| _version_ | 1866902316077023232 |
|---|---|
| author | Chiappone, Giovanni |
| author_facet | Chiappone, Giovanni |
| contents | <p>DESCRIPTION<br>This article develops a plane-aware QFT (Omniverse QFT, O-QFT) on fixed sector–planes (s,b). Only uncut internal propagators are multiplied by a positive, completely monotone (Bernstein/Laplace) damper Φ(a² D̅_sb²) with Φ(0)=1; external LSZ legs and all vertices are unchanged. A plane binder P_{sb|s′b′}=δ_{ss′}δ_{bb′} enforces plane-diagonal propagation, cuts, and discontinuities, so unitarity (optical theorem) holds separately on each plane. Minkowski boundary values are taken vectorially with a common per-plane sign σ_s∈{±1} used consistently in numerator tags and pole shifts: k² → k² + i σ_s ε̂, with ε̂ = ε·p̂₀ (ε→0⁺ and p̂₀ the unit geometric-time vector). On a cut line, perform the δ-replacement and omit Φ; discontinuities are the standard phase space times the external plane projector. Under mild conditions on the damping measure ρ, loop integrals are UV-finite to all orders, while BRST/Slavnov–Taylor identities, analyticity/crossing, and OS reflection positivity remain intact per plane.</p> <p>KEY RESULTS<br>- All-orders UV finiteness (per (s,b)):<br> (G) Gap: supp(ρ) ⊂ [τ*, ∞) ⇒ uniform finiteness at all loop orders.<br> (M) Near-zero moment: ∫₀¹ τ^(−β) ρ(dτ) < ∞ with β > 1 + D(Γ)/2, where D(Γ) is the tensor degree from numerators.<br> Canonical Φ: heat-kernel shifts (ρ = δ_{τ*}); power kernels (1 + q/Λ²)^(−ν) with ν > 1 + D(Γ)/2; finite mixtures.<br>- Symmetries exact per plane:<br> BRST-exact gauge fixing + Algebraic Renormalization ⇒ Zinn–Justin/Slavnov–Taylor functional, background Ward identities (BFM), and Nielsen identity without symmetry-restoring counterterms. Φ is a class function of D̅_sb² and is BRST-inert.<br>- Vectorial boundary values and unitarity:<br> Use i σ_s ε̂ with the same σ_s everywhere in the (s,b) block. Cut rule: 1/(k² − m² + i σ_s ε̂) → −iπ δ(k² − m²) θ(σ_s k⁰) and no Φ on the cut line. Optical theorem holds per plane.<br>- Analyticity, crossing, causality:<br> Φ is analytic on ℂ \ (−∞, 0]; no new poles/pinches or Landau singularities. The heat-kernel representation preserves analyticity/crossing. OS reflection positivity holds on the gauge-invariant (BRST-cohomology) subalgebra.</p> <p>METHOD<br>Use the positive Laplace–Stieltjes representation Φ(q) = ∫₀^∞ e^(−τ q) ρ(dτ) (ρ ≥ 0), rewrite each internal covariance as a heat-kernel convolution e^{tΔ_sb} * C_std, and apply Gram/Gaussian bounds with dominated convergence to transport analyticity/crossing and OS positivity.</p> <p>SCOPE AND ASSUMPTIONS<br>- Φ multiplies uncut internal propagators only; external LSZ legs and vertices are unchanged.<br>- Plane binder P_{sb|s′b′} eliminates cross-plane interference.<br>- Vectorial prescription uses i σ_s ε̂ with unit geometric-time axis p̂₀.<br>- Optional growth control for boundary values: existence of ε₀ > 0 with ∫ e^{ε₀ τ} ρ(dτ) < ∞.</p> <p>ILLUSTRATIVE EXAMPLE<br>- ϕ⁴ bubble (s-channel): Disc B_{sb}^Φ(s) = P^{(4)}_{sb|ext} · (i/16π) β(s) θ(σ_s P⁰) θ(s − 4m²), β(s) = √(1 − 4m²/s). Imaginary part is vectorial ((σ_s i p̂₀) tag) times a scalar magnitude; real part is purely scalar. The same mechanism extends to YM/QED.</p> <p>VERSION HIGHLIGHTS<br>- Explicit scope: no Φ on cut lines; clarified uncut vs cut at first use and in the Cutkosky section.<br>- Standardized vectorial boundary-value notation: i σ_s ε̂ with p̂₀ in both numerator tags and pole shifts.<br>- Tightened OS-positivity statement (gauge-invariant subalgebra) and UV routes (gap/moment); listed canonical kernel families.</p> <p>RELATED IDENTIFIERS<br>- Concept monograph: 10.5281/zenodo.15833365<br>- Companion notes: 10.5281/zenodo.17096024, 10.5281/zenodo.16990196</p> <p>HOW TO CITE<br>G. J. Chiappone, "Proving All-Orders UV Finiteness: Non-Abelian BRST Closure, Analyticity, and Crossing in Omniverse Quantum Field Theory," 34-page article, Zenodo (2025). DOI assigned on release.</p> <p>KEYWORDS<br>BRST; Slavnov–Taylor; LSZ; Cutkosky; optical theorem; plane-aware QFT; vectorial i ε̂; unit geometric-time axis p̂₀; sector–plane (s,b); OS reflection positivity; Bernstein/Laplace damper; background-field method; analyticity; crossing; all-orders UV finiteness.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17308907 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Proving All-Orders UV Finiteness: Non-Abelian BRST Closure, Analyticity, and Crossing in Omniverse Quantum Field Theory Chiappone, Giovanni <p>DESCRIPTION<br>This article develops a plane-aware QFT (Omniverse QFT, O-QFT) on fixed sector–planes (s,b). Only uncut internal propagators are multiplied by a positive, completely monotone (Bernstein/Laplace) damper Φ(a² D̅_sb²) with Φ(0)=1; external LSZ legs and all vertices are unchanged. A plane binder P_{sb|s′b′}=δ_{ss′}δ_{bb′} enforces plane-diagonal propagation, cuts, and discontinuities, so unitarity (optical theorem) holds separately on each plane. Minkowski boundary values are taken vectorially with a common per-plane sign σ_s∈{±1} used consistently in numerator tags and pole shifts: k² → k² + i σ_s ε̂, with ε̂ = ε·p̂₀ (ε→0⁺ and p̂₀ the unit geometric-time vector). On a cut line, perform the δ-replacement and omit Φ; discontinuities are the standard phase space times the external plane projector. Under mild conditions on the damping measure ρ, loop integrals are UV-finite to all orders, while BRST/Slavnov–Taylor identities, analyticity/crossing, and OS reflection positivity remain intact per plane.</p> <p>KEY RESULTS<br>- All-orders UV finiteness (per (s,b)):<br> (G) Gap: supp(ρ) ⊂ [τ*, ∞) ⇒ uniform finiteness at all loop orders.<br> (M) Near-zero moment: ∫₀¹ τ^(−β) ρ(dτ) < ∞ with β > 1 + D(Γ)/2, where D(Γ) is the tensor degree from numerators.<br> Canonical Φ: heat-kernel shifts (ρ = δ_{τ*}); power kernels (1 + q/Λ²)^(−ν) with ν > 1 + D(Γ)/2; finite mixtures.<br>- Symmetries exact per plane:<br> BRST-exact gauge fixing + Algebraic Renormalization ⇒ Zinn–Justin/Slavnov–Taylor functional, background Ward identities (BFM), and Nielsen identity without symmetry-restoring counterterms. Φ is a class function of D̅_sb² and is BRST-inert.<br>- Vectorial boundary values and unitarity:<br> Use i σ_s ε̂ with the same σ_s everywhere in the (s,b) block. Cut rule: 1/(k² − m² + i σ_s ε̂) → −iπ δ(k² − m²) θ(σ_s k⁰) and no Φ on the cut line. Optical theorem holds per plane.<br>- Analyticity, crossing, causality:<br> Φ is analytic on ℂ \ (−∞, 0]; no new poles/pinches or Landau singularities. The heat-kernel representation preserves analyticity/crossing. OS reflection positivity holds on the gauge-invariant (BRST-cohomology) subalgebra.</p> <p>METHOD<br>Use the positive Laplace–Stieltjes representation Φ(q) = ∫₀^∞ e^(−τ q) ρ(dτ) (ρ ≥ 0), rewrite each internal covariance as a heat-kernel convolution e^{tΔ_sb} * C_std, and apply Gram/Gaussian bounds with dominated convergence to transport analyticity/crossing and OS positivity.</p> <p>SCOPE AND ASSUMPTIONS<br>- Φ multiplies uncut internal propagators only; external LSZ legs and vertices are unchanged.<br>- Plane binder P_{sb|s′b′} eliminates cross-plane interference.<br>- Vectorial prescription uses i σ_s ε̂ with unit geometric-time axis p̂₀.<br>- Optional growth control for boundary values: existence of ε₀ > 0 with ∫ e^{ε₀ τ} ρ(dτ) < ∞.</p> <p>ILLUSTRATIVE EXAMPLE<br>- ϕ⁴ bubble (s-channel): Disc B_{sb}^Φ(s) = P^{(4)}_{sb|ext} · (i/16π) β(s) θ(σ_s P⁰) θ(s − 4m²), β(s) = √(1 − 4m²/s). Imaginary part is vectorial ((σ_s i p̂₀) tag) times a scalar magnitude; real part is purely scalar. The same mechanism extends to YM/QED.</p> <p>VERSION HIGHLIGHTS<br>- Explicit scope: no Φ on cut lines; clarified uncut vs cut at first use and in the Cutkosky section.<br>- Standardized vectorial boundary-value notation: i σ_s ε̂ with p̂₀ in both numerator tags and pole shifts.<br>- Tightened OS-positivity statement (gauge-invariant subalgebra) and UV routes (gap/moment); listed canonical kernel families.</p> <p>RELATED IDENTIFIERS<br>- Concept monograph: 10.5281/zenodo.15833365<br>- Companion notes: 10.5281/zenodo.17096024, 10.5281/zenodo.16990196</p> <p>HOW TO CITE<br>G. J. Chiappone, "Proving All-Orders UV Finiteness: Non-Abelian BRST Closure, Analyticity, and Crossing in Omniverse Quantum Field Theory," 34-page article, Zenodo (2025). DOI assigned on release.</p> <p>KEYWORDS<br>BRST; Slavnov–Taylor; LSZ; Cutkosky; optical theorem; plane-aware QFT; vectorial i ε̂; unit geometric-time axis p̂₀; sector–plane (s,b); OS reflection positivity; Bernstein/Laplace damper; background-field method; analyticity; crossing; all-orders UV finiteness.</p> |
| title | Proving All-Orders UV Finiteness: Non-Abelian BRST Closure, Analyticity, and Crossing in Omniverse Quantum Field Theory |
| url | https://doi.org/10.5281/zenodo.17308907 |