V2.232 - Dimensional Dependence of alpha_s — Does 1/(24*sqrt(pi)) Generalize?
V2.232: Dimensional Dependence of alpha_s — Does 1/(24*sqrt(pi)) Generalize?
Executive Summary
The area-law coefficient alpha_4 = 1/(24*sqrt(pi)) = 0.02351 is specific to D=4 spacetime dimensions. If alpha_D follows a recognizable pattern across dimensions, this would reveal the formula and potentially prove the D=4 result.
This experiment generalizes the Srednicki/Lohmayer radial chain to arbitrary spacetime dimension D and computes alpha_D for D=3,4,5,6.
Key Results:
- D=4: Confirmed. Richardson extrapolation gives k_4 = 1/(alpha_4 * sqrt(pi)) = 24.0 to 0.024%. The generalized chain correctly reproduces the known result.
- D=3: New result. k_3 = 7.62 +/- 0.04, well-converged but NOT a simple integer or factorial.
- D=5,6: Not converged. The C→∞ convergence is too slow (degeneracies grow as l^{D-3}). Richardson extrapolation is unreliable.
- No simple formula found that works across all D. The dimensional pattern is obscured by convergence issues in D >= 5.
Method
Generalized radial chain
For D spacetime dimensions (D_s = D-1 spatial), the free scalar decomposes into angular momentum channels on S^{D_s-1}. Each channel l reduces to a 1D radial chain with:
- Centrifugal potential: l*(l + D_s - 2) / j^2 (reduces to l(l+1)/j^2 for D=4)
- Jacobian factors: j^{D_s-1} from the radial measure (reduces to j^2 for D=4)
- Degeneracy: dimension of l-th harmonic on S^{D_s-1}
| D | D_s | Surface | Degeneracy d_l | Centrifugal | Area A(n) |
|---|---|---|---|---|---|
| 3 | 2 | S^1 | 2 (l>=1), 1 (l=0) | l^2/r^2 | 2pin |
| 4 | 3 | S^2 | 2l+1 | l(l+1)/r^2 | 4pin^2 |
| 5 | 4 | S^3 | (l+1)^2 | l(l+2)/r^2 | 2pi^2n^3 |
| 6 | 5 | S^4 | (l+1)(l+2)(2l+3)/6 | l(l+3)/r^2 | 8pi^2n^4/3 |
Extraction via finite differences
alpha_D is extracted from the (D-2)-th finite difference of S(n):
- D=3: dS/dn = 2pialpha, using first central difference
- D=4: d^2S/dn^2 = 8pialpha, using second central difference
- D=5: d^3S/dn^3 = 12pi^2alpha, using third forward difference
- D=6: d^4S/dn^4 = 64pi^2alpha, using fourth central difference
Richardson extrapolation in C
Following V2.184, alpha is extrapolated to C→∞ using Neville’s polynomial interpolation in 1/C at multiple C values.
Part 1: D=4 Verification
| C | alpha_4(C) | Error from 1/(24√π) |
|---|---|---|
| 5 | 0.02123 | -9.69% |
| 10 | 0.02278 | -3.10% |
| 15 | 0.02315 | -1.54% |
| 20 | 0.02329 | -0.92% |
| 25 | 0.02336 | -0.62% |
| 30 | 0.02340 | -0.44% |
| Richardson | 0.02351 | +0.024% |
The generalized chain reproduces V2.184’s result. The Richardson extrapolation at n=12 gives 0.024% accuracy, validating the D-dimensional generalization.
Part 2: D=3
The 2+1D case has the simplest degeneracies (d_l = 2 for l >= 1) and the fastest C convergence.
n-convergence at C=15
| n | alpha_3 |
|---|---|
| 8 | 0.07419 |
| 12 | 0.07403 |
| 15 | 0.07397 |
| 20 | 0.07391 |
The n-dependence is mild (~0.4% from n=8 to n=20).
C-convergence at n=15
| C | alpha_3(C) |
|---|---|
| 5 | 0.07344 |
| 10 | 0.07391 |
| 15 | 0.07397 |
| 20 | 0.07399 |
| 25 | 0.07400 |
| Richardson | 0.07400 |
C convergence is fast — alpha barely changes from C=15 to C=25.
Result: k_3 = 1/(alpha_3 * sqrt(pi)) = 7.624 +/- 0.04
This is close to 24/pi = 7.639 (0.2% away), but the match is uncertain at our precision level. A double-limit extrapolation (n→∞ and C→∞ simultaneously) would be needed to settle this.
Part 3: D=5
The 4+1D case has quadratically growing degeneracies (d_l = (l+1)^2), causing very slow C convergence.
n-convergence at C=10
| n | alpha_5 |
|---|---|
| 6 | 0.01333330 |
| 8 | 0.01333310 |
| 10 | 0.01333305 |
| 12 | 0.01333303 |
The n-dependence is negligible (< 0.002% variation). At fixed C, the area law is perfectly established even at n=6.
C-convergence at n=8
| C | alpha_5(C) |
|---|---|
| 5 | 0.00999 |
| 10 | 0.01333 |
| 15 | 0.01475 |
| 20 | 0.01554 |
| 25 | 0.01604 |
| Richardson | 0.01847 |
The C convergence is extremely slow — alpha is still growing rapidly at C=25. The Richardson extrapolation (0.01847) is unreliable because we are far from the asymptotic regime.
D=5 is NOT converged. Would require C > 100 for meaningful results.
Part 4: D=6
Even worse convergence than D=5:
| C | alpha_6(C, n=6) |
|---|---|
| 5 | 0.00584 |
| 10 | 0.01175 |
| 15 | 0.01607 |
| 20 | 0.01949 |
| Richardson | 0.03985 |
Alpha doubles from C=5 to C=20 and is clearly nowhere near convergence. The Richardson result is meaningless.
Part 5: Pattern Analysis
Using the converged values (D=3,4 only):
| D | alpha_D | k_D = 1/(alpha*sqrt(pi)) | Converged? |
|---|---|---|---|
| 3 | 0.07400 | 7.62 | Yes (~0.5%) |
| 4 | 0.02351 | 24.0 | Yes (0.024%) |
| 5 | ? | ? | No |
| 6 | ? | ? | No |
Ratio alpha_3/alpha_4 = 3.147
This is tantalizingly close to pi = 3.1416 (0.2% away), which would mean:
alpha_3/alpha_4 = pi
If true: alpha_3 = pi * alpha_4 = pi/(24*sqrt(pi)) = sqrt(pi)/24 = 0.07385
Our measured alpha_3 = 0.0740, which is 0.2% above sqrt(pi)/24 = 0.07385. This is within the expected precision of our single-limit (C-only) Richardson extrapolation.
Conjectured formula
If alpha_3 = sqrt(pi)/24 and alpha_4 = 1/(24*sqrt(pi)), then:
alpha_D = pi^{(3-D)/2} / 24 ???
This gives:
- D=3: pi^0/24 = 1/24 = 0.04167 — NO, this doesn’t match.
Or: alpha_D = 1/(24 * pi^{(D-3)/2})
- D=3: 1/24 = 0.04167 — NO.
The simple power-of-pi scaling doesn’t work. The relationship alpha_3/alpha_4 ≈ pi might be a coincidence, or the formula might involve additional D-dependent factors.
Candidate formulas tested
| Formula | D=3 err | D=4 err |
|---|---|---|
| 1/(D! * Gamma((D-1)/2)) | -125% | -100% |
| 1/(2*(D-1)! * Gamma(D/2)) | -281% | -254% |
| Gamma((D-1)/2)/(4pi^((D-1)/2)(D-2)!) | -7.5% | +15.4% |
None of the simple formula candidates match both D=3 and D=4.
What This Means
The D=4 generalization works
The D-dimensional radial chain correctly reproduces the known D=4 result via Richardson extrapolation (0.024% accuracy). This validates the generalization.
D=3 provides a new data point
k_3 ≈ 7.62 is NOT a simple integer or factorial, unlike k_4 = 24 = 4!. This suggests the formula alpha_D = 1/(D! * sqrt(pi)) does NOT hold in general. The D=4 coincidence k_4 = 4! = 24 may be special to D=4.
Higher dimensions need much larger C
The C convergence becomes dramatically slower in higher dimensions:
- D=3: converged by C=20 (0.1% accuracy)
- D=4: converged by C=30 (0.4% accuracy), Richardson gives 0.024%
- D=5: NOT converged at C=25 (still growing rapidly)
- D=6: NOT converged at C=20
This is because the degeneracy d_l grows as l^{D-3}, so modes at very large l contribute increasingly to the area law. A dedicated high-C computation for D=5 would be valuable but computationally expensive (C > 100 needed).
The ratio alpha_3/alpha_4 ≈ pi is suggestive
If confirmed by a double-limit analysis, this would imply alpha_3 = sqrt(pi)/24 = pi * alpha_4. This pi factor could arise from the difference in entangling surface geometry (S^1 in D=3 vs S^2 in D=4), but the theoretical explanation remains unclear.
Limitations
- Only D=3 and D=4 are converged. The D=5,6 results are preliminary upper bounds at best.
- Single-limit Richardson. A double-limit (n,C)→(∞,∞) analysis (as in V2.184) would improve D=3 accuracy from ~0.5% to ~0.01%.
- No pattern identified. With only two converged data points, we cannot distinguish between competing formulas.
- D=5 would be the decisive test — but requires C > 100 at n ~ 8, meaning l_max > 800 modes, each requiring a chain eigenvalue problem. This is computationally feasible but would take ~10x longer.