User:JabberWok/Sandbox

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Let

p_x=|\vec{p}| \sin{\theta} \,
p_y=0 \,
p_z=|\vec{p}| \cos{\theta} \,

Thus,

p_T = \sqrt{p_x^2+p_y^2} = |\vec{p}| |\sin{\theta}| \,


The polarization vectors then are

\epsilon^\mu(p,x) \, = \frac{1}{|\vec{p}| p_T} \left(0,p_x p_z, p_y p_z, -p_T^2 \right) \,
= \frac{1}{|\vec{p}|^2 |\sin{\theta}| } \left( 0, |\vec{p}|^2 \sin{\theta} \cos{\theta}, 0, - |\vec{p}|^2 |\sin{\theta}|^2 \right) \,
= \left(0, \cos{\theta}, 0, -|\sin{\theta}| \right) \,
\epsilon^\mu(p,y) \, = \frac{1}{p_T} \left( 0, -p_y, p_x, 0 \right) \,
=\frac{1}{|\vec{p}| \sin{\theta} } \left(0,0,|\vec{p}| \sin{\theta}, 0 \right) \,
=\left(0,0,1,0 \right) \,
\epsilon^\mu(p,z) \, =\frac{E}{m|\vec{p}|} \left(\frac{|\vec{p}|^2}{E}, p_x, p_y, p_z \right) \,
=\frac{E}{m|\vec{p}|} \left(\frac{|\vec{p}|^2}{E}, |\vec{p}| \sin{\theta}, 0, |\vec{p}| \cos{\theta} \right) \,


So the helicity states are

\epsilon^\mu (p, \pm) \, = \frac{1}{\sqrt{2}} \left[\mp\epsilon(p,x) - i\epsilon(p,y) \right] \,
=\frac{1}{\sqrt{2}} \left(0, \mp \cos{\theta}, -i, \pm \sin{\theta} \right)\,

And for longitudinal

\epsilon^\mu(p, 0) \, = \epsilon^\mu(p, z) \,
= ? \,