Finding helicity eigenstates

Question

Question:
Give the mode expansion  of the $A_i$ in terms of plane wave
begin{equation}
    epsilon^{pm}_i(p)e^{-ip cdot x}      text{ and }       epsilon^{*pm}_i(p)e^{-ip cdot x}
end{equation}
where the polarisation vectors for the helicity eigenstates satisfy
begin{equation}
    epsilon_{ijk}hat{p}_j epsilon_k^{pm}(p)  = pm i epsilon_i^{pm}(p)       text{ with }       hat{p}_i equiv frac{p_i}{|p|}.
end{equation}
Find the helicity eigenstates for $hat{p} = hat{e}_z$.
Solution
I know that the mode expansion is given by
$$
A_i = int frac{d^3 p}{(2pi)^3} frac{1}{sqrt{2E_{mathbf{p}}}} sum_{lambda = pm} left[ epsilon_i^{lambda}(mathbf{p})a_{mathbf{p}}^{lambda}e^{-ipcdot x} +  epsilon_i^{*lambda}(mathbf{p})a_{mathbf{p}}^{lambda dagger}e^{+ipcdot x} right],
$$
but I have no idea how to determine the helicity eigenstates.

amilton moreira · Answer

The Helicity operator is given by
begin{equation}
hat{Lambda}=int mathrm{d}^{3} kleft(hat{a}_{k+}^{dagger} hat{a}_{k+}-hat{a}_{k-}^{dagger} hat{a}_{k-}right)
end{equation}
where we have
begin{equation}
begin{aligned}
hat{a}_{k+} &=frac{1}{sqrt{2}}left(hat{a}_{k 1}-mathrm{i} hat{a}_{k 2}right) 
hat{a}_{k-} &=frac{1}{sqrt{2}}left(hat{a}_{k 1}+mathrm{i} hat{a}_{k 2}right)
end{aligned}
end{equation}
for the usual creation and annihilation operators $hat{a}_{klambda}^{dagger}, hat{a}_{klambda}$. The operators $hat{a}_{k -}^{dagger},hat{a}_{k +}^{dagger}, hat{a}_{k-},hat{a}_{k,+}$
begin{equation}
left[hat{a}_{k^{prime}+}, hat{a}_{k+}^{dagger}right]=left[hat{a}_{k^{prime}-}, hat{a}_{k-}^{dagger}right]=delta^{3}left(k-k^{prime}right)
end{equation}
Then an helicity eigenstate $ |k,+rangle $ for example,is given by $hat{a}_{k,+}^{dagger} |0rangle $

Finding helicity eigenstates

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