Coupling Coefficients in SO(4)

Question

I have two equations (from two distinct authors) for the decomposition of a coupling coefficient of SO(4) (i.e. Wigner 3j-symbol for SO(4)). In the first:

\begin{equation} \left( \begin{array}{ccc} l_1 & l_2 & l_3 \\ \left(l'_1, N_1\right) & \left(l'_2, N_2\right) & \left(l'_3, N_3\right) \end{array} \right)_{SO(4)} = \left( \begin{array}{ccc} l_1 & l_2 & l_3 \\ l'_1 & l'_2 & l'_3 \end{array} \right)_{\left(SO(4):SO(3)\right)} \left( \begin{array}{ccc} l'_1 & l'_2 & l'_3 \\ N_1 & N_2 & N_3 \end{array} \right) \end{equation}

The left hand side is the coupling (Wigner) coefficient for SO(4) and the right hand side has an isoscalar factor with the label $SO(4):SO(3)$, and a normal Wigner coefficient for SO(3).

In the second equation the author factors the SO(4) coupling coefficient into the product of two SO(3) coupling coefficients as:

$$\left(\begin{array}{ccc}\left(X_1Y_1\right)&\left(X_2Y_x\right)&\left(XY\right)\\\left(M_{X_1}M_{Y_1}\right)&\left(M_{X_2}M_{Y_2}\right)&\left(M_XM_Y\right)\end{array}\right)_{SO(4)}=\left(\begin{array}{ccc}X_1&X_2&X\\M_{X_1}&M_{X_2}&M_X\end{array}\right)\left(\begin{array}{ccc}Y_1&Y_2&Y\\M_{Y_1}&M_{Y_2}&M_Y\end{array}\right)$$

In this case SO(4) is from the direct product of two SO(3)s: $(X_1Y_1)\bigotimes(X_2Y_2)\rightarrow(XY)$

$\bf QUESTION$: I want to set these equations equal to each other and solve for the isoscalar factor, but I am confused by the fact that the first author only uses a scalar for the upper arguments whereas the second author uses a tuple. How do the parameters of the SO(4) coupling coefficients equate? (e.g. Is there a way to get $X_1,Y_1$ from $l_1$?)

$\bf SUPPLEMENTARY\ INFO$:

The first equation is eq. 4.6b from ftp://ftp.physics.uwa.edu.au/pub/Clebsch-Gordan/Papers/SO%28n%29.pdf

The second equation is eq. 22 from http://jmp.aip.org/resource/1/jmapaq/v51/i9/p093518_s1

The parameters of the second equation are defined as follows:

$L_{rs}\equiv-i(x_r\partial_s-x_s\partial_r)$

$J_r\equiv\frac{1}{2}\varepsilon_{rst}L_{st}$, $N_r\equiv L_{r4}$, i.e. $$\begin{array}{ccc}J_1=L_{23}&J_2=L_{31}&J_3=L_{12}\\N_1=L_{14}&N_2=L_{24}&N_3=L_{34}\end{array}$$

$X_k\equiv\frac{1}{2}(J_k+N_k)$
$Y_k\equiv\frac{1}{2}(J_k-N_k)$

$M_X=-X,...,X-1,X$
$M_Y=-Y,...,Y-1,Y$

$X=\left|X_1-X_2\right|,\left|X_1-X_2\right|+1,...,X_1+X_2$
$Y=\left|Y_1-Y_2\right|,\left|Y_1-Y_2\right|+1,...,Y_1+Y_2$

Answer 1

Your first set of LaTeX typesets do not work but it is not clear that Eq.(4.6b) of Alisauskas and Eq.(22) of Caprio are calculations done using the same set of states constructed using the same subgroup chains.

To be precise: Caprio et al use two copies of $SO(3)$, i.e. the $SO(3)\otimes SO(3)$ subgroup chain - and thus states labelled by $\vert L_X,M_X;L_Y,M_Y\rangle$ , whereas Alisauskas uses a single $SO(3)$ - with states labelled (presumably) as $\vert (l_2,0);LM\rangle$. Thus the CG technology and the resulting reduced CGs are very likely different.

In addition, it seems Alisauskas' expression is limited to fully symmetry irreps, ie irreps of the type $(l_n,0,\ldots,0)$ of $SO(2n)$ - hence the $0$ in $(l_2,0)$ in my labelling of his states, so it's even more unlikely that the two results can be compared in general.

To proceed, it seems you will need to construct $so(4)$ states explicitly in an $so(3)$ basis. I suggest you refer to the work of

• S.C. Pang and K.T. Hecht, J. Math. Phys 8 (1967) 1233
• M.K.F. Wong, J.Math.Phys 8 (1967) 1899
• id., J.Math.Phys. 10 (1969) 1065

as starting points if you need the $SO(4)\downarrow SO(3)$ construction (which may become technical as you will discover).

The $SO(4)\downarrow SO(3)\otimes SO(3)$ basis is much easier, but you would have to specify which one of the subgroup you wish to use to reduce your CGs.