Is \(|p,\lambda\rangle\) an Eigenstate of the Helicity Operator?

[CharlieCW]

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 $\alpha=sinh^{-1}(\frac{1}{2}(p^0-1/p^0))$. 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

$$J_3|p_R,\lambda\rangle=\lambda|p_R,\lambda\rangle$$

$$\vec{p}=|\vec{p}|(sin\theta cos\phi, sin\theta sin\phi, cos\theta )$$

$$U(\Lambda_a)U(\Lambda_b)=U(\Lambda_a \Lambda_b)$$

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 non-reducible 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?

 

[jambaugh]

I think your approach is going to need to be a bit more algebraic. Rather than trying to expand those exponential representation operators, you'll want to map the question back to a question about the reference vector, and take advantage of the fact that we're talking here about the Lorentz group (with inverses and everything!).

Given:
|p,\lambda\rangle=U(L(p))|p_R,\lambda\rangle
then
U(L(p))^{-1}|p,\lambda\rangle=|p_R,\lambda\rangle
now try to frame the eigen-value question you're being asked to the transformed question on \lvert p_R, \lambda\rangle.

This is the basic transform--solve--transform-back method.