Semiclassical analysis of the Wigner $9j$ symbol with small and large angular momenta

We derive an asymptotic formula for the Wigner $9j$ symbol, in the limit of one small and eight large angular momenta, using a gauge-invariant factorization for the asymptotic solution of a set of coupled wave equations. Our factorization eliminates the geometric phases completely, using gauge-invariant noncanonical coordinates, parallel transports of spinors, and quantum rotation matrices. Our derivation generalizes to higher $3\mathit{nj}$ symbols. We display without proof some asymptotic formulas for the $12j$ symbol and the $15j$ symbol in the Appendices. This work contributes an asymptotic formula of the Wigner $9j$ symbol to the quantum theory of angular momentum and serves as an example of a general method for deriving asymptotic formulas for $3\mathit{nj}$ symbols.

Figure 1

The tetrahedron constructed from the six edge lengths $J_r$, $r=1,2,4,5,12,24$.

Figure 2

The vectors configuration from Fig. 1 are rearranged from ${\mathbf{J}}_1+{\mathbf{J}}_2+{\mathbf{J}}_4+{\mathbf{J}}_5=0$ to ${\mathbf{J}}_1+{\mathbf{J}}_4+{\mathbf{J}}_2+{\mathbf{J}}_5=0$. The intermediate vector ${\mathbf{J}}_{12}$ no longer appears. Instead, we have ${\mathbf{J}}_{14}={\mathbf{J}}_1+{\mathbf{J}}_4$.

Figure 3

The vectors configuration from Fig. 1 are rearranged from ${\mathbf{J}}_1+{\mathbf{J}}_2+{\mathbf{J}}_4+{\mathbf{J}}_5=0$ to ${\mathbf{J}}_4+{\mathbf{J}}_1+{\mathbf{J}}_2+{\mathbf{J}}_5=0$. The intermediate vector ${\mathbf{J}}_{24}$ no longer appears. Instead, we have ${\mathbf{J}}_{14}={\mathbf{J}}_1+{\mathbf{J}}_4$.

Figure 4

Illustration for the intersection of the Lagrangian manifolds and for the paths ${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_3$.

Figure 5

The configuration corresponding to the reference point $z_0$.

Figure 6

The holonomy $H_4$ is the area bounded by the curve from $\mathrm{ẑ}$ to ${\mathrm{ĵ}}_4$ along a great circle, the curve from $U_b$ that rotates ${\mathrm{ĵ}}_4$ to ${\mathrm{ĵ}}_4^{\prime }$, and the curve that rotates ${\mathrm{ĵ}}_4^{\prime }$ back to $\mathrm{ẑ}$ along another great circle.

Figure 7

Comparison of the exact $9j$ symbol (vertical sticks and dots) and the asymptotic formula Eq. (89) in the classically allowed region, for the values of $j$’s shown in Eq. (90).

Figure 8

Absolute value of the error of the asymptotic formula Eq. (89) for (a) the case in Eq. (90) and (b) the case in Eq. (91). The error is defined as the difference between the approximate value and the exact value.

Figure 9

Comparison of the exact $9j$ symbol (vertical sticks and dots) and the asymptotic formula Eq. (89) in the classically allowed region away from the caustics, for the values of $j$’s shown in Eq. (91).

Figure 10

Comparison of the exact $9j$ symbol (vertical sticks and dots) and the asymptotic formula Eq. (89) near a caustic, for the values of $j$’s shown in Eq. (91).

Figure 11

The angle $\theta$ is defined as the exterior angle between the edges $J_1$ and $J_4$ in a triangle having the three edge lengths $J_1,J_4,$ and $J_5$.

Figure 12

Comparison of exact $9j$ symbol (vertical sticks) and the asymptotic formula Eq. (A4), for the values of $j$’s shown in Eq. (A6).

Figure 13

The volume $V$ and the external dihedral angles ${\psi }_i$ are defined on the tetrahedron with the six edge lengths $J_3,J_5,J_2,J_6,J_{24},$ and $J_{34}$.

Figure 14

Comparison of the exact $12j$ symbol (vertical sticks and dots) and the asymptotic formula Eq. (B4), in the classically allowed region away from the caustics. The values used are those in Eq. (B8).

Figure 15

The volume $V$ and the external dihedral angles ${\psi }_i$ are defined on the tetrahedron with the six edge lengths $J_1,J_2,J_4,J_7,J_{12},$ and $J_{24}$.

Figure 16

Comparison of the exact $15j$ symbol (vertical sticks and dots) and the asymptotic formula Eq. (C4), for the values of $j$’s shown in Eq. (C8).