation} I am already struggling from the start with setting up the integral. The only thing I have written down is that the Fourier transformation (modulo a factor of $2\pi$) is that I think it should be \begin{equation} \int\prod^n_{i=1}d^2|i\rangle e^{i\langle i|_{\dot{a}}|\tilde{\mu}_i\rangle^{\dot{a}}} \end{equation} where the product is over $i=1,2,...,n$ particles. I think it is partially the notation confusing me since, for example, if I had an ordinary function that I wanted to Fourier transform in $d=3$, $|\mathbf{q}|^{-2}$ then I would just throw it into $\int\frac{d^3q}{(2\pi)^3}e^{i\mathbf{q}\cdot\mathbf{r}}|\mathbf{q}|^{-2}$ and plug-in $d^3\mathbf{q} = q^2d\cos(\theta)d\phi$ with proper integration bounds of course and defining $\mathbf{q}\cdot\mathbf{r} = qr\cos(\theta)$. But in this case, I am lost. (The discussion on this is pages 99-100 in the textbook referenced prior (or lecture notes).)

My guess would be to multiply by a suitable test function as one method. However another method would be to insert a complete basis of states (which is weird to say since these are spinors), or even hit the state with the translation operator$P^{\dot{a}b} = -\sum_i|i\rangle^{\dot{a}}[i|^b$, then insert a complete basis of states in order to perform the proper integration (but I do not know what the "basis of states" would be in this case).

Or, if I am dead wrong in anything stated, I of course ask for any help or guidance (thank you).

MathZilla
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1 Answers1

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In order to set up the correct Fourier transform, we can start from the usual one: $$ f(x) = \int d^4p ~e^{ip\cdot x} f(p) $$ The first thing we need to do is write things in terms of spinors using $$p_{a\dot{a}} = |i\rangle_a|i]_{\dot{a}}.$$ This gives $$ \int d^4p ~e^{ip\cdot x} f(p) = \int d^2|i\rangle_a d^2|i]_{\dot{a}}~|J| e^{i[i|x|i\rangle} f(|i],|i\rangle), $$ where $|J|$ is some Jacobian.

However, we note that this needs to be evaluated on the incidence relation: $$ \langle\tilde{\mu}_i|_{a} + [i|^{\dot{a}}x_{a\dot{a}} = 0. $$ So the fourier transform we actually need is

$$ f(x) = \int d^2|i\rangle_a~|J| d^2|i]_{\dot{a}} e^{i[i|x|i\rangle} \delta^2(\langle\tilde{\mu}_i|_{a} + [i|^{\dot{a}}x_{a\dot{a}})~ f(|i],|i\rangle) $$ Eliminating the square bracket using the delta function and performing two of the integrals then gives $$ f(x) = \int d^2|i\rangle_a~|J| e^{-i\langle i \tilde{\mu}_i\rangle} f(\langle{\tilde{\mu}}|,|i\rangle) $$

Akoben
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