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Eigenstates of helicity operator

  • #1
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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^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?
 

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Answers and Replies

  • #2
jambaugh
Science Advisor
Insights Author
Gold Member
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245
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:
[tex]|p,\lambda\rangle=U(L(p))|p_R,\lambda\rangle[/tex]
then
[tex]U(L(p))^{-1}|p,\lambda\rangle=|p_R,\lambda\rangle[/tex]
now try to frame the eigen-value question you're being asked to the transformed question on [itex]\lvert p_R, \lambda\rangle[/itex].

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

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