Extracted from Varshalovich, Moskalev, and Khersonskii - "Quantum Theory of Angular Momentum"
The Wigner 6j symbols are related to the coefficients of transformations between different coupling schemes of three angular momenta. The angular momenta $j_1$, $j_2$, $j_3$ may be coupled to give a resultant angular momentum $j$ and its projection $m$ in three different ways:
I) $j_1 + j_2 = j_{12}$, $j_{12} + j_3 = j$
II) $j_2 + j_3 = j_{23}$, $j_1 + j_{23} = j$
III) $j_1 + j_3 = j_{13}$, $j_{13} + j_2 = j$
States belonging to each coupling scheme form a complete set of states. A transition from one coupling scheme to another is performed by some unitary transformation which relates the states with the same total angular momentum $j$ and projection $m$.
One defines the Wigner 6j symbols $\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right}$ by the relation:
$\langle j_1j_2(j_{12})j_3jm|j_1j_3(j_{13})j_2jm\rangle = \delta_{jj'}\delta_{mm'}(-1)^{j_1+j_2+j_3+j} \sqrt{(2j_{12} + 1)(2j_{13} + 1)} \left{\begin{array}{ccc} j_1 & j_2 & j_{12} \ j_3 & j & j_{13} \end{array}\right}$
According to this definition, the 6j symbols may be given in terms of the Clebsch-Gordan coefficients. Here the sum is over $m_1,m_2,m_3,m_{12},m_{23}$ while $m$ and $m'$ are fixed. This relation completely determines absolute values and phases of the 6j symbols. The 6j symbols turn out to be real just as the Clebsch-Gordan coefficients are.
Instead of the Wigner 6j symbols the Racah coefficients are often used, especially in spectroscopy theory. These coefficients differ from the 6j symbols only by a phase factor:
$\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} = (-1)^{a+b+d+e} W(abed;cf)$
The Racah coefficients were introduced independently of the 6j symbols. The phase of the Racah coefficients coincides with the phase of the coefficients which describe the transformation between I and II coupling schemes.
The 6j symbols and the Racah coefficients may be written in the form of a 3×4 array $||R_{ia}||$ ($i = 1,2,3$; $\alpha = 1,2,3,4$) which is called the R-symbol:
$\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} = (-1)^{r} \prod_{i=1}^{3} \prod_{\alpha=1}^{4} \frac{(R_{i\alpha})!}{\prod_{i=1}^{3} \prod_{\alpha=1}^{4} (R_{i\alpha})!}$
where the elements $R_{i\alpha}$ are:
$R_{11} = -c + d + e$, $R_{12} = b + d - f$, $R_{13} = a + c - f$, $R_{14} = a + b - c$
$R_{21} = -b + d + f$, $R_{22} = c + d - e$, $R_{23} = a - b + c$, $R_{24} = a - e + f$
$R_{31} = -a + e + f$, $R_{32} = -a + b + c$, $R_{33} = c - d + e$, $R_{34} = b - d + f$
All 12 elements $R_{i\alpha}$ are integer nonnegative numbers.
The quantum-mechanical rules of vector addition impose some restrictions on possible values of momenta which are arguments of the 6j symbol $\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right}$:
(a) All momenta are integer or half-integer nonnegative numbers (with one exception considered in Sec. 9.4).
(b) Each triad ${j_1j_2j_{12}}$, ${j_{23}j_3j}$, ${j_1j_3j_{13}}$ and ${j_{23}j_1j}$ should satisfy the triangular condition.
The 6j symbols $\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right}$ vanish if at least one of the triads $(abc)$, $(cde)$, $(aef)$ and $(bdf)$ does not obey the triangular conditions.
The 6j symbols turn out to be real just as the Clebsch-Gordan coefficients are. The unitarity of the recoupling transformations implies the orthogonality and normalization conditions of the 6j symbols.
The symmetry properties of the 6j symbols may be formulated using the R symbols. The value of the R symbol is invariant under any permutation of its rows or columns. These symmetry relations involve $3! \times 4! = 144$ generally different Racah coefficients.
Classical Symmetries: The 6j symbol is invariant under any permutation of its columns or under interchange of the upper and lower arguments in each of any two columns:
$\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} = \left{\begin{array}{ccc} a & c & b \ d & f & e \end{array}\right} = \left{\begin{array}{ccc} b & a & c \ e & d & f \end{array}\right} = \cdots$
These relations involve $3! \times 4 = 24$ different 6j symbols.
Regge Symmetries: The relations below are functional ones, i.e. in general they cannot be obtained by interchanging the 6j symbol arguments:
$\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} = \left{\begin{array}{ccc} a & s_1-b & s_1-c \ d & s_1-e & s_1-f \end{array}\right} = \left{\begin{array}{ccc} s_2-a & b & s_2-c \ s_2-d & e & s_2-f \end{array}\right}$
where $s_1 = a + b + c$, $s_2 = a + d + e$, $s_3 = b + d + f$.
Combining the Regge symmetries and the classical symmetries, one gets all 144 symmetry relations.
$\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} = \Delta(abc)\Delta(cde)\Delta(aef)\Delta(bdf) \sum_n \frac{(-1)^{n+s}}{[n - a - b - c]![n - c - d - e]![n - a - e - f]![n - b - d - f]![a + b + d + e - n]!} \times \frac{1}{[a + c - d + f - n]![b + c + e + f - n]!}$
where the sums are over all integer nonnegative values of $n$ so that no factorial in denominators has a negative argument. The quantities $\Delta(abc)$ are defined by:
$\Delta(abc) = \sqrt{\frac{(a+b-c)!(a-b+c)!(-a+b+c)!}{(a+b+c+1)!}}$
This is the fundamental Racah formula for computing 6j symbols.
By the replacement $n \to a + b + d + e - n$ one can rewrite the Racah formula in the form:
$\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} = (-1)^{a+b+d+e} \Delta(abc)\Delta(cde)\Delta(aef)\Delta(bdf) \sum_n \frac{(-1)^n (a + b + d + e + 1 - n)!}{n![a + b - c - n]![-c + d + e - n]![a + e - f - n]![b + d - f - n]!} \times \frac{1}{[-a + c - d + f + n]![-b + c - e + f + n]!}$
Bargmann Formula:
$\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} = \prod_{i=1}^{3} \prod_{\alpha=1}^{4} \frac{(R_{i\alpha})!}{\sum_{n} (-1)^n (n+1)! \prod_{i=1}^{3} \prod_{\alpha=1}^{4} \frac{1}{(x_i + y_\alpha - n)!}}$
where $R_{i\alpha}$ are elements of the R symbol, $x_i, y_\alpha$ are summation indices, $n = \sum_{i=1}^{3} x_i + \sum_{\alpha=1}^{4} y_\alpha$.
Relations in which arguments are changed by 1/2:
$[(a + b + c + 1)(-a + b + c)(c + d + e + 1)(c + d - e)]^{1/2} \left{\begin{array}{ccc} a & b-\frac{1}{2} & c-\frac{1}{2} \ d & e & f \end{array}\right}$
$= -2c[(b + d + f + 1)(b + d - f)]^{1/2} \left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right}$
$+ [(a + b - c + 1)(a - b + c)(-c + d + e + 1)(c - d + e)]^{1/2} \left{\begin{array}{ccc} a & b+\frac{1}{2} & c-\frac{1}{2} \ d & e & f \end{array}\right}$
Relations in which arguments are changed by 1:
$(2c + 1){2[a(a + 1)d(d + 1) + b(b + 1)e(e + 1) - c(c + 1)f(f + 1)] - [a(a + 1) + b(b + 1) - c(c + 1)][d(d + 1) + e(e + 1) - c(c + 1)]} \left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right}$
$= -c[(a + b + c + 2)(-a + b + c + 1)(a - b + c + 1)(a + b - c) \times (d + e + c + 2)(-d + e + c + 1)(d - e + c + 1)(d + e - c)]^{1/2} \left{\begin{array}{ccc} a & b & c+1 \ d & e & f \end{array}\right}$
$- (c + 1)[(a + b + c + 1)(-a + b + c)(a - b + c)(a + b - c + 1) \times (d + e + c + 1)(-d + e + c)(d - e + c)(d + e - c + 1)]^{1/2} \left{\begin{array}{ccc} a & b & c-1 \ d & e & f \end{array}\right}$
One of Arguments is Equal to Zero:
$\left{\begin{array}{ccc} 0 & b & c \ d & e & f \end{array}\right} = (-1)^{b+e+d} \frac{\delta_{bc}\delta_{ef}}{\sqrt{(2b + 1)(2e + 1)}}$
$\left{\begin{array}{ccc} a & 0 & c \ d & e & f \end{array}\right} = (-1)^{a+d+e} \frac{\delta_{ac}\delta_{df}}{\sqrt{(2a + 1)(2d + 1)}}$
$\left{\begin{array}{ccc} a & b & 0 \ d & e & f \end{array}\right} = (-1)^{a+b+e} \frac{\delta_{ab}\delta_{de}}{\sqrt{(2a + 1)(2d + 1)}}$
One of Arguments is Equal to the Sum of Two Others:
If one of the 6j symbol arguments is equal to sum of two others from the same triad $(abc)$, $(cde)$, $(aef)$, $(bdf)$, one may use the classical symmetries to express it in the form:
$\left{\begin{array}{ccc} a & b & a+b \ d & e & a+b \end{array}\right} = (-1)^{a+b+d+e} \frac{(2a)!(2b)!(a + b + d + e + 1)!(a + b - d + e)!(a + b + d - e)!(-a + e + f)!(-b + d + f)!}{(2a + 2b + 1)!(-a - b + d + e)!(a + e - f)!(a - e + f)!(a + e + f + 1)!(b + d - f)!(b - d + f)!(b + d + f + 1)!}$
The asymptotic behaviour of the 6j symbols $\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right}$ for large angular momenta is closely associated with geometric properties of the tetrahedron whose edges are $a + \frac{1}{2}$, $b + \frac{1}{2}$, etc.
The tetrahedron edges are defined as:
The Ponzano-Regge Formula (semiclassical approximation to the 6j symbols): If $a, b, c, d, e, f \gg 1$, then
$\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} \approx \frac{1}{\sqrt{12\pi V}} \cos\left(\sum_{i<k} j_{ik} \Theta_{ik} + \frac{\pi}{4}\right)$
where $V$ is the volume of the tetrahedron, $\Theta_{ik}$ is the angle between two external normals to the planes adjacent to the edge $j_{ik}$.
This formula is valid only if $V^2 > 0$ (classically allowed domain). If $V^2 < 0$ (classically forbidden domain, when an associated tetrahedron does not exist), the asymptotic expression becomes:
$\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} \approx \frac{2}{\sqrt{12\pi|V|}} \cos\Phi \exp\left(-\sum_{i,k=1}^{4} j_{ik}|\text{Im}\Theta_{ik}|\right)$
where $V^2 < 0$, and $\Phi = \sum_{i<k} (\pi - \frac{\pi}{2})\text{Re}\Theta_{ik}$.
In this case the 6j symbols are exponentially small even if the triangular condition is satisfied.
The tetrahedron volume is equal to:
$288 V^2 = \begin{vmatrix} 0 & j_{12}^2 & j_{13}^2 & j_{14}^2 & 1 \ j_{12}^2 & 0 & j_{23}^2 & j_{24}^2 & 1 \ j_{13}^2 & j_{23}^2 & 0 & j_{34}^2 & 1 \ j_{14}^2 & j_{24}^2 & j_{34}^2 & 0 & 1 \ 1 & 1 & 1 & 1 & 0 \end{vmatrix}$
The angles $\Theta_{ik}$ are given by: $5 S_i S_k \sin \Theta_{ik} = -F_{ik}$
where $S_i$ is the area of the triangle opposite to the vertex $p_i$. One can evaluate $S_i$ using the standard formulas. For example:
$16 S_1^2 = (j_{12} + j_{13} + j_{14})(j_{12} + j_{13} - j_{14})(j_{12} - j_{13} + j_{14})(-j_{12} + j_{13} + j_{14})$
Wigner Formula: In particular, for large angular momenta, one has:
$\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} \approx \frac{1}{24\pi V}$
This formula is valid only on the average because the 6j symbols oscillate rapidly with momentum variations in the region of large angular momenta.
The expressions for the 6j symbols given above are valid if all triangular conditions are satisfied. When implementing these formulas, care must be taken with:
Factorial calculations: For large arguments, factorials can overflow. Use logarithmic representation or Stirling's approximation when necessary.
Alternating sums: The Racah formula involves alternating sums which can lead to numerical instability due to cancellation of large terms.
Square root calculations: Ensure arguments of square roots are non-negative, which should be guaranteed by the triangular conditions.
Use recursion relations: The recursion relations can be more numerically stable than direct calculation for certain parameter ranges.
Special cases first: Always check for special cases (zero arguments, equal arguments, etc.) before applying general formulas.
Symmetry relations: Use the 144 symmetry relations to reduce computational effort by transforming arguments to a canonical form.
Caching: Since 6j symbols appear frequently in calculations, implement caching mechanisms.
The 144 symmetry relations of the 6j symbols can be used to:
The 6j symbols may be written as sums of products of the Clebsch-Gordan coefficients or 3jm symbols. The relations between the 6j symbols and the 3jm symbols are:
$\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} = \sum_{\alpha,\beta,\gamma,\delta,\epsilon,\phi} (-1)^{s-\alpha-\beta-\gamma} \begin{pmatrix} a & b & c \ \alpha & \beta & \gamma \end{pmatrix} \begin{pmatrix} c & d & e \ \gamma & \delta & \epsilon \end{pmatrix} \begin{pmatrix} a & f & e \ \alpha & \phi & -\epsilon \end{pmatrix} \begin{pmatrix} b & d & f \ \beta & -\delta & -\phi \end{pmatrix}$
In this equation the sum is over all possible values of $\alpha,\beta,\gamma,\delta,\epsilon,\phi$ with only three summation indices being independent.
The 6j symbols appear naturally in the evaluation of spin network vertices. In particular, they are essential for:
The primary application of 6j symbols is in angular momentum coupling theory:
In quantum geometry and loop quantum gravity:
Racah Formula (most fundamental): $\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} = \Delta(abc)\Delta(cde)\Delta(aef)\Delta(bdf) \sum_n \frac{(-1)^{n+s}}{[n - a - b - c]![n - c - d - e]![n - a - e - f]![n - b - d - f]![a + b + d + e - n]!} \times \frac{1}{[a + c - d + f - n]![b + c + e + f - n]!}$
Delta function: $\Delta(abc) = \sqrt{\frac{(a+b-c)!(a-b+c)!(-a+b+c)!}{(a+b+c+1)!}}$
Triangular condition check: The 6j symbol vanishes unless all four triads $(abc)$, $(cde)$, $(aef)$, $(bdf)$ satisfy: $|j_1 - j_2| \leq j_3 \leq j_1 + j_2$
Phase relation to Racah coefficients: $\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} = (-1)^{a+b+d+e} W(abed;cf)$
Zero argument cases: $\left{\begin{array}{ccc} 0 & b & c \ d & e & f \end{array}\right} = (-1)^{b+e+d} \frac{\delta_{bc}\delta_{ef}}{\sqrt{(2b + 1)(2e + 1)}}$
Sum cases: $\left{\begin{array}{ccc} a & b & a+b \ d & e & a+b \end{array}\right} = (-1)^{a+b+d+e} \frac{(2a)!(2b)!(a + b + d + e + 1)!(a + b - d + e)!(a + b + d - e)!}{(2a + 2b + 1)!(-a - b + d + e)!(a + e - f)!(a - e + f)!(a + e + f + 1)!} \times \frac{(-a + e + f)!(-b + d + f)!}{(b + d - f)!(b - d + f)!(b + d + f + 1)!}$
Equal pairs: When $a = b$ and $d = e$: $\left{\begin{array}{ccc} a & a & c \ d & d & f \end{array}\right} = (-1)^{2a+c+f} \frac{\sqrt{(2a - c)!(2d - c)!}}{[(2a + c + 1)!(2d + c + 1)!]^{1/2}} V_c(a,f,d)$
where $V_c(a,f,d) = V_c(d,f,a)$ and satisfies specific recursion relations.
Ponzano-Regge Formula (large arguments): $\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} \approx \frac{1}{\sqrt{12\pi V}} \cos\left(\sum_{i<k} j_{ik} \Theta_{ik} + \frac{\pi}{4}\right)$
Edmonds' Formula (specific large argument case): If $f, m, n$ are arbitrary and $a, b, c \gg f, m, n$: $\left{\begin{array}{ccc} a & b & c \ f & m & n \end{array}\right} \approx \frac{(-1)^{a+b+c+f+m+n}}{\sqrt{(2a + 1)(2b + 1)}} d^f_{mn}(\theta)$
where $d^f_{mn}(\theta)$ is the rotation matrix, and $\theta$ is the angle between tetrahedron edges.
Racah Formula (one large argument): If $a, b, c \gg f$ and $f$ is an arbitrary integer: $\left{\begin{array}{ccc} a & b & c \ b & a & f \end{array}\right} \approx \frac{(-1)^{a+b+c+f}}{\sqrt{(2a + 1)(2b + 1)}} P_f(\cos\theta)$
where $P_f$ is the Legendre polynomial and $\cos\theta = \frac{a(a + 1) + b(b + 1) - c(c + 1)}{2\sqrt{a(a + 1)b(b + 1)}}$.
Wigner Formula (all large arguments): $\left{\begin{array}{ccc} a & b & c \ d & e & f \end{array}\right} \approx \frac{1}{24\pi V}$
This formula is valid only on the average due to rapid oscillations.