T22 — Spin Foam Amplitudes

SU(2) four-leg group-average toy calculation

Last updated: 2026-07-14 13:48 IST

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Objective

Compute and validate a normalized SU(2) four-leg group average as an analytic exercise and code check. This is explicitly a toy calculation, not a physical spin-foam vertex amplitude.

Status

✅ T22a Complete — Reclassified as SU(2) Group-Average Toy Calculation

2026-07-08: The T22a implementation has been reclassified. It computes a normalized SU(2) four-leg group average, NOT a complete FK/EPRL spin-foam vertex amplitude. The analytic result is exact; Monte Carlo is used only for validation.

⚠️ Important Correction

The original T22a code mislabeled the result as an “FK vertex amplitude” and incorrectly applied an extra squaring step, converting the ratio ~0.45 into an unsupported ~0.20 “suppression factor.” This extra squaring was an error. The correct ratio is R = 4/9 ≈ 0.444. See T32 for the full correction record.

Results (T22a)

Normalized SU(2) four-leg group average:

Quantity	Exact Value	MC Validation (2M samples)
G(j=1/2)	1/4 = 0.250000	0.250303 ± 1.5×10⁻⁴
G(j=1)	1/9 ≈ 0.111111	0.111243 ± 8.9×10⁻⁵
G(j=1) / G(j=1/2)	4/9 ≈ 0.444444	0.444474 ± 6.3×10⁻⁴
Power law G(j) ~ j^(-α)	α = 2 (asymptotic)	α ≈ 1.54 (fit j≥1)

Analytic derivation: Using the Clebsch–Gordan decomposition χ^j(h)² = Σ_{J=0}^{2j} χ^J(h) and orthogonality of characters, the exact result is G(j) = 1/(2j+1)².

What this is NOT: - A complete FK or EPRL spin-foam vertex amplitude - Evidence for j=1/2 dominance in any physical model - A substitute for the full intertwiner, Immirzi, and face-amplitude structure

Conclusion: This is a useful analytic exercise and code check, but it does not by itself justify j=1/2 truncation. The ~0.45 ratio is exact for this toy integral; there is no additional ~0.20 suppression factor.

Theory

The normalized SU(2) four-leg group average for uniform spin \(j\):

\[ G(j) = \frac{1}{(2j+1)^2} \]

This follows from the Clebsch–Gordan decomposition of the squared character, \(\chi^j(h)^2 = \sum_{J=0}^{2j} \chi^J(h)\), and orthogonality of SU(2) characters under the Haar measure. The result is exact; Monte Carlo integration serves only as validation.

This is not a complete FK or EPRL spin-foam vertex amplitude, which would require intertwiners, Immirzi parameter dependence, face amplitudes, and a full boundary state. The \(G(j)\) ratio does not by itself establish \(j=1/2\) dominance in any physical model.

Tasks

• Compute SU(2) four-leg group average \(G(j)\) analytically via character orthogonality
• Validate with Monte Carlo integration (2M samples) for \(j \in \{1/2, 1, 3/2, 2, 5/2, 3\}\)
• Extract power-law decay exponent \(\alpha\) from fit to \(j \geq 1\)
• Generate amplitude plot and corrected task documentation
• Future: implement full EPRL/FK vertex with intertwiners, Immirzi parameter, and face amplitudes (requires ts-quantum 6j symbols and boundary state construction)
• Future: verify \(j=1/2\) dominance claim with complete vertex, not toy group average

References

• Paper: Section 3.3
• Donà et al. (2020/2022) reference implementations