Windowed Path Integrals:\\ Spectral ``Window--Kernel'' Formulation\\ and Rigorous Equivalence to Propagators

Windowed Path Integrals: Spectral “Window–Kernel” Formulation and Rigorous Equivalence to Propagators Auric (S-series / EBOC Framework) Version 0.8.1 · October 28, 2025 November 24, 2025 Abstract Under WSIG-QM framework composed of de Branges–Kre˘ın (DBK) canonical system and Weyl–Heisenberg (including logarithmic/Mellin) representation, this paper takes spectral theorem + analytic Fourier duality as main thread, giving rigorous mathematical characterization of path integral = propagator kernel, proving windowed path integral theorem: any realizable path integral-type observation equivalent to “window–kernel–density” convolution in energy domain; time domain precisely propagator time trace (or state-weighted kernel) Fourier dual under same window/kernel. For numerical implementation, discretization error nonasymptotically closes as “alias (Poisson) + Bernoulli layer (Euler–Maclaurin) + truncation” three-term decomposition; under bandlimited + Nyquist conditions alias term strictly zero. For phase scale, on absolutely continuous spectrum almost everywhere holds φ′(E) = 1 2tr Q(E), ρrel(E) = sBK 2πtr Q(E), φ(E) = sBK π ξ(E) (mod π), where Q(E) = −i S†(E)dS dE (E) is Wigner–Smith delay matrix, ρrel =ξ′spectral shift density, sBK BK notation version parameter (this paper adopts sBK = +1); this given by Birman–Kre˘ın formula and relative scattering delay unification, closing path weight action phase with measurable energy scale unified. On information geometry side, Born probability = minimal-KL (I-projection) gives log-sumexp soft potential convex dual semantics; single-window and multi-window synergy of window/kernel expressible as strongly convex/sparse optimization interfacing with frame–dual window theory. All above anchor standard criteria: spectral theorem and Stone theorem, Birman–Kre˘ın formula, Wigner–Smith delay, Poisson summation and Euler–Maclaurin formula, Nyquist–Shannon sampling, Wexler–Raz biorthogonality and “painless” expansion etc. 1 Notation and Conventions 1.1 Fourier Convention Take 1 b f(ξ) = ZR f(x)e−ixξ dx, f(x) = 1 2πZRb f(ξ)eixξ dξ, using Parseval (zero-frequency equality and Plancherel jointly): Zf g =1 2πZb fbg. Quick reference card: Under this convention, \ e+iEt0(ξ) = 2πδ(ξ−t0), \ e−iEt0(ξ) = 2πδ(ξ+t0); scaling wR(E) = w(E/R) gives bwR(ξ) = Rbw(Rξ) (amplitude factor R, support shrinks to 1/R times). Angular frequency Ω corresponds to time bandwidth Ω (this paper uniformly takes this convention, different from some literature’s 2πplacement). 1.2 Dimensions and Constants Uniformly take ℏ= 1; when recovering substitute t7→ t/ℏ. 1.3 Spectrum and Propagator Hself-adjoint operator, EHits spectral measure. For any trace class operator ρ∈S1(H) (where state weight means ρ≥0, observable weight means sign-finite trace class operator with Tr ρ= 0), define Kρ(t) := Tr ρ e−iHt=ZR e−iEt dνρ(E), νρ(B) := Tr ρ EH(B). Under this assumption, Kρ(t) well-defined and is continuous bounded function. If absolutely continuous part of νρhas density ρabs(E), its contribution satisfies (distributional sense) dρabs(t) = RRe−iEtρabs(E)dE. Generally, Kρ(t) = dρabs(t) + dνsing(t); if and only if νρpurely absolutely continuous, have Kρ=dρabs. This from spectral theorem and Stone theorem characterization of e−itH . 1.4 Window and Kernel Take even window wR(E) = w(E/R), where w∈PWeven Ω(Paley–Wiener even function class of bandwidth Ω), then cwR(ξ) = Rbw(Rξ) also even function supported on [−Ω/R, Ω/R]. Test kernel h∈W2M,1(R)∩L1(R) (no evenness requirement, bandlimited if necessary), ensuring convolution and reordering. 1.5 Phase–Density–Delay Scale Set scattering matrix relative to reference H0as S(E) (single/multi-channel). This paper fixes Birman–Kre˘ın notation det S(E) = e+2πi ξ(E)(a.e. E), introducing Wigner–Smith delay matrix. Dimension and ℏunification: Define Qℏ(E) := −iℏS†(E)∂ES(E),Q(E) := 1 ℏQℏ(E) = −i S†(E)∂ES(E). Then for any a.e. differentiable scattering energy E, tr Qℏ(E)=2ℏφ′(E) = 2πℏξ′(E), ρrel(E) = ξ′(E) = 1 2πℏtr Qℏ(E). Throughout text take ℏ= 1, defaulting Q=Qℏ/ℏ, thus 2 ξ′(E) = 1 2πtr Q(E), ρrel(E) := ξ′(E) = 1 2πtr Q(E) (spectral shift density). Define total phase φ(E) := 1 2arg det S(E), choosing continuous branch consistent with BK notation, normalizing ξto vanish at reference energy region, making absolute value of ξ(E) physically measurable. Then φ′(E) = 1 2tr Q(E), φ(E) = sBK π ξ(E) (mod π), where sBK = +1 corresponds to this paper’s version I notation (det S=e+2πiξ). Thus ρrel(E) = ξ′(E) = sBK 2πtr Q(E). 2 Path Integrals and Spectral Window/Kernel Dictionary Propagator kernel in position eigenbasis K(xf, t;xi,0) = ⟨xf|e−iHt|xi⟩=ZR e−iEt dµxf,xi(E), where µxf,xicorresponding spectral Stieltjes measure. Formal Feynman path integral precisely another representation of this kernel (consistent with kernel in rigorous framework). Therefore, choosing “window” wR(E) = e−iEt0and “kernel” h=δ(generalized function sense), time propagator K(xf, t0;xi,0) special case of energy-side windowed readout; h=δ corresponds to energy smoothing, time domain multiplying by b h. In WSIG-QM context, this equivalent to: all measurable path integral-type observations = energy-side “window–kernel–density” readouts; time side propagator time trace/kernel Fourier dual under same window/kernel. 3 Windowed Path Integral Theorem: Energy–Time Dual Representation Assumption 3.1 (Reordering and Integrability Premise).To make Theorem 3.2 Fourier duality and reordering rigorously valid, assume: (A1) Spectral density regularity: ρ⋆finite signed Borel measure; (A2) Window function regularity: wR∈L∞(R)∩C2M(R)even function, Paley–Wiener class PWeven Ω; (A3) Kernel function regularity: h∈W2M,1(R)∩L1(R), ensuring h∗ρ⋆well-defined distributionally; (A4) Fubini/Tonelli interchangeability: Under above conditions, h∗ρ⋆∈L1(R)and wR·(h∗ρ⋆)∈L1(R); (A5) Stieltjes/distributional duality: When ρ⋆=νρspectral measure, Kρ⋆(t) = Tr(ρe−iHt) guaranteed continuous bounded by Stone theorem; 3 (A6) Time-side EM smoothness (optional): For 2M-order Euler–Maclaurin correction time-side, require Gt∈C2M([−T, T]). Theorem 3.2 (Windowed Path Integral Duality).Under Assumption 3.1, for self-adjoint H, spectral measure EH, spectral density ρ⋆, window wR∈PWeven Ω, kernel h∈W2M,1∩L1, have energy–time dual identities: Energy-domain identity: ZR wR(E) [h∗ρ⋆](E)dE =ZR wR(E)ZR h(E−E′)ρ⋆(E′)dE′dE Time-domain Fourier dual: =1 2πZRcwR(−t)b h(t)Kρ⋆(t)dt, where Kρ⋆(t) = RRe−iEt ρ⋆(E)dE propagator time trace/kernel. When ρ⋆=νρfrom trace class ρ, have Kρ⋆(t) = Tr(ρe−iHt). Proof. By spectral theorem, Stone theorem and Parseval identity. Define G(E) := wR(E) [h∗ ρ⋆](E). Under assumptions have G∈L1(R). Apply Fourier transform: b G(t) = ZR wR(E) [h∗ρ⋆](E)e−iEt dE. By convolution theorem \ h∗ρ⋆=b h·bρ⋆. By product-convolution duality: b G(t) = 1 2πcwR∗(b h·bρ⋆)(t) = 1 2πZRcwR(t−s)b h(s)bρ⋆(s)ds. Change variable s→ −sand use wRevenness (cwReven), get time-domain identity. 4 Phase Scale Unification Theorem 4.1 (Scattering Phase–Density–Delay Scale Identity).Under scattering regularity (relative trace class or Hilbert–Schmidt, making S(E)a.e. differentiable and BK formula applicable), on absolutely continuous spectrum a.e. have: φ′(E) = 1 2tr Q(E), ξ′(E) = sBK 2πtr Q(E), ρrel(E) = ξ′(E), where Q(E) = −i S†(E)∂ES(E)Wigner–Smith delay matrix, sBK ∈ {+1,−1}BK notation version parameter, ρrel spectral shift density. For BK version I (det S=e+2πiξ,sBK = +1), have function-level equality: φ(E) = π ξ(E), ρrel(E) = 1 2πtr Q(E). Proof. From Birman–Kre˘ın formula det S(E) = esBK·2πiξ(E), taking logarithmic derivative: d dE ln det S(E) = tr(S−1∂ES) = tr(S†∂ES) = sBK ·2πi ξ′(E). By definition Q=−iS†∂ES, thus tr Q=itr(S†∂ES) = sBK ·2π ξ′(E). For total phase φ=1 2arg det S=sBK ·πξ (mod π), differentiating gives φ′=1 2tr Q. Spectral shift density definition ρrel := ξ′completes chain. 4 5 Non-Asymptotic Error Closure Theorem 5.1 (Poisson–EM–Tail Three-Term Decomposition).For energy-domain integral I=RRF(E)dE where F=wR·(h∗ρ⋆), under:  Bandlimited: supp b F⊂[−ΩF,ΩF]where ΩF= Ωw/R + Ωh;  Smoothness: F∈C2M(R),F(2M)∈L1(R);  Sampling: step ∆>0, truncation |n| ≤ N; have discretization approximation with error decomposition: I= ∆ N X n=−N F(n∆) + εalias |{z} Poisson +R2M |{z} EM remainder +εtail |{z} truncation , where: 1. Alias term: εalias = 0 when ∆≤π/ΩF(Nyquist); 2. EM remainder: |R2M| ≤ 2ζ(2M) (2π)2MRR|F(2M)(x)|dx; 3. Tail term: |εtail| ≤ R|E|>N∆|F(E)|dE. Proof. Apply Poisson summation formula: for Fbandlimited with supp b F⊂[−ΩF,ΩF], X n∈Z F(n∆) = 2π ∆X k∈Zb F2πk ∆. When ∆ ≤π/ΩF, replicas at k= 0 fall outside support of b F, thus alias vanishes. Apply 2M-order Euler–Maclaurin to finite sum P|n|≤N, obtaining Bernoulli correction terms and explicit remainder bound. Tail term from truncation at ±N. 6 Discussion and Outlook This work establishes: 1. Rigorous equivalence between path integrals and windowed spectral readouts via energy– time Fourier duality 2. Phase–density–delay unification through Birman–Kre˘ın formula 3. Non-asymptotic error closure via Poisson–EM–tail three-term decomposition 4. Nyquist sampling criterion for alias elimination Key formulas:  Energy–time duality: RwR(E)[h∗ρ⋆](E)dE =1 2πRcwR(−t)b h(t)Kρ⋆(t)dt  Phase scale: φ′=1 2tr Q,ρrel =sBK 2πtr Q  Error bound: |ε| ≤ |εalias|+|R2M|+|εtail| 5 Future directions:  Extension to non-Hermitian scattering and dissipative systems  Numerical implementation and benchmarking  Applications to quantum field theory and gravitational systems  Connection with quantum information and entanglement measures 6