Plan

Equation for the Photon

photon

RHCP Photon E Field and H Field (Eq. 4.6):

\begingroup
\renewcommand*{\arraystretch}{2} \begin{bmatrix}x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix}\dfrac{1}{2}+\dfrac{\cos \theta}{2} & \dfrac{1}{2}-\dfrac{\cos \theta}{2} & -\dfrac{\sin \theta}{\sqrt{2}}\\ \dfrac{1}{2}-\dfrac{\cos \theta}{2} & \dfrac{1}{2}+\dfrac{\cos \theta}{2} & \dfrac{\sin \theta}{\sqrt{2}} \\  \dfrac{\sin \theta}{\sqrt{2}} &  -\dfrac{\sin \theta}{\sqrt{2}} & \cos \theta \end{bmatrix} \bullet \left \lgroup \begin{bmatrix}0 \\ r_n \cos \phi \\ r_n \sin \phi \end{bmatrix}_\textrm{Red} + \begin{bmatrix}r_n \cos \phi \\ 0 \\ r_n \sin \phi \end{bmatrix}_\textrm{Blue} \right \rgroup
\endgroup

Equation for the Electron

electron

BECVF Matrices (R_{(-i_x,i_y,0i_z)}(\theta)) (Eq. 1.84) :

\begingroup \renewcommand*{\arraystretch}{2}
\begin{bmatrix}x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix}\dfrac{1}{2}+\dfrac{\cos \theta}{2} & -\dfrac{1}{2}+\dfrac{\cos \theta}{2} & -\dfrac{\sin \theta}{\sqrt{2}}\\ -\dfrac{1}{2}+\dfrac{\cos \theta}{2} & \dfrac{1}{2}+\dfrac{\cos \theta}{2} & -\dfrac{\sin \theta}{\sqrt{2}} \\  \dfrac{\sin \theta}{\sqrt{2}} &  \dfrac{\sin \theta}{\sqrt{2}} & \cos \theta \end{bmatrix} \bullet \left \lgroup \begin{bmatrix}0 \\ r_n \cos \phi \\ -r_n \sin \phi \end{bmatrix} + \begin{bmatrix}r_n \cos \phi \\ 0 \\ - r_n \sin \phi \end{bmatrix} \right \rgroup
\endgroup

Equation for the Neutrino

(Eq. 39.15)

\begingroup e \ \& \ mvf = (E_0 +B_0)\cos^2 \theta \begin{bmatrix}x' \\ y' \\ z' \end{bmatrix}
\endgroup

\begingroup
\renewcommand*{\arraystretch}{2} = \cos^2 \theta \delta (r-r_{photon})\begin{bmatrix}\dfrac{1}{2}+\dfrac{\cos \theta}{2} & \dfrac{1}{2}-\dfrac{\cos \theta}{2} & -\dfrac{\sin \theta}{\sqrt{2}}\\ \dfrac{1}{2}-\dfrac{\cos \theta}{2} & \dfrac{1}{2}+\dfrac{\cos \theta}{2} & \dfrac{\sin \theta}{\sqrt{2}} \\  \dfrac{\sin \theta}{\sqrt{2}} &  -\dfrac{\sin \theta}{\sqrt{2}} & \cos \theta \end{bmatrix} \bullet \left \lgroup E_0 \begin{bmatrix}0 \\ r_n \cos \phi \\ r_n \sin \phi \end{bmatrix}_\textrm{Red} + B_0 \begin{bmatrix}r_n \cos \phi \\ 0 \\ r_n \sin \phi \end{bmatrix}_\textrm{Blue} \right \rgroup
\endgroup