Semiclassical analysis of the Wigner $12j$ symbol with one small angular momentum

We derive an asymptotic formula for the Wigner $12j$ symbol, in the limit of one small and 11 large angular momenta. There are two kinds of asymptotic formulas for the $12j$ symbol with one small angular momentum. We present the first kind of formula in this paper. Our derivation relies on the techniques developed in the semiclassical analysis of the Wigner $9j$ symbol [L. Yu and R. G. Littlejohn, Phys. Rev. A 83, 052114 (2011)], where we used a gauge-invariant form of the multicomponent WKB wave functions to derive asymptotic formulas for the $9j$ symbol with small and large angular momenta. When applying the same technique to the $12j$ symbol in this paper, we find that the spinor is diagonalized in the direction of an intermediate angular momentum. In addition, we find that the geometry of the derived asymptotic formula for the $12j$ symbol is expressed in terms of the vector diagram for a $9j$ symbol. This illustrates a general geometric connection between asymptotic limits of the various $3nj$ symbols. This work contributes an asymptotic formula for the $12j$ symbol to the quantum theory of angular momentum, and serves as a basis for finding asymptotic formulas for the Wigner $15j$ symbol with two small angular momenta.

Figure 1

The spin network of the Wigner $12j$ symbol.

Figure 2

Decomposition of the spin network of the $12j$ symbol.

Figure 3

Configuration of a point on ${\mathcal{L}}_a^{9j}$, projected onto the angular momentum space ${\Lambda }_{5j}$, and viewed in a single ${\mathbb{R}}^3$.

Figure 4

Configuration of a point on the intersection $I_{11}$ set, projected onto the angular momentum space ${\Lambda }_{5j}$, and viewed in a single ${\mathbb{R}}^3$.

Figure 5

The loop from a point $p\in I_{11}$ to $q\in I_{12}$ along ${\mathcal{L}}_a^{9j}$, and then to $q^{\prime }\in I_{12}$ along $I_{12}$, and then to $p^{\prime }\in I_{11}$ along ${\mathcal{L}}_b^{9j}$, and finally back to $p$ along $I_{11}$.

Figure 6

The loop from Fig. 5 projected onto a loop in ${\Lambda }_{5j}$, as viewed in a single ${\mathbb{R}}^3$.

Figure 7

Decomposition of the angles ${\phi }_1$ and ${\phi }_6$ into dihedral angles in two tetrahedra.

Figure 8

The angles ${\phi }_r$ is the angle between the normals of the adjacent triangles sharing the edge $J_r$, where the normals are defined by the orientation of the triangles shown. This is essentially Fig. 2 in [16], with an unsymmetrical labeling of the $9j$.

Figure 9

The vector configuration at the point $z_{11}$ in $I_{11}$.

Figure 10

The phase difference between two gauge choices can be expressed as an area around a closed loop on the unit sphere.

Figure 11

The angles ${\phi }_{12}$ and ${\phi }_{13}$ are internal dihedral angles in the tetrahedron with the six lengths ${\mathbf{J}}_6,{\mathbf{J}}_{12},{\mathbf{J}}_{34},{\mathbf{J}}_{13},{\mathbf{J}}_{24}$, ${\mathbf{J}}_{2^{\prime }3}$, where ${\mathbf{J}}_{2^{\prime }3}={\mathbf{J}}_3-{\mathbf{J}}_2$. The angle $\theta$ is the angle between ${\mathbf{J}}_{12}$ and ${\mathbf{J}}_{13}$.

Figure 12

Comparison of the exact $12j$ symbol (vertical sticks and dots) and the asymptotic formula (80) in the classically allowed region away from the caustics, for the values of the $j$’s shown in Eq. (85).

Figure 13

Comparison of the exact $12j$ symbol (vertical sticks and dots) and the asymptotic formula (80) in the classically allowed region away from the caustics, for the values of the $j$’s shown in Eq. (86).

Figure 14

Absolute value of the error of the asymptotic formula (80) for (a) the case shown in Eq. (85), and (b) the case shown in Eq. (86). The error is defined as the difference between the approximate value and the exact value.