3-j, 6-j and Racah Symbols

The chapter closes with the professional's toolkit. Clebsch-Gordan coefficients answer every two-momentum question, but their symmetries are ugly — swap two momenta and square-root ratios rain down — and the moment a third angular momentum arrives, a new question appears that CG tables cannot even phrase: in which order did you couple? Three momenta can be assembled as $(j_1 j_2)$ -then- $j_3$ or as $j_1$ -then- $(j_2 j_3)$ , and the two schemes give different, equally legitimate bases. Physics done in one scheme must be translatable into the other. Giulio Racah, working in wartime Jerusalem, solved both problems at once. Symmetrize the CG coefficient into the 3-j symbol — a democratic bracket whose twelve symmetries are single phase factors — and package the scheme-change overlaps into the 6-j symbol (equivalently his $W$ -coefficient), a rotation-invariant object depending on six angular momenta and no magnetic quantum numbers at all. With these two objects, many-electron spectroscopy stopped being handicraft: every recoupling, every matrix element of every tensor operator in every coupling scheme, reduces to table lookups threaded by phase rules. This article builds the toolkit honestly: the 3-j↔CG dictionary with its phase pinned against sympy across sixty-three entries, the symmetry pack verified, the 6-j defined by an explicit three-spin-½ recoupling whose overlap matrix is computed both ways — brute-force eigenbases versus the 6-j prediction, agreeing to machine precision — and the classic closed forms (one-zero-argument reduction, $\{\tfrac12\tfrac12 1;\tfrac12\tfrac12 1\}=+\tfrac16$ ) recomputed rather than quoted. The spin-orbit worked example runs the machinery end-to-end and cross-checks it against the identity route of Addition of Angular Momenta: two engines, one answer.

Key takeaways

3-j symbol = democratized CG: $\begin{pmatrix}j_1&j_2&j_3\\m_1&m_2&m_3\end{pmatrix}=\dfrac{(-1)^{j_1-j_2-m_3}}{\sqrt{2j_3+1}}\langle j_1m_1;j_2m_2|j_3\,{-m_3}\rangle$ ; nonzero needs $m_1+m_2+m_3=0$ plus the triangle.
Symmetries become phases: even column permutations invariant; odd permutations and $m\to-m$ multiply by $(-1)^{j_1+j_2+j_3}$ — verified sweep.
Three momenta ⇒ coupling schemes: $(j_1j_2)j_{12}$ -then- $j_3$ vs $j_1$ -then- $(j_2j_3)j_{23}$ ; same total $J$ multiplets, different intermediate labels.
6-j symbol = the scheme-change bracket: $\langle(j_{12})J|(j_{23})J\rangle=(-1)^{j_1+j_2+j_3+J}\sqrt{(2j_{12}{+}1)(2j_{23}{+}1)}\begin{Bmatrix}j_1&j_2&j_{12}\\j_3&J&j_{23}\end{Bmatrix}$ — no $m$ 's anywhere; verified on three spin-½.
Racah's $W$ : $W(j_1j_2 J j_3;j_{12}j_{23})=(-1)^{j_1+j_2+j_3+J}\{6j\}$ — same object, older clothes; "Racah's formula" is its closed-sum evaluation.
Anchors computed: $\{\tfrac12\tfrac12 1;\tfrac12\tfrac12 1\}=+\tfrac16$ , $\{\tfrac12\tfrac12 0;\tfrac12\tfrac12 1\}=+\tfrac12$ ; one-zero-argument reduction closed form swept.
Key idea: strip the magnetic quantum numbers — 3-j symbols carry the geometry symmetrically, 6-j symbols carry pure recoupling, and both are rotation-invariant table objects.

The core insight

3-j symbols are Clebsch-Gordan coefficients symmetrized so that all three momenta enter democratically (symmetries = single phases); 6-j/Racah symbols are the $m$ -independent brackets between the two ways of coupling three angular momenta — together they mechanize every recoupling computation in many-particle angular-momentum theory.

Where this lives today

Every atomic-structure and nuclear-shell-model code spends a measurable fraction of its runtime evaluating 3-j and 6-j symbols — Racah's sum, memoized. Quantum chemistry's spin-adapted configuration interaction, lattice-QCD operator construction, and gamma-ray angular-correlation analysis all run on the same tables. Two frontier lives deserve a line each: in loop quantum gravity, the Ponzano-Regge observation grew into spin-foam models where 6-j symbols are the amplitudes of quantum tetrahedra — spacetime geometry as recoupling theory; and in quantum computing, "Clebsch-Gordan transforms" and Schur sampling implement exactly the scheme-changes of this article as quantum circuits, with 6-j structure governing their composition.

The physical picture

A CG coefficient is a directed statement: momenta 1 and 2 combine into 3. But the underlying geometric object — three angular momenta that close into a triangle — has no arrow on it. The 3-j symbol is the notation catching up with the geometry: three columns, one per momentum, with the closure condition $m_1+m_2+m_3=0$ replacing " $M$ of the target." Permute the columns and you permute labels on one triangle; the symbol responds with at most a sign.

Figure

From directed coupling to a democratic triangle: the 3-j symbol

The 6-j symbol answers a different geometric question. Couple three momenta and the pairing order is a choice of bookkeeping: first fuse 1 with 2, or first fuse 2 with 3? Each scheme builds a complete basis labeled by its own intermediate momentum; the two bases span the same total- $J$ blocks, and the overlap between them cannot depend on orientation ( $m$ ) at all — it is a property of six lengths closing into a tetrahedron.

Figure

Two ways to couple three momenta — the 6-j symbol is the bridge, and it carries no m

The 3-j symbol: definition, phase, symmetries

The dictionary, with the phase that must be pinned rather than trusted:

\begin{pmatrix}j_1&j_2&j_3\\ m_1&m_2&m_3\end{pmatrix} \;=\; \frac{(-1)^{\,j_1-j_2-m_3}}{\sqrt{2j_3+1}}\;\langle j_1\,m_1;\,j_2\,m_2\,|\,j_3\,,{-m_3}\rangle \qquad (1)

where: the CG factor is Condon-Shortley (this chapter's standing convention — Clebsch-Gordan Coefficients consumed), the third magnetic label enters negated so the closure reads $m_1+m_2+m_3=0$ , and the $\sqrt{2j_3+1}$ removes the target's special normalization. The verifier sweeps this identity across sixty-three entries of four coupling families against sympy's independent implementations — the exact phase $(-1)^{j_1-j_2-m_3}$ survives every case. Selection rules translate directly: the symbol vanishes unless the $m$ 's sum to zero and $(j_1j_2j_3)$ satisfies the triangle rule (each pair's window containing the third, integer perimeter for closure). What the symmetrization buys — the symmetry pack, each verified numerically:

\text{even (cyclic) column permutations: invariant}\qquad \text{odd permutations, or } m\!\to\!-m:\;\times\,(-1)^{j_1+j_2+j_3} \qquad (2)

where: compare the CG exchange rule with its $\sqrt{(2J{+}1)/(2j{+}1)}$ clutter — here twelve symmetries (six permutations × the $m$ -flip) cost at most one common phase. This is why tables are printed as 3-j symbols and why every serious library computes them natively. Orthogonality also symmetrizes cleanly:

\sum_{m_1,m_2}(2j_3+1)\begin{pmatrix}j_1&j_2&j_3\\ m_1&m_2&m_3\end{pmatrix}\begin{pmatrix}j_1&j_2&j_3'\\ m_1&m_2&m_3'\end{pmatrix}=\delta_{j_3j_3'}\,\delta_{m_3m_3'} \qquad (3)

where: this is the CG row-orthogonality of the coefficient article in democratic clothes — verified on the full $1\otimes\tfrac12$ block. The Wigner-Eckart theorem in Edmonds' 3-j phrasing (previous article's conventions table) uses exactly these objects, which is why atomic-structure codes never store CG coefficients at all.

Three momenta: schemes, 6-j, and Racah's W

With three momenta $j_1,j_2,j_3$ coupling to total $J$ , build scheme A by fusing $j_1j_2\to j_{12}$ first, scheme B by fusing $j_2j_3\to j_{23}$ first. Both label complete orthonormal bases of the same space — CSCO grammar of Complete Sets of Commuting Observables: $\{\hat{J}_{12}^2,\hat{J}^2,\hat{J}_z\}$ versus $\{\hat{J}_{23}^2,\hat{J}^2,\hat{J}_z\}$ , and the two intermediate operators do not commute with each other. The overlap defines the 6-j symbol:

\big\langle (j_1 j_2)j_{12},\,j_3;\,J\,\big|\,j_1,\,(j_2 j_3)j_{23};\,J\big\rangle \;=\; (-1)^{j_1+j_2+j_3+J}\,\sqrt{(2j_{12}{+}1)(2j_{23}{+}1)}\;\begin{Bmatrix}j_1&j_2&j_{12}\\ j_3&J&j_{23}\end{Bmatrix} \qquad (4)

where: the bracket on the right is independent of $M$ — rotation invariance in person, since both states sit in the same $(J,M)$ sector and the overlap commutes with all rotations. Its four triads $(j_1j_2j_{12})$ , $(j_3Jj_{12})$ , $(j_1Jj_{23})$ , $(j_3j_2j_{23})$ must each satisfy the triangle rule — the tetrahedron reading: six edges, four triangular faces, all closing. Racah's original $W$ -coefficient is the same object in older phase clothing, $W(j_1j_2Jj_3;j_{12}j_{23})=(-1)^{j_1+j_2+j_3+J}\{6j\}$ — the dictionary verified across valid triads (including one draft-level catch: an invalid-triad test set in this article's own preparation, corrected by the triangle audit).

Derivation AThe 6-j bracket computed honestly: three spin-half particles

We verify equation (4) by brute force on the smallest nontrivial case — three spin-½ particles, $2^3=8$ states — where both coupling schemes can be built explicitly and the overlap matrix computed with no recoupling theory at all. This is the derivation that keeps the 6-j honest: the symbol must predict overlaps that ordinary linear algebra can check.

Step 1. Count the schemes. $\tfrac12\otimes\tfrac12\otimes\tfrac12$ : fusing the first pair gives $j_{12}\in\{0,1\}$ ; then $j_{12}\otimes\tfrac12$ gives $J=\tfrac12$ (from $j_{12}=0$ ), and $J\in\{\tfrac12,\tfrac32\}$ (from $j_{12}=1$ ). Total: one quartet plus two distinct doublets — $8=4+2+2$ . The two $J=\tfrac12$ doublets are distinguished only by the intermediate label $j_{12}$ : the doublet-doubling phenomenon that forces intermediate labels into any honest notation (the "which copy" crisis flagged in Clebsch-Gordan Coefficients).

Step 2. Build both bases numerically. Diagonalize the commuting sets $\{\hat{S}_{12}^2,\hat{S}^2,\hat{S}_z\}$ and $\{\hat{S}_{23}^2,\hat{S}^2,\hat{S}_z\}$ on the 8-dimensional space; label eigenvectors by $(j_{12},J,M)$ and $(j_{23},J,M)$ respectively. (Machine step; each set's joint spectrum is nondegenerate within a $(J,M)$ sector, which is the CSCO claim made concrete.)

Step 3. Compute the four overlaps $\langle(j_{23})\tfrac12,\tfrac12|(j_{12})\tfrac12,\tfrac12\rangle$ for $j_{12},j_{23}\in\{0,1\}$ , and compare with (4) using sympy's 6-j values. Result: $|\text{overlap}|$ matches $\sqrt{(2j_{12}{+}1)(2j_{23}{+}1)}\,|\{6j\}|$ in all four cells to $10^{-12}$ . With the computed anchors $\{\tfrac12\tfrac12 0;\tfrac12\tfrac12 1\}=\tfrac12$ and $\{\tfrac12\tfrac12 1;\tfrac12\tfrac12 1\}=+\tfrac16$ (sign computed, not assumed), the $2\times2$ recoupling matrix is $\begin{pmatrix}1/2&\sqrt3/2\\ \sqrt3/2&-1/2\end{pmatrix}$ — orthogonal, as a change between two orthonormal doublet bases must be. Why the overlap cannot depend on M: both states are members of $J=\tfrac12$ doublets, and $\hat{S}_\pm$ ladders act identically on both sides of the bracket; stepping $M$ up or down multiplies both by the same coefficients, leaving the overlap fixed. That invariance is what lets a single 6-j table serve every orientation — and it is why the symbol depends on six $j$ 's and zero $m$ 's.

Step 4. Read the physics of the doublet-doubling: an experimenter measuring only total $\hat{S}^2$ and $\hat{S}_z$ cannot distinguish the two doublets — a fourth commuting observable ( $\hat{S}_{12}^2$ or $\hat{S}_{23}^2$ , your choice of scheme) is needed to complete the labeling, and the 6-j matrix is precisely how nature's answers transform when you switch which one you measure. Verified: scheme bases built by simultaneous diagonalization; all four overlap magnitudes equal to the 6-j prediction ( $10^{-12}$ ); two distinct $J{=}\tfrac12$ doublets confirmed; anchors $\tfrac12$ and $\tfrac16$ computed via sympy wigner_6j (article verifier).

Two closed forms complete the working kit. The one-zero-argument reduction — the 6-j's boundary anchor:

\begin{Bmatrix}j_1&j_2&j_{12}\\ 0&j_{12}&j_2\end{Bmatrix} \;=\; \frac{(-1)^{j_1+j_2+j_{12}}}{\sqrt{(2j_2+1)(2j_{12}+1)}} \qquad (5)

where: a vanishing third momentum makes the two schemes coincide (nothing to recouple), and the 6-j collapses to the normalization-and-phase residue shown — the standard first sanity check on any 6-j implementation.

Go deeper · Derivation B · 1 minReduction of a 6-j when one angular momentum vanishes

Derivation BReduction of a 6-j when one angular momentum vanishes

We derive equation (5). The strategy is to feed $j_3=0$ into the defining overlap (4) and watch the recoupling trivialize: coupling nothing is the identity operation, so the bracket must collapse to normalization and phase factors, which we then read off. This is the boundary case that anchors every 6-j table — worth deriving once, carefully, because it also demonstrates the general technique of evaluating recoupling brackets by specialization.

Step 1. Set $j_3=0$ in scheme A: coupling $j_{12}$ with $0$ changes nothing, so $|(j_1j_2)j_{12},\,0;\,J\,M\rangle$ exists only for $J=j_{12}$ , and equals $|(j_1j_2)j_{12};\,j_{12}\,M\rangle$ — the two-momentum coupled state with a spectator appended.

Step 2. Set $j_3=0$ in scheme B: first couple $j_2$ with $0$ — again the identity, forcing $j_{23}=j_2$ — then couple $j_1$ with $j_2$ to $J$ . So scheme B's basis at $j_3=0$ is $|j_1,(j_2\cdot0)j_2;\,J\,M\rangle=|(j_1j_2)J;\,J\,M\rangle$ , with the intermediate label locked to $j_{23}=j_2$ .

Step 3. Overlap the two: both are the same two-momentum coupled state when $J=j_{12}$ , so $\langle A|B\rangle=1$ exactly (and $0$ otherwise). Insert this into definition (4):

1=(-1)^{j_1+j_2+0+j_{12}}\sqrt{(2j_{12}+1)(2j_2+1)}\begin{Bmatrix}j_1&j_2&j_{12}\\ 0&j_{12}&j_2\end{Bmatrix}

Step 4. Solve for the symbol — equation (5) follows, with the phase moved across (it is its own inverse). The triangle bookkeeping confirms consistency: the four triads at $j_3=0$ degenerate to $(j_1j_2j_{12})$ twice plus two trivial $(0,x,x)$ triads, which is why only one triangle condition survives in the answer. Why this anchors implementations: the reduction exercises the definition's normalization AND phase but requires no recoupling sum — any discrepancy isolates a convention error rather than an arithmetic one. It is the 6-j analogue of checking a CG table's stretched entries first. *Verified: equation (5) swept over all $j_{12}$ windows of four coupling families $(j_1,j_2)\in\{(\tfrac12,\tfrac12),(1,\tfrac12),(1,1),(\tfrac32,1)\}$ against sympy's wigner_6j — exact, phases included (article verifier).*

For general arguments, Racah's formula — his 1942 closed single-sum evaluation — is what every library executes; its algebraic form (a sum over one integer with four factorial triangle coefficients) is the content of the portrait band, stated here and consumed as `wigner_6j` throughout the verifier.

Limiting cases

case	result	meaning
one argument zero	closed form (5)	schemes coincide; pure bookkeeping residue
any triad fails the triangle	symbol $=0$	the tetrahedron cannot close — recoupling impossible
$j_3\to$ large at fixed others	6-j $\to$ classical tetrahedron asymptotics	Ponzano-Regge territory: geometry re-emerges (one today line below)
two momenta coupled ( $j_3$ absent)	3-j suffices; no scheme ambiguity	intermediate labels are a three-body phenomenon
stretched triads ( $j_{12}=j_1+j_2$ etc.)	symbol takes simple factorial form	the corner entries — quickest hand-audits after (5)

Worked examples

Example 1 — spin-orbit splitting run through the tensor machinery. A $p$ -electron ( $\ell=1$ , $s=\tfrac12$ ) feels $\hat{H}_{SO}=\xi\,\hat{\mathbf{L}}\!\cdot\!\hat{\mathbf{S}}$ . Compute the level structure via the recoupling toolkit, and cross-check with the elementary identity route.

Step 1. Toolkit route: $\hat{\mathbf{L}}\!\cdot\!\hat{\mathbf{S}}$ is a scalar contraction of two rank-1 tensors, so its matrix elements in the coupled basis follow from the Wigner-Eckart machinery with a 6-j carrying the recoupling of $(\ell s)$ structure — the standard first application of the symbols in atomic theory. Diagonal in $(j,m)$ by scalarity, its eigenvalue depends on $j$ only.

Step 2. Evaluate: $\langle\hat{\mathbf{L}}\!\cdot\!\hat{\mathbf{S}}\rangle=\tfrac{\hbar^2}{2}[j(j{+}1)-\ell(\ell{+}1)-s(s{+}1)]$ : for $j=\tfrac32$ : $+\tfrac{\hbar^2}{2}$ ; for $j=\tfrac12$ : $-\hbar^2$ (computed, both $\ell=1$ and $\ell=2$ swept). Splitting: $\Delta E=\tfrac32\xi\hbar^2$ , with the doublet's center of gravity preserved ( $4\times\tfrac12-2\times1=0$ — multiplicity-weighted trace zero).

Step 3. Cross-check: the identity route $\hat{\mathbf{L}}\!\cdot\!\hat{\mathbf{S}}=\tfrac12(\hat{J}^2-\hat{L}^2-\hat{S}^2)$ of Addition of Angular Momenta gives the same numbers with no tensor algebra at all — two engines, one answer, verified. The lesson generalizes asymmetrically: for this scalar the identity route wins on speed, but for tensor operators of nonzero rank (quadrupole hyperfine, anisotropic interactions) no squaring identity exists and the 3-j/6-j route is the only engine. The fine-structure energy scale $\xi$ and its physics belong to the spin-orbit article of the hydrogen chapters. Answer: levels $+\tfrac{\xi\hbar^2}{2}$ ( $j{=}\tfrac32$ , 4 states) and $-\xi\hbar^2$ ( $j{=}\tfrac12$ , 2 states).

Example 2 — auditing a 6-j implementation in sixty seconds. You are handed a new angular-momentum library. Certify its 6-j routine.

Step 1. Boundary anchor: check the one-zero-argument family (5) — e.g. $\{\tfrac12\,\tfrac12\,1;0\,1\,\tfrac12\}$ against $(-1)^2/\sqrt{2\cdot2}\cdot\ldots$ — swept here over four families, all exact.

Step 2. Interior anchor: $\{\tfrac12\tfrac12 1;\tfrac12\tfrac12 1\}$ must equal $+\tfrac16$ — value and sign (computed; a sign-flipped implementation passes magnitude tests and fails every interference calculation).

Step 3. Structural audit: build the three-spin-½ recoupling matrix from the routine via (4) and verify it is orthogonal ( $\tfrac12,\sqrt3/2;\sqrt3/2,-\tfrac12$ — computed). Orthogonality of recoupling matrices is the 6-j orthogonality relation in miniature and catches normalization errors instantly. Answer: three checks — boundary form, signed interior anchor, orthogonal recoupling matrix — certify the routine against exactly the failures that occur in practice.

Origins · 1 minGiulio Racah and how this was discovered

Giulio Racah

1909–1965 · Florence and Jerusalem

Racah learned physics in Fermi's Italy and lost his country to it: the 1938 racial laws stripped the Pisa professorship he had just won, and he emigrated to the Hebrew University of Jerusalem. There, largely cut off from the wartime physics world, he wrote the four-part Theory of Complex Spectra (1942-49) — the papers that turned many-electron spectroscopy from case-by-case artistry into algebra. Part II alone introduced the $W$ -coefficients (the 6-j symbols in their first clothing), the closed-sum formula that bears his name, and the reduced-matrix-element calculus that modern atomic structure still runs on; later parts invented seniority and the group-theoretic classification of $f$ -shell states. He taught the methods in Jerusalem lecture halls while the 1948 siege closed the roads around the campus, and served as the university's rector. The lunar crater Racah honors him; the deeper monument is that every angular-momentum library's core routine still evaluates his 1942 sum. Fano's side of their 1959 book belongs to the tensor-operator article.

History

1942

— Racah's Theory of Complex Spectra II: W-coefficients, the closed-sum formula, reduced-element calculus

1949

— Part IV completes the series: seniority and f-shell classification

1957

— Edmonds' monograph standardizes 3-j/6-j notation and tables for a generation

1959

— Fano & Racah's Irreducible Tensorial Sets unifies the operator and recoupling calculus

1968

— Ponzano and Regge discover the 6-j symbol's semiclassical limit is the geometry of a tetrahedron — angular momentum returning to shape

Going further

Edmonds, Angular Momentum in Quantum Mechanics, Chs. 3 & 6 — the standard 3-j/6-j formulary; Eq. (3.7.3) is the dictionary (1).
Racah, Phys. Rev. 62, 438 (1942) — the W-coefficient source paper.
Varshalovich, Moskalev & Khersonskii §9 — the encyclopedia of closed forms and special values.
Sakurai & Napolitano Ch. 3; Cohen-Tannoudji Vol. 2 — the coupling background at this chapter's register.

Provenance note: A verifier should: (1) confirm the 3-j↔CG definition and phase against Edmonds Eq. (3.7.3) and Sakurai Ch. 3; (2) recompute the zero-entry and $\{1/2,1/2,1;1/2,1/2,1\}=1/6$ 6-j values from standard tables (Varshalovich §9, Edmonds Ch. 6); (3) check the spin-orbit eigenvalues $j(j+1)-l(l+1)-s(s+1)$ giving $+1/2,-1$ ( $\hbar^2$ ) for $j=3/2,1/2$ ; (4) verify the orthogonality and Racah- $W$ relations against Cohen-Tannoudji Vol. 2. No nonstandard numbers are introduced. — All four verifier items are executed in this article's verify script (the 3-j dictionary swept over 63 entries; the 6-j anchors computed with signs; the recoupling matrix cross-checked against explicit eigenbases; the $W$ dictionary verified on valid triads). Per the chapter map this article also carries the PDG-phase audit item banked in Clebsch-Gordan Coefficients' provenance (its item 4): the $(-1)^{j_1-j_2+M}\sqrt{2J+1}$ relation is equation (1) here with $M=-m_3$ , verified in the sweep.

When should I use 3-j symbols instead of CG coefficients?

Derivations and single couplings: CG — the directed form matches the physics question "what does this state contain." Computation at scale, symmetry manipulations, anything involving sums over magnetic quantum numbers: 3-j — the phases are manageable and the orthogonality relations are clean. The conversion is one line (equation 1); professionals typically think in CG and compute in 3-j.

Is there a 9-j symbol, and when would I meet it?

Yes — it bridges coupling schemes for four momenta, most famously the LS↔jj transformation for two electrons (couple $(\ell_1\ell_2)(s_1s_2)$ versus $(\ell_1s_1)(\ell_2s_2)$ ). It is expressible as a single sum over products of three 6-j symbols. This article stops at three momenta by design; the 9-j machinery belongs to the many-electron chapters where LS-versus-jj becomes physics rather than bookkeeping.

Do the two coupling schemes describe different physical states?

No — same Hilbert space, same total- $J$ sectors, two complete bases. "Which scheme" becomes physical only when dynamics prefers one: a Hamiltonian with a strong $\hat{\mathbf{J}}_1\!\cdot\!\hat{\mathbf{J}}_2$ term is nearly diagonal in scheme A, one with strong $\hat{\mathbf{J}}_2\!\cdot\!\hat{\mathbf{J}}_3$ in scheme B. That is the two-momentum basis-choice doctrine of Coupled and Uncoupled Bases, one storey higher.

Why is the phase in the 3-j definition (−1)^(j1−j2−m3) and not something simpler?

It is the unique choice that makes the column-permutation symmetries come out as the single factor $(-1)^{j_1+j_2+j_3}$ — simplicity was spent where it pays. Any other phase leaves the symbol well-defined but re-clutters the symmetry pack. The verifier's 63-entry sweep is the practical answer: whatever the aesthetics, this is the phase every standard implementation uses, and interoperability is the game.

How do I know which triads of a 6-j must satisfy triangles?

Draw the tetrahedron: vertices are the four "kets" (three particles and the total), edges the six momenta, faces the four triads $(j_1j_2j_{12})$ , $(j_{12}j_3J)$ , $(j_2j_3j_{23})$ , $(j_1j_{23}J)$ . Every face must close. In the curly-bracket notation the four triads are: top row; first column with bottom-right pair... — honestly, use the tetrahedron. It is also the object whose geometry returns in the large- $j$ Ponzano-Regge limit.

Are 6-j symbols ever negative, and does the sign matter?

Frequently — $\{\tfrac12\tfrac12 0;\tfrac12\tfrac12 1\}=+\tfrac12$ but many neighbors are negative, and the one-zero form (5) alternates sign with $(-1)^{j_1+j_2+j_{12}}$ . The sign matters wherever recoupling amplitudes interfere: transition amplitudes summed over intermediate states, hyperfine-structure diagonalizations, spin-foam amplitudes. Hence the certification advice: anchor value and sign.

Is there a closed formula, or are the symbols defined by recursion like CG tables?

Both exist. Racah's 1942 formula evaluates any 6-j as a single finite sum of factorial ratios (his "formula" in the glossary); 3-j symbols have the analogous Racah/Van der Waerden sum. Recursions (in one momentum at fixed others) are numerically stabler for very large $j$ and are what production libraries actually iterate. For anything in this chapter's range, the closed sums are instant and exact.