V2.287: Spectral Measure Decomposition — What Determines α_s?

Motivation

V2.236 showed the continuum Bessel/WKB approximation for α_s fails by 20×, establishing that α_s is a “lattice quantity.” But this negative result left open the question: what about the lattice produces 1/(24√π)?

This experiment performs the first mode-by-mode decomposition of the boundary correlators X_nn and P_nn that determine α_s, to understand:

1. Is α_s IR-dominated or UV-dominated?
2. How does the centrifugal barrier l(l+1)/r² affect the spectral measure?
3. Can a matched asymptotic (exact boundary + uniform bulk) converge?

Method

For each angular momentum channel l, the Srednicki coupling matrix K’_l is a tridiagonal N×N matrix with:

• Diagonal: a_j = (2j² + 0.5 + l(l+1))/j²
• Off-diagonal: b_j = -(j+0.5)²/(j(j+1))

The boundary site (j = n_sub) correlators are:

• X_nn = (1/2) Σ_m |V[n,m]|²/ω_m (position)
• P_nn = (1/2) Σ_m |V[n,m]|²·ω_m (momentum)
• ν_site = √(X_nn · P_nn) (site-level symplectic eigenvalue)

We decompose these sums into:

1. Mode-resolved contributions: which modes m dominate?
2. IR/UV split: cut at the classical turning point ω = x = l/n
3. Barrier vs uniform: compare Srednicki chain (with l(l+1)/r²) to flat chain (no barrier)
4. Progressive barrier removal: keep barrier at j ≤ K, remove for j > K
5. Continued fraction / matched asymptotic: replace interior with uniform Green’s function

Key Results

1. X_nn is 96% UV-Dominated

For each channel l, splitting the mode sum at the centrifugal turning point ω = x:

• IR fraction of X_nn: 4.1% (weighted sum over all l)
• UV fraction: 95.9%

The position correlator X_nn is overwhelmingly determined by high-frequency modes. This explains why the WKB continuum approximation fails: it captures the correct IR behavior (below the barrier) but misses the UV spectral weight at the boundary.

For l = 30 (x = 1.0):

• 50% of X_nn comes from modes with ω < 1.49 (49% of modes by count)
• 90% from ω < 2.22 (95% of modes)
• 99% from ω < 2.26 (95% of modes)

The top contributor is a single UV mode near ω ≈ 2.22 (the spectral edge), providing 8.3% of X_nn.

2. The Centrifugal Barrier Reduces α by Factor ~37

Comparing Srednicki chain (with barrier) to a uniform chain (no barrier):

l	x = l/n	ν_Srednicki	ν_uniform	ratio	h_sred/h_unif
0	0.00	0.782	0.783	1.000	0.999
10	0.33	0.577	0.783	0.737	0.410
30	1.00	0.519	0.783	0.662	0.137
60	2.00	0.504	0.783	0.643	0.036
120	4.00	0.500	0.783	0.639	0.005

• l = 0 (no barrier): Srednicki matches uniform exactly — confirming the s-wave is barrier-free.
• l > 0: The barrier progressively suppresses ν_site and hence h(l).
• Uniform chain α: 0.876 (diverges with angular cutoff C)
• Srednicki α: 0.0235 (converges — barrier regulates the angular sum)

The centrifugal barrier is what makes α_s FINITE. Without it, the angular sum Σ(2l+1)h(l) grows as C² (proportional to the angular cutoff squared), and there is no area law coefficient. With the barrier, h(l) decays exponentially at high l, regularizing the sum.

3. Barrier Effect is Non-Local (Global Eigenvector Modification)

Progressive barrier removal shows the effect is NOT localized at the barrier:

For l = 30, keeping barrier at j ≤ K, uniform for j > K:

K	ν_site	Fraction of full effect captured
0 (uniform)	0.783	0%
5	0.771	4.3%
10	0.757	9.7%
20	0.718	24.4%
30 = l	0.532	94.7%
60 = 2l	0.519	100%
full	0.519	100%

Key finding: ~95% of the barrier effect is captured by including the barrier up to j = l. Beyond j = 2l, adding more barrier makes no difference. This is physically sensible: the centrifugal term l(l+1)/j² is negligible for j ≫ l.

But the effect is NOT local to the barrier region. The barrier at j ≤ l affects the eigenvectors EVERYWHERE, including at j = n_sub ≫ l. This is because the eigenvalue equation is a global BVP — modifying the potential near the origin changes the eigenvector normalization at the boundary.

4. The Matched Asymptotic Converges Trivially Above the Spectrum

The Green’s function G_nn(z) for z above the spectrum converges to the uniform result at K = 0 (relative error < 10⁻⁷). This means the Green’s function is insensitive to the barrier for z > λ_max.

The barrier effect is encoded entirely in the spectral weight distribution WITHIN the spectrum, not in the Green’s function above it. The matched asymptotic must be done at the level of the spectral measure Im G(ω + iε), not at a single z-value.

5. Alpha Decomposition by Angular Momentum

50% of α comes from l < 1.24n (channels with moderate angular momentum). 90% from l < 3.32n. 99% from l < 4.76n.

The area law coefficient is dominated by the low-to-moderate l channels where the barrier effect is partial (ν_sred/ν_unif = 0.4–0.7). Very high l channels (x > 3) contribute negligibly because h(ν) ~ exp(-const/x) decays rapidly.

6. n_sub Scaling

The site-level α is n_sub-independent to within 1% across n_sub = 10–30, confirming that α_s is a universal lattice constant. The uniform chain ν_site grows slowly with n_sub (0.709 → 0.782), but the Srednicki result, after the barrier kills the high-l channels, converges to a fixed value.

Physical Interpretation

Why α_s = 1/(24√π) — A Structural Decomposition

The area law coefficient α_s arises from a precise balance:

1. Angular degeneracy: The factor (2l+1) weights high-l channels heavily, pushing toward large α.

2. Centrifugal suppression: The barrier l(l+1)/r² creates a spectral gap proportional to (l/n)², which exponentially suppresses the per-channel entropy h(l) at high l.

3. UV spectral weight: 96% of the boundary correlator X_nn comes from UV modes (ω > x), confirming that α_s is a UV quantity determined by short-distance physics at the entangling surface.

4. Global eigenvector modification: The barrier at j ≤ l modifies eigenvectors globally. Even at the distant boundary site, |V[n_sub, m]|² is reduced compared to the barrier-free case. The reduction factor at x = 1 is 0.66 for ν and 0.14 for h.

The number 1/(24√π) = 1/(4! × Γ(3/2)) must encode:

• 4!: Related to the 3+1D structure (D=4 gives the area law in the radial decomposition)
• Γ(3/2) = √π/2: Related to the Gaussian ground state and the spherical harmonic measure

Why the Continuum Fails

V2.236’s Bessel/WKB approximation replaces the discrete eigenvectors V[j,m] with continuum Bessel functions J_{l+1/2}(kr). This fails because:

1. UV modes dominate X_nn: The continuum approximation is worst at high frequencies, where lattice discreteness matters most.

2. Spectral weight redistribution: The barrier REDISTRIBUTES spectral weight within the spectrum (moving weight from IR to UV or vice versa). The Bessel approximation correctly captures the spectral DENSITY (eigenvalue distribution) but incorrectly predicts the spectral WEIGHT (|V[n,m]|² distribution).

3. Boundary normalization: The lattice eigenvector at the boundary site has amplitude 3-8× smaller than the Bessel prediction (V2.236), but this mismatch varies non-uniformly across modes, with some UV modes enhanced and others suppressed.

Implications for Proving α_s

The decomposition reveals that an analytic proof of α_s = 1/(24√π) must:

1. Handle the UV spectrum: The proof cannot use IR/continuum methods alone (they give 20× too small).

2. Account for the centrifugal barrier globally: The barrier’s effect propagates from j ~ 0 to j = n_sub through the eigenvector structure. A local analysis at the boundary is insufficient.

3. Sum over angular momenta: The l-sum involves a precise cancellation between the (2l+1) growth and the centrifugal suppression. The proof must evaluate this sum exactly.

Most promising route: Study the spectral measure μ_n(dλ) at the boundary site as a function of l, in the scaling limit l = xn, n → ∞. The measure determines X_nn and P_nn as moments, and α_s as an integral over x. If μ_n has a known asymptotic form (e.g., from the theory of Jacobi matrices with polynomial potentials), the moments might be computable.

Honest Assessment

What this experiment achieves:

• First quantitative decomposition of α_s into mode contributions (IR vs UV)
• Quantifies the centrifugal barrier’s role: reduces α by factor ~37, makes it finite
• Confirms UV dominance: 96% of the boundary correlator from high-frequency modes
• Maps the barrier’s spatial range: 95% of effect from j ≤ l, fully converged at j ≤ 2l
• Shows matched asymptotic fails above spectrum: barrier effect encoded in spectral weight within the spectrum

What it does NOT achieve:

• No analytic proof of α_s = 1/(24√π)
• No closed-form expression for the spectral weight at the boundary
• Site-level α overestimates by ~45% (known limitation — nu_site > nu_max)
• The “factor of 37” between uniform and Srednicki α depends on C and is not universal

Connection to the overall science:

This experiment clarifies WHY α_s = 1/(24√π) is hard to prove: it depends on the UV spectral weight of a tridiagonal matrix with a polynomial potential, modified globally by a centrifugal barrier. The proof requires either:

1. Exact asymptotics of the spectral measure for Jacobi matrices with l(l+1)/j² potential
2. A combinatorial identity for the angular momentum sum
3. A heat kernel or zeta function technique that directly gives the moments of the spectral measure

The result α_s = 1/(24√π) = 1/(4! × Γ(3/2)) remains the single unproven link in the derivation chain. Once proven, the cosmological constant prediction Ω_Λ = |δ|/(6α) = 149√π/384 becomes a theorem of quantum field theory on a lattice.