# Confusing "Fourier coefficients" to represent 2d spinors in the helicity spinor formalism

+ 0 like - 0 dislike
526 views

Using the following notation (in the spin helicity formalism) when represnting energy-momentum 4-vectors as bispinors

$\lambda^{\alpha} = p\rangle, \quad \lambda_{\alpha} = \langle p, \quad \tilde{\lambda}_{\dot{\alpha}} = p], \quad \tilde{\lambda}^{\dot{\alpha}} = [p$

such that

$p^{\alpha\dot{\alpha}} = p\rangle[p, \quad p_{\alpha\dot{\alpha}} = p]\langle p$

a two-dimensional spinor $1\rangle$ can be represented as a sum of two other spinors two-dimensional  $2\rangle$ and $3\rangle$ as

$1\rangle = \frac{\langle1 3 \rangle}{\langle23\rangle} 2\rangle - \frac{\langle1 2 \rangle}{\langle23\rangle} 3\rangle$

I dont understand the "Fourier coefficients" in this this case, as I would rather have expected them to be $\frac{\langle 12 \rangle}{\langle 11 \rangle}$ to project on the base spinor $2\rangle$ and $\frac{\langle 13 \rangle}{\langle 11 \rangle}$ to project on the base spinor $3 \rangle$.

Can anybody explain what is going on here? Generally, to me this notation is very confusing and not well enough explained in the source I am reading this from ...

asked Jul 17, 2015 in Chat
edited Jul 17, 2015

The notation is strange (and the formula you cite seems simply wrong).

If I find a paper confusing to read I usually try to read the same material from another more readable source. (Often I read several related papers or books in parallel.) One can usually find the same stuff treated in multiple places, except when the material is very new or very specialized.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysic$\varnothing$OverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.