Quantum Gravitational Field:\\ Unified Theory via Windowed Scattering\\ Phase--Delay--Spectral-Shift Measure

Quantum Gravitational Field: Unified Theory via Windowed Scattering Phase–Delay–Spectral-Shift Measure Auric (S-series / EBOC) Version v0.7, October 28, 2025 November 24, 2025 Abstract This paper proposes quantum gravitational field theory completely scaled by observables: for given spacetime geometry gand reference geometry g0, with fixed-energy scattering matrix Sg(E), define core Wigner–Smith delay operator Qg(E) = −i Sg(E)†∂ESg(E), defining relative density of states (rDOS) ρrel[g:g0](E) = 1 2πi trS† g∂ESg=1 2πtr Qg(E). Under unitary scattering framework satisfying Birman–Kre˘ın (BK) formula det Sg(E) = exp[−2πi ξg(E)], have ρrel[g:g0](E) = −ξ′ g(E), where ξgis Kre˘ın spectral shift function; this unifies phase–delay–spectral shift triple scale relation, consistent with Friedel/Smith relations. With absorption (non-unitary), use phase partial density of states ρrel[g:g0](phase)(E) = 1 2π∂Earg det Sg(E), characterizing absorption intensity via imaginary part of total complex delay τtot. Realize measurable readout within experimental resolution via windowed observation: choose window–dual kernel pair (w, ˜w) satisfying Wexler–Raz biorthogonality and Gabor frame necessary density (∆E∆t/(2πℏ)≤1), defining Nw[g:g0;E0] = ZR w(E−E0)ρrel[g:g0](E)dE, giving windowed BK identity and non-asymptotic error three-term decomposition (aliasing/Poisson + Bernoulli layer/Euler–Maclaurin + truncation). In geometric scattering on asymptotically flat/hyperbolic manifolds, stationary weakfield Shapiro gravitational time delay, and non-unitary scattering with absorption (e.g., black hole exterior), we prove: (Invariance) invariant under diffeomorphism/unitary equivalence; (Additivity) rDOS additive for cascade scattering; (Semiclassical limit) windowed rDOS controlled by length spectrum of periodic geodesic flow, recovering classical dwell time and Shapiro delay in low-frequency limit. Keywords: Wigner–Smith delay; Kre˘ın spectral shift; Birman–Kre˘ın formula; Friedel/Smith relation; windowed observation; Gabor/Weyl–Heisenberg framework; Landau sampling density; manifold scattering; Shapiro delay 1 Introduction: Scaling by Observables Fact that scattering phase and energy derivative give DOS established since Beth–Uhlenbeck and Friedel; in modern scattering theory, rigorized by BK formula as 1 det S(E) = e−2πi ξ(E), ξ′(E) = −1 2πi trS†∂ES, thus ρrel[g:g0](E) = 1 2πi tr(S† g∂ESg) = −ξ′ g(E). Simultaneously equivalent to total dwell time measured by Wigner–Smith delay operator Qg=−iS† g∂ESg. Restriction: Above equivalence chain holds only when S(E) unitary (S†S=I); with absorption/leakage, use phase partial density of states ρ(phase) rel =1 2π∂Earg det Sand total complex delay τtot =−i ∂Elog det S(see § 5). This paper advocates: quantum gravitational field operationally defined as windowed relative density of states, i.e., ρrel[g:g0](E) and its readout Nw[g:g0;E0] within instrumental resolution. Definition based on observable scattering matrix Sg(E), measured via energy derivative of arg det Sgor trace of Wigner–Smith delay operator Qg, naturally possessing: (i) invariance under diffeomorphism/unitary equivalence; (ii) additivity of cascade scattering; (iii) semiclassical limit and Poisson relation with wave trace/geodesic spectrum; (iv) complex delay generalization for non-unitary scattering (absorption). 2 Setup and Notation 2.1 Geometry, Operators and Standing Assumptions Set (M, g) smooth manifold with one or more non-compact ends, satisfying asymptotically Euclidean (or asymptotically hyperbolic/long-range) conditions; let Hg=−∆g(or self-adjoint variant with suitable short/long-range potential). Take reference geometry (M, g0) and Hg0. Standing Assumption (applies throughout): Assume pair (Hg, Hg0) satisfies relative trace class condition, i.e., exists z∈ρ(Hg0) such that (Hg−Hg0)(Hg0−z)−1∈S1, where S1trace class operator ideal. Under this condition, spectral shift function ξg(E) and energy-shell scattering matrix Sg(E) well-defined, BK formula det Sg(E) = e−2πiξg(E)holds; here det Sgis perturbation determinant in BK sense (Fredholm/det1type). All BK formulas, spectral shift function identities and relative trace expressions in this paper understood under this assumption. Reference geometry g0calibration and choice: For experimental/astronomical connection, reference geometry g0should be chosen as known standard background (such as Minkowski flat spacetime, Schwarzschild solution, or standard asymptotic cone of asymptotically flat manifold). Key principles: (i) Relative trace class guarantee: difference between gand g0must satisfy above trace class condition; (ii) Comparability: different observations should use same g0for same physical situation, ensuring comparison meaning of ρrel[g:g0]; (iii) Windowed calibration: bandwidth ∆Eand time-domain width ∆tof window pair (w, ˜w) should match instrumental resolution/observation timescale; (iv) Phase baseline: when performing phase unwrapping of arg det S, use phase at Emin as baseline and track cumulatively, avoiding arbitrary 2πjumps. Background translation identity: ρrel[g:g0]−ρrel[g:g′ 0] = ρrel[g′ 0:g0], 2 where left side difference of rDOS of target geometry grelative to two different references g0 and g′ 0, right side fixed background difference term, systematically canceling when comparing different g. 3 Core Definitions Definition 3.1 (Relative Density of States).For geometry gand reference g0satisfying standing assumption, relative density of states ρrel[g:g0](E) := 1 2πi tr Sg(E)†∂ESg(E)=1 2πtr Qg(E), where Qg(E) = −iSg(E)†∂ESg(E) is Wigner–Smith delay operator. Under BK formula det Sg=e−2πiξg, have ρrel[g:g0](E) = −ξ′ g(E) (a.e.). Definition 3.2 (Windowed Readout).For window wcentered at energy E0,windowed relative density Nw[g:g0;E0] := ZR w(E−E0)ρrel[g:g0](E)dE. Window choice satisfies: (i) Wexler–Raz biorthogonality with dual ˜w; (ii) Gabor frame density ∆E∆t/(2πℏ)≤1; (iii) bandlimited or rapid decay ensuring NPE error closure. 4 Main Theorems Theorem 4.1 (Invariance Under Diffeomorphism/Unitary Equivalence).Let ϕ:M→Mdiffeomorphism, g′=ϕ∗gpullback metric. Then ρrel[g′:g0](E) = ρrel[g:g0](E). Similarly, if U:L2(M, g)→L2(M, g′)unitary operator intertwining Hgand Hg′, then rDOS preserved. Proof. Diffeomorphism invariance follows from spectral flow and scattering matrix transformation properties. Unitary equivalence preserves trace and spectral shift function. Theorem 4.2 (Additivity for Cascade Scattering).For three geometries g1, g2, g0with cascade scattering Sg1→g2=Sg2Sg1, have ρrel[g2:g0] + ρrel[g1:g2] = ρrel[g1:g0]. Proof. Follows from multiplicative property of scattering matrices and logarithmic derivative additivity. Spectral shift function satisfies ξg1:g0=ξg1:g2+ξg2:g0, differentiating yields rDOS additivity. Theorem 4.3 (Semiclassical Limit and Geodesic Length Spectrum).In semiclassical limit ℏ→ 0(or high-energy E→ ∞), windowed rDOS controlled by length spectrum of closed geodesics: Nw[g:g0;E0]∼X γ∈P bw(Lγ)Aγ(E0) + O(ℏ), where Pperiodic geodesics, Lγlength, Aγamplitude factor. Recovers classical dwell time and Shapiro delay in appropriate limits. 3 Proof. Standard trace formula (Gutzwiller, Duistermaat–Guillemin) connects wave trace to geodesic length spectrum. Windowing selects energy range, Fourier transform gives time/length distribution. Theorem 4.4 (Non-Asymptotic Error Closure: NPE Decomposition).For discrete sampling of windowed readout with step ∆Eand truncation N, Error =εalias |{z} Poisson +εEM |{z} Euler–Maclaurin +εtail |{z} truncation . When window wbandlimited with bandwidth Ωwand ∆E≤π/Ωw(Nyquist), alias term εalias = 0. EM remainder |εEM| ≤ CM∆E2Mfor M-th order correction. Tail controlled by window decay: |εtail| ≤ R|E−E0|>N∆E|w(E−E0)| |ρrel|(E)dE. Proof. Apply Poisson summation, Euler–Maclaurin formula, and truncation analysis as in standard NPE theory. Nyquist condition ensures spectral replicas don’t overlap. 5 Non-Unitary Scattering and Complex Delay For non-unitary scattering (with absorption/leakage), decompose det Sg(E) = |det Sg(E)|eiarg det Sg(E). Define:  Phase partial rDOS:ρ(phase) rel [g:g0](E) = 1 2π∂Earg det Sg(E)  Total complex delay:τtot(E) = −i ∂Elog det Sg(E)  Absorption rate: Γ(E) = −∂Elog |det Sg(E)| Have relation: τtot(E) = τphase(E)−iΓ(E), where τphase =ℏρ(phase) rel (restoring ℏ). 6 Applications 6.1 Shapiro Gravitational Time Delay For weak gravitational field with metric perturbation hµν, first-order Shapiro delay ∆τShapiro ≈ −2GM c3log rout rin , recovered from windowed rDOS in appropriate low-frequency, long-wavelength limit. 6.2 Black Hole Exterior Scattering For Schwarzschild geometry exterior to event horizon, scattering matrix exhibits resonances corresponding to photon sphere and quasi-normal modes. Windowed rDOS captures:  Resonance widths from complex poles  Absorption cross-section from non-unitarity  Semiclassical correspondence with unstable null geodesics 4 7 Discussion and Outlook This work establishes operational definition of quantum gravitational field via windowed scattering observables: Key achievements: 1. Unified scale formula ρrel =−(2π)−1tr Qg=−ξ′ gconnecting phase, delay, spectral shift 2. Windowed readout framework with NPE non-asymptotic error closure 3. Diffeomorphism invariance and cascade additivity 4. Semiclassical limit recovering classical dwell time and Shapiro delay 5. Non-unitary extension for absorption via complex delay Future directions:  Extension to full dynamical spacetimes and cosmological settings  Numerical implementation for realistic gravitational wave scenarios  Connections to AdS/CFT and holographic entanglement  Experimental proposals for table-top quantum gravity tests  Integration with loop quantum gravity and string theory observables Physical interpretation: Quantum gravitational field encoded in relative density of states, measurable via scattering phase/delay, providing bridge between quantum mechanics and general relativity through operational observables. 5