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Homework Statement
For massless particles, we can take as reference the vector ##p^{\mu}_R=(1,0,0,1)## and note that any vector ##p## can be written as ##p^{\mu}=L(p)^{\mu}_{\nu}p^{\nu}_R##, where ##L(p)## is the Lorentz transform of the form
$$L(p)=exp(i\phi J^{(21)})exp(i\theta J^{(13)})exp(i\alpha J^{(30)})$$
Where ##(\theta,\phi)## are the spherical coordinates of ##\vec{p}## and ##\alpha=sinh^{1}(\frac{1}{2}(p^01/p^0))##. This allows to define the general state for the massless particle as:
$$p,\lambda\rangle=U(L(p))p_R,\lambda\rangle$$
Where ##p_R,\lambda\rangle## is an eigenstate with value ##\lambda## of the operator ##J_3##. Show that ##p,\lambda\rangle## is an eigenstate of the helicity operator ##\frac{\vec{p}}{\vec{p}}\cdot\vec{J}##.
Homework Equations
$$J_3p_R,\lambda\rangle=\lambdap_R,\lambda\rangle$$
$$\vec{p}=\vec{p}(sin\theta cos\phi, sin\theta sin\phi, cos\theta )$$
$$U(\Lambda_a)U(\Lambda_b)=U(\Lambda_a \Lambda_b)$$
The Attempt at a Solution
For the last week, I've been trying to verify this last statement by expanding the exponentials or using commutators. For example, by using the commutation relationship
$$[J_i,J_k]=i\epsilon_{ijk}J_k$$
But I only end with nonreducible expressions. I also tried expanding the exponentials of the operators using the relationship
$$e^{A}=1+A+\frac{1}{2}A^2+\frac{1}{6}A^3+...$$
Without arriving at a result. Particulary, I don't understand how to act using the unitary transformations, as when I even try to start by calculating:
$$p,\lambda\rangle=U(L(p))p_R,\lambda\rangle)=U(exp(i\phi J^{(21)})exp(i\theta J^{(13)})exp(i\alpha J^{(30)}))p_R,\lambda\rangle$$
Or even the direct calculation:
$$(\frac{\vec{p}}{\vec{p}}\cdot\vec{J})p_R,\lambda\rangle=(\frac{\vec{p}}{\vec{p}}\cdot\vec{J})U(L(p))p_R,\lambda\rangle)$$
I don't know how to reduce terms. Do you have any suggestions?
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