This question is fundamental enough and I probably should know the answer at this point, but for some reason I am confused. I know that helicity states should go into each other under parity transform like that $P[e_+] = - e_{-}$. But if I write out the helicity states as:

$e_{\pm} = \mp \frac{1}{\sqrt{2}}(e_x \pm ie_y)$

I don't see it! Since $e_x$ and $e_y$ are just ordinary unit vector components, what I get under parity is this:

$P[e_{\pm}] = P[\mp \frac{1}{\sqrt{2}}(e_x \pm ie_y)]= \mp \frac{1}{\sqrt{2}}(-e_x \mp ie_y) = \pm \frac{1}{\sqrt{2}}(e_x \pm ie_y) =-e_{\pm} $

Where do I get wrong?


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