Representation of Angular Momentum Operator in the (j,j')

[a2009]

Hello All,

I'm trying to understand how the (j,j') representation of the Lorentz group. Following Ryder, I can see why we define A=J+iK and B=J-iK, which each form an SU(2) group. So it's clear to me what the rep of these generators is when acting on a state (j,j'): Rep(A)\otimes1+1\otimes Rep(B). Where Rep(A) and Rep(B) are the appropriate j and j' reps.

My question is this: given the rep (j,j')\oplus(j',j), what is the induced rep on the generators? For example how do I act with A or J on this state?

Thanks a whole bunch

 

[fzero]

If you have a representation r and associated generators R, then the generators for the representation r\oplus r' are

\begin{pmatrix} R & 0 \\ 0& R'\end{pmatrix}.

As a sort of converse, if a representation \tilde{r} is reducible to a sum r\oplus r', then the generators \tilde{R} are block diagonal, as above.

 

[a2009]

Thanks for the quick reply. But I didn't understand. In the (1/2,1)\oplus(1,1/2) of Lorentz, does the operator A (the left SU(2)) look like this

<br /> \begin{pmatrix}<br /> A_{2\times2} &amp; 0 &amp; 0 &amp; 0 \\<br /> 0 &amp; 1_{3\times3} &amp; 0 &amp; 0 \\<br /> 0 &amp; 0 &amp; A_{3\times3} &amp; 0 \\<br /> 0 &amp; 0 &amp; 0 &amp; 1_{2\times2} <br /> \end{pmatrix}<br />

 

[fzero]

Thanks for the quick reply. But I didn't understand. In the (1/2,1)\oplus(1,1/2) of Lorentz, does the operator A (the left SU(2)) look like this

<br /> \begin{pmatrix}<br /> A_{2\times2} &amp; 0 &amp; 0 &amp; 0 \\<br /> 0 &amp; 1_{3\times3} &amp; 0 &amp; 0 \\<br /> 0 &amp; 0 &amp; A_{3\times3} &amp; 0 \\<br /> 0 &amp; 0 &amp; 0 &amp; 1_{2\times2} <br /> \end{pmatrix}<br />

No, there's separate blocks for the right and left-generators. The block decomposition is only for the sums, not the tensor products. So you'd have

<br /> \begin{pmatrix}<br /> A_{2\times2} &amp; 0 \\<br /> 0 &amp; A_{3\times3} <br /> \end{pmatrix}<br />

 

[a2009]

Sorry I still don't understand. Each (j,j') is a (2j+1)X(2j'+1) dimensional vector space. So in the case of (1/2,1)\oplus(1,1/2) it should be a twelve dimensional vector space. What you wrote is five dimensional. Maybe the answer is A_{2\times2}\otimes 1_{3\times3} \oplus A_{3\times3}\otimes 1_{2\times2}? This would give

\begin{pmatrix}<br /> \left( A_{2\times2}\otimes 1_{3\times3} \right)_{6\times 6} &amp; 0 \\<br /> 0 &amp; \left( A_{3\times3}\otimes 1_{2\times2} \right)_{6\times 6}<br /> \end{pmatrix}<br />

which is a twelve dim operator like I'd expect. Does this make any sense?

Thank!

 

[weejee]

Sorry I still don't understand. Each (j,j') is a (2j+1)X(2j'+1) dimensional vector space. So in the case of (1/2,1)\oplus(1,1/2) it should be a twelve dimensional vector space. What you wrote is five dimensional. Maybe the answer is A_{2\times2}\otimes 1_{3\times3} \oplus A_{3\times3}\otimes 1_{2\times2}? This would give

\begin{pmatrix}<br /> \left( A_{2\times2}\otimes 1_{3\times3} \right)_{6\times 6} &amp; 0 \\<br /> 0 &amp; \left( A_{3\times3}\otimes 1_{2\times2} \right)_{6\times 6}<br /> \end{pmatrix}<br />

which is a twelve dim operator like I'd expect. Does this make any sense?

Thank!

This is right.