Convention

Normalization constants

• $signature=1 - the default metric is \((-,+,+,+)\).

• $polar=$polarn=$polar0=1 - the default normalization of polarization vectors.

• $CPWFGauge=1 - the default CPWF gauge parameter.

Under the default settings, for \(d=2\) the components of vectors are

\[\begin{align} & \qhat=(z \bar{z}+1,\bar{z}+z,-i (z-\bar{z}),1-z \bar{z}) \, , \\ & \phat=\frac{1}{2y}(z \bar{z}+y^2+1,\bar{z}+z,-i (z-\bar{z}),-z \bar{z}-y^2+1) \, , \\ & \epsilon=\pp_{z}\qhat = (\bar{z},1,-i,-\bar{z}) \, , \\ & \epsilonb=\pp_{\zb}\qhat = (z,1,i,-z) \, , \\ & \epsilon_{0}=y\pp_{y}\phat = \frac{1}{2y}(-z \bar{z}+y^2-1,-\bar{z}-z,i (z-\bar{z}),z \bar{z}-y^2-1) \, , \\ & n = (1,0,0,-1) \, . \end{align}\]

For higher dimensions, they are

\[\begin{align} & \qhat=(| x|^2+1,2 x,1-| x|^2) \, , \\ & \phat=\frac{1}{2y}(| x|^2+y^2+1,2 x,-| x|^2-y^2+1) \, , \\ & \epsilon_a=\pp_{x_{a}}\qhat \textInMath{for} 1\leq a\leq d \, , \\ & \epsilon_{0}=y\pp_{y}\phat = \frac{1}{2y}(-| x|^2+y^2-1,-2 x,| x|^2-y^2-1) \, , \\ & n= (1,0,\cdots,0,-1) \, . \end{align}\]

Two dimensional polarization

• Massless polarization vectors: \(\set{\epsilon,\epsilonb}\) for \(J\in\set{1,-1}\).

  • Complete basis: \(\epsilon(q)_{a}=\set{\epsilon,\epsilonb,n,\qhat}\) with the inner product

    \[\begin{equation} \epsilon(q)_{a}^{*}\cdot \epsilon(q)_{b} = \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & -2 \\ 0 & 0 & -2 & 0 \\ \end{bmatrix} \, . \end{equation}\]

• Massive polarization vectors: \(\set{\epsilon,\epsilonb,\epsilon_{0}}\) for \(J\in \set{1,-1,0}\).

  • Complete basis: \(\epsilon(p)_{a}=\set{\epsilon,\epsilonb,\epsilon_{0},\phat}\) with the inner product

    \[\begin{equation} \epsilon(p)_{a}^{*}\cdot \epsilon(p)_{b} = \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix} \, . \end{equation}\]