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  1. section{Motivation and Notation}
  2.  
  3.  
  4. begin{wrapfigure}{r}{0textwidth}
  5.     vspace{-25cm}
  6.     begin{tikzpicture}[rotate=90,scale=1.5]
  7.     vspace{-5cm}
  8.     hspace{0.3cm}
  9.     foreach a/l in {0/$x_1$,60/$x_0$,120/$x_5$,180/$x_4$,240/$x_3$,300/$x_2$} { %a is the angle variable
  10.         draw[line width=.7pt,black,fill=black] (a:1.5cm) coordinate (aa) circle (2pt);
  11.         node[anchor=202.5+a] at ($(aa)+(a+22.5:3pt)$) {l};
  12.     }
  13.     draw [line width=.4pt,black] (a0) -- (a60) -- (a120) -- (a180) -- (a240) -- (a300) -- cycle;
  14.  
  15.  
  16.     node [label={[red,xshift=0.1cm, yshift=0.0cm]$p_2$}] (m1) at ($(a0)!0.65!(a300)$){};
  17.     draw[->] (a0) -- (m1);
  18.  
  19.     node [label={[red,xshift=0.35cm, yshift=-0.2cm]$p_3$}] (m2) at ($(a300)!0.65!(a240)$){};
  20.     draw[->] (a300) -- (m2);
  21.  
  22.     node [label={[red,xshift=0.5cm, yshift=-0.5cm]$p_4$}] (m3) at ($(a240)!0.65!(a180)$){};
  23.     draw[->] (a240) -- (m3);
  24.  
  25.     node [label={[red,xshift=0.15cm, yshift=-0.8cm]$p_5$}] (m4) at ($(a180)!0.65!(a120)$){};
  26.     draw[->] (a180) -- (m4);
  27.  
  28.     node [label={[red,xshift=-0.35cm, yshift=-0.6cm]$p_6$}] (m5) at ($(a120)!0.65!(a60)$){};
  29.     draw[->] (a120) -- (m5);
  30.  
  31.     node [label={[red,xshift=-0.3cm, yshift=-0.3cm]$p_1$}] (m6) at ($(a60)!0.65!(a0)$){};
  32.     draw[->] (a60) -- (m6);
  33.     end{tikzpicture}
  34.     setlength{belowcaptionskip}{-5pt}
  35.     captionsetup{justification=centering,margin=5cm}
  36.     vspace*{-5cm}
  37.     hspace{0.5cm}
  38.     caption{A $n$ = 6 representation of $p$-conservation, where the momenta $p^{mu}$ form a closed contour in dual space.}
  39.     label{fig:Diagram_Mom_Con}
  40. end{wrapfigure}
  41.  
  42. We have described the spinor-helicity formalism as a natural way to encode massless scattering amplitudes. However, we have to impose momentum conservation by hand, since spinor-helicity is derived from a Lorentz invariant foundation, which can be thought of as a subgroup of Poincar'e invariance. The 10-dimensional Poincar'e group includes translations (3 spatial and 1 time) as well as the 6-dimensional Lorentz group, consisting of 3 boosts and 3 rotations. Hence, spinor variables are not invariant under spatial translations, and momentum is not automatically conserved footnotemark.
  43. Since all scattering processes naturally conserve momentum, we would like to have a formalism where both the on-shell massless condition, $p^2 =0$ and momentum conservation, $sum p^{mu} = 0$ are manifest. This comes in the form of momentum twistors, developed by Hodges as an extension of Penrose's twistor geometry.
  44.  
  45. footnotetext{This is a well-known consequence of Noether's Theorem. See REFS REMOVED For more explicit details.}
  46.  
  47. %
  48. par
  49. We take inspiration by considering a different geometrical interpretation of momentum conservation. We start by drawing an $n$-sided polygon in dual space, as shown by Figure ref{fig:Diagram_Mom_Con}.
  50. There are two ways to consider defining the polygon; either through the edges or the vertices. Considering the edges, we obtain the traditional statement of momentum conservation; the $n$ edges form a closed contour, which corresponds to the net sum of momenta equalling zero, and no new intuition has been obtained.
  51. par
  52. Let us now define the polygon through the vertices, using a new set of dual coordinates $x_i$ where $i={ 1,dots,n}$. To ensure our contour is closed, we demand the periodic boundary $x_{0} equiv x_{n}$. The momenta in dual space may now be defined as the difference of these dual coordinates
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