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\title{EB definition/convention}
\author{}
\date{\today,~ $ $Revision: 1.1 $ $}
\begin{document}
\maketitle

\section{}

Kamionkowski:
\EQ
{\cal P}_{ab}^{\rm K}={1\over\sqrt{2}}\pmatrix{Q & U\cr U & -Q}
\EN
Durrer:
\EQ
{\cal P}_{ab}^{\rm D}={1\over2}\pmatrix{Q & U\cr U & -Q}
\EN
Ignore prefactor altogether.\\
Kamionkowski+Durrer:
\EQ
\nabla^2 E={\cal P}_{ab,ab}=(\partial_x^2-\partial_y^2)Q+2\partial_x\partial_y U
\EN
Kamionkowski:
\EQA
\nabla^2 B&=&\epsilon_{ac}{\cal P}_{ab,bc} \nonumber \\
&=&P_{11,12}+P_{12,22}-P_{21,11}-P_{22,21} \nonumber \\
&=&Q_{,12}+U_{,22}-U_{,11}+Q_{,21}\\
&=&2\partial_x\partial_y Q-(\partial_x^2-\partial_y^2)U
\ENA
Durrer:
\EQA
\nabla^2 B&=&\epsilon_{ij}\epsilon_{lm}{\cal P}_{jm,il}\\
&=&\epsilon_{12}\epsilon_{12}{\cal P}_{22,11}
  +\epsilon_{12}\epsilon_{21}{\cal P}_{21,12} \nonumber \\
&+&\epsilon_{21}\epsilon_{12}{\cal P}_{12,21}
  +\epsilon_{21}\epsilon_{21}{\cal P}_{11,22} \nonumber \\
&=&P_{11,22}-P_{12,21}-P_{21,12}+P_{22,11} \nonumber \\
&=&Q_{,22}-U_{,21}-U_{,12}-Q_{,11} \nonumber \\
&=&-2\partial_x\partial_y U-(\partial_x^2-\partial_y^2)Q
\ENA
Kamionkowski:
\EQA
\nabla^2 (E+\ii B)
&=&[(\partial_x^2-\partial_y^2)+2\ii\partial_x\partial_y](Q-\ii U) \nonumber \\
&=&(\partial_x+\ii\partial_y)^2(Q-\ii U)\\
\ENA
\EQA
(\tilde E+\ii \tilde B)
&=&[(\hat k_x^2-\hat k_y^2)+2\ii \hat k_x \hat k_y](\tilde Q-\ii \tilde U) \nonumber \\
&=&(\hat k_x+\ii \hat k_y)^2(\tilde Q-\ii \tilde U)
\ENA
or
\EQA
(\tilde E-\ii \tilde B)
=(\hat k_x-\ii \hat k_y)^2(\tilde Q+\ii \tilde U)
\ENA
Durrer:
\EQA
\nabla^2 (E+\ii B)
&=&(\partial_x^2-\partial_y^2)Q+2\partial_x\partial_y U\\
&-&2\ii \partial_x\partial_y U-(\partial_x^2-\partial_y^2)\ii Q\\
&=&(\partial_x^2-\partial_y^2)Q-2\ii\partial_x\partial_y \ii U\\
&-&2\ii \partial_x\partial_y U-(\partial_x^2-\partial_y^2)\ii Q
\ENA

\section{}

Kamionkowski:
\EQ
\pmatrix{E\cr B}={1\over\sqrt{2}}\pmatrix{
\cos2\phi & \sin2\phi \cr
-\sin2\phi & \cos2\phi }
\pmatrix{Q\cr U}
\EN
Thus
\EQ
E={1\over\sqrt{2}}\Rey\left[e^{-2\ii\phi}(Q+\ii U)\right]
\EN
\EQ
B={1\over\sqrt{2}}\Imag\left[e^{-2\ii\phi}(Q+\ii U)\right]
\EN
so
\EQ
E+\ii B={1\over\sqrt{2}} e^{-2\ii\phi}(Q+\ii U)
\EN

%r e f
\begin{thebibliography}{}

\bibitem[Kamionkowski \& Kovetz(1016)]{KK16}
Kamionkowski, M., \& Kovetz, E. D.\yaraa{2016}{54}{227}
{269}{The quest for B modes from inflationary gravitational waves}

\end{thebibliography}

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