Restricted SU(2) Yang–Mills via spectral whitening

This page records, in the form of a referee report, what the spectral‑whitening construction of Issue #964 actually establishes when transferred from the abelian U(1) setting of Sevostyanov (arXiv:2102.03224) to a restricted, toy version of SU(2) Yang–Mills on ℝ³. The full SU(2) Yang–Mills mass gap with a constructively defined measure on Map(ℝ³, SU(2))‑equivalence classes of connections is a Clay Millennium problem; nothing on this page asserts a proof of that problem. What is recorded is exactly: the U(1) input that is rigorously proved, the structural transfer that is well defined, and the propositions that the construction needs but that are not established by the documents as written.

1. Definitions used on this page

Hardy Z‑function. Z(t) := e^{iϑ(t)} ζ(½ + it), with ϑ(t) = arg Γ(¼ + it/2) − (t/2) log π the Riemann–Siegel theta function. Z is real on ℝ.

Time‑changed process. For T₀ = 200 and u = ϑ(t) on [T₀, ∞),

H(u) := Z(ϑ⁻¹(u)) ⁄ √(ϑ′(ϑ⁻¹(u))).

Cramér spectral measure. ν is the spectral measure of H, of the form ν = S(λ) dλ + w·δ_{−1} on [−1, 1], with S ≥ 0 continuous on [−1, 1] and w ≥ 0; total mass ν([−1, 1]) = σ_H² = 2.

Stone generator. A is the unique self‑adjoint operator on the closed L²(Ω, ℙ) span of {H(u) : u ∈ ℝ} for which the shift group U_τ H(u) := H(u + τ) satisfies U_τ = e^{iτA}.

Gauge data. G = SU(2); 𝒜 the affine space of smooth 𝔰𝔲(2)‑valued connections on ℝ³; 𝒢 = Map(ℝ³, SU(2)) the gauge group. T ⊂ G the diagonal U(1) maximal torus; W = ℤ/2 the Weyl group; the adjoint orbit of g = exp(iφ·σ₃/2) ∈ T in G is the conjugacy class Tr g/2 = cos(φ/2). Class functions on SU(2) integrate by Weyl:

∫_{SU(2)} f(g) dg = (2/π) ∫₀^π f(diag(e^{iφ}, e^{−iφ})) sin² φ dφ.

Two‑point function on ℝ³. W(x − y) := ⟨A(x), A(y)⟩_{𝒜/𝒢} the gauge‑invariant two‑point function of the would‑be SU(2) vacuum, assumed SO(3)‑radial.

Radial Fourier kernel on ℝ³. j₀(x) := sin(x)/x, the spherical Bessel function of order zero, the reproducing kernel of the Paley–Wiener space PW₁ of entire functions of exponential type ≤ 1 that are L² on ℝ.

Akhiezer non‑negativity criterion. An entire function H of exponential type ≤ 1, real on ℝ, with non‑negative compactly supported power spectrum S = |Ĥ|², has all its zeros real, and they form the spectrum of multiplication by ξ on L²([−1, 1], S(ξ) dξ).

2. The U(1) input that is established

The Hardy‑Z analysis of Issue #964 (Comment 1) establishes the following, taken here as input:

(i) H is weakly stationary on its domain. (ii) The shift group {U_τ : τ ∈ ℝ} on the L² closure of {H(u)} is strongly continuous unitary; ‖U_τ H − H‖² = D_H(τ) → 0. (iii) Stone's theorem produces A self‑adjoint with σ(A) ⊆ [−1, 1], ‖A‖ ≤ 1; under the Cramér isomorphism ℋ ≅ L²([−1, 1], dν), A is multiplication by λ. (iv) C_H(τ) = ⟨H, U_τ H⟩ = ∫{−1}^{1} e^{iτλ} dν(λ); D_H(τ) = 4 ∫{−1}^{1} sin²(λτ/2) dν(λ). (v) The second‑quantized Hamiltonian dΓ(A) on Fock(ℋ) is bounded self‑adjoint per chaos sector, with σ(dΓ(A)) concentrated on n‑fold sums of points in [−1, 1].

These facts are operator‑theoretic consequences of stationarity and Stone's theorem and do not depend on any Yang–Mills input. They are the U(1) side of the construction, in the sense that A is a bounded self‑adjoint operator on a single‑torus L² space.

3. Sevostyanov's U(1) construction

Sevostyanov, arXiv:2102.03224, constructs a Gaussian probability measure on the space of gauge equivalence classes of U(1) connections on ℝ³, depending on parameters m > 0, c ≠ 0, and proves that the resulting quantized Hamiltonian is a self‑adjoint operator on a Fock space with spectrum {0} ∪ [m/2, ∞). For non‑abelian G the abstract gives a formally self‑adjoint expression for the Hamiltonian on the corresponding L²; the measure on 𝒜/𝒢 and the self‑adjoint realization are not constructed. This is the gap that the spectral‑whitening proposal of Issue #964 is meant to close.

4. The radial‑Fourier reduction on ℝ³

For W ∈ L¹(ℝ³) ∩ L²(ℝ³) radial, the standard SO(3) Fourier identity

Ŵ(|k|) = 4π ∫₀^∞ W(r) j₀(|k| r) r² dr

holds, with j₀(x) = sin(x)/x. This identity is not in question; it is the standard reduction of the 3D Fourier transform on radial functions to a one‑dimensional Hankel transform of order ½. The reproducing‑kernel property j₀ ∈ PW₁ is also standard.

5. The whitening map on the U(1) side

Define, for S(ξ) > 0 on [−1, 1],

dW(ξ) = dĤ(ξ) ⁄ √(S(ξ)) = e^{iΦ(ξ)} dξ,

where Φ is the argument of the saddle‑sum amplitude Ĥ(ξ) on the Hardy‑Z side. As an identity on the open set {ξ ∈ [−1, 1] : S(ξ) > 0} this is well defined. The map sends the L²([−1, 1], S dξ) inner product to the flat Lebesgue inner product on the same set, and the multiplication operator M_ξ to itself.

Finding 5.1 — fixable, not established as written. The map dW = dĤ/√S is undefined at points of the zero set Z := {ξ ∈ [−1, 1] : S(ξ) = 0}, which Issue #964 does not specify. What would establish a complete definition: an explicit statement that Z is a Lebesgue null set, together with a fixed extension of Φ to Z (for example, by symmetric continuation, or by declaring the image of Z to carry zero measure on the whitened side). Either choice suffices; the issue selects neither.

6. The proposed transfer to SU(2)

The Issue #964 transfer is structured as five steps:

(T1) Identify the gauge‑invariant two‑point function W(x − y) of an SU(2) connection on ℝ³ as SO(3)‑radial; pass to its radial Fourier transform Ŵ(|k|) via the j₀ kernel.

(T2) Posit a spectral density S of an entire H of exponential type ≤ 1, real on ℝ, with S ≥ 0 and compactly supported on [−1, 1], obtained from the Hardy‑Z stationary‑phase locus.

(T3) Identify Ŵ(|k|) with S via the rescaling ξ = |k|/m, so that the support condition supp S ⊆ [−1, 1] corresponds to supp Ŵ ⊆ [−m, m] in ξ‑variable, and to a spectral gap |k| ≥ m on the YM side after the unit change.

(T4) Define the YM measure on 𝒜/𝒢 by pulling back Sevostyanov's U(1) Gaussian measure along the adjoint‑orbit reduction SU(2) → T = U(1), with whitened phase e^{iΦ(ξ)} living in the maximal torus T.

(T5) Read off the SU(2) Hamiltonian as M_ξ on L²([−1, 1], S dξ) and apply the Akhiezer closure of §1 to conclude reality of spectrum and the gap.

Step (T1) is standard. Steps (T2) and (T5) are the U(1)‑side input of §2. Step (T3) is a unit choice. The substantive non‑abelian content is concentrated in (T4) and in the precise location of the gap interval after rescaling.

7. Findings

Each finding is stated together with what would establish the proposition. None of the findings is fatal; all are fixable.

Finding 7.1 — fixable, not established as written. (Adjoint‑orbit reduction of the gauge group.) The transfer step (T4) asserts that the SU(2) gauge‑invariant two‑point function on ℝ³ coincides with the U(1)‑torus two‑point function obtained by adjoint‑orbit reduction. The Weyl integration formula of §1 reduces class functions on the constant group SU(2) to U(1) integrals against sin²φ dφ. The gauge group in Sevostyanov's setup is 𝒢 = Map(ℝ³, SU(2)), not the constant group; reducing 𝒢‑equivalence classes of connections to a single‑copy U(1) torus problem is not an instance of the Weyl integration formula. The off‑diagonal components of an 𝔰𝔲(2)‑valued connection (the W‑bosons) are charged under the residual U(1) and contribute to gauge‑invariant correlators. Issue #964 asserts this reduction; it does not derive it. What would establish it: a derivation, explicit on ℝ³, of the equality

⟨Tr A(x) A(y)⟩{𝒜{SU(2)}/𝒢_{SU(2)}} = push‑forward under T ↪ SU(2) of ⟨a(x) a(y)⟩{𝒜{T}/𝒢_{T}}

including a treatment of the off‑diagonal sector (e.g. an argument that the off‑diagonal modes decouple from the gauge‑invariant two‑point function, or are integrated out exactly).

Finding 7.2 — fixable, not established as written. (Location of the gap interval after rescaling.) Step (T3) sets ξ = |k|/m. Under this map the YM gap interval [m, ∞) corresponds to ξ ∈ [1, ∞), not to a subset of [−1, 1]; conversely the Hardy‑Z spectral interval [−1, 1] is mapped to |k| ∈ [−m, m], which contains 0. The U(1)‑side spectrum σ(M_ξ) on L²([−1, 1], S dξ) equals supp S, which contains the atom at ξ = −1. The body of the issue states "spectrum bounded below by inf supp(S) = m > 0," which is consistent only when 0 ∉ supp S. Issue #964 does not derive 0 ∉ supp S from the Hardy‑Z stationary‑phase data. What would establish it: a proof that the saddle‑sum amplitudes 𝒜_{n,σ}(ξ) of the Hardy‑Z stationary‑phase locus vanish on a neighbourhood of ξ = 0, together with a precise rescaling that maps the resulting positive‑separation interval to [m, ∞) on the YM side.

Finding 7.3 — fixable, not established as written. (Whitening at zeros of S.) The map dW = dĤ/√S is recorded in §5 above; the proposition is well posed only after a convention for the zero set Z = {S = 0} is fixed.

Finding 7.4 — fixable, not established as written. (Akhiezer closure on the YM side.) Akhiezer's theorem applies to a single entire H of exponential type ≤ 1 real on ℝ with S = |Ĥ|² ≥ 0. In the SU(2) transfer, the would‑be H is the radial profile of the gauge‑invariant correlator Ŵ(|k|), and its membership in PW₁ is asserted via the j₀ kernel. The j₀ kernel is the reproducing kernel of PW₁, and a function whose 3D Fourier transform is supported in {|k| ≤ 1} has its 1D radial profile in PW₁ — but here the support condition is supp Ŵ ⊆ [m, ∞), which is not a band‑limit at exponential type ≤ 1. What would establish the closure: an explicit identification of an entire function H ∈ PW₁, real on ℝ, whose squared modulus on ℝ equals the YM radial spectral density after rescaling, and a proof that the rescaling preserves PW₁ membership.

Finding 7.5 — fixable, not established as written. (Saddle‑sum formula for the whitened phase.) The expression Φ(ξ) = arg ∑{n,σ} |𝒜{n,σ}(ξ)| e^{iφ_{n,σ}(ξ)} of §5 is the U(1)‑side stationary‑phase output, taken from the SpectralNonNegativity files as input. The wiki entry for this page does not re‑derive it; it inherits its truth from those files. Anyone wishing to cite this page for a non‑abelian conclusion needs first to establish 7.1 and 7.2.

8. What is proved on this page

Reading §§1–6 strictly and applying only the Akhiezer criterion of §1 to the U(1) side:

(P‑U1) On the U(1) side, the map M_ξ : L²([−1, 1], S dξ) → L²([−1, 1], S dξ), (M_ξ f)(ξ) = ξ f(ξ), is bounded self‑adjoint with ‖M_ξ‖ ≤ 1 and σ(M_ξ) = supp S ⊆ [−1, 1].

(P‑U2) On the U(1) side, when S is the power spectrum |Ĥ|² of an entire H of exponential type ≤ 1 real on ℝ, with S ≥ 0 supported on [−1, 1], the zeros of H are real and form the spectrum of M_ξ on the closed span of {e^{iξu} : u ∈ ℝ} ⊂ L²([−1, 1], S dξ). This is Akhiezer.

(P‑Rad) On ℝ³, for W ∈ L¹ ∩ L² radial, Ŵ(|k|) = 4π ∫₀^∞ W(r) j₀(|k| r) r² dr, and j₀ is the reproducing kernel of PW₁.

(P‑Sev) Sevostyanov's U(1) Yang–Mills construction on ℝ³ produces a self‑adjoint Hamiltonian on Fock space with spectrum {0} ∪ [m/2, ∞). This is taken from arXiv:2102.03224.

These are the propositions for which the construction provides full chains of derivation. Their conjunction is a structural transfer template, not a proof of the SU(2) gap.

9. What is not proved on this page

The propositions of §7 — the SU(2) torus reduction (7.1), the location of the gap interval after rescaling (7.2), the convention at zeros of S (7.3), Akhiezer transfer to the YM radial profile (7.4) — are the steps that would have to be filled to convert §6 from a transfer template into a proof of a restricted SU(2) mass gap. As of this page they are not filled.

10. References

• Alexey Sevostyanov, Towards non‑perturbative quantization and the mass gap problem for the Yang–Mills field, arXiv:2102.03224, 2021.
• Issue #964, body and Comment 1.
• Files: SpectralTilingCorrected.tex, StationaryPhaseLocusAndRemainderAtom.tex, SpectralNonNegativity.md (input on U(1) side).
• N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner, 1965, on entire functions of exponential type and reality of zeros.
• L. de Branges, Hilbert spaces of entire functions, Prentice‑Hall, 1968, on PW₁ and reproducing kernels.

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