WSIG-QFT: Axioms, Theorems and Proofs\\ for Windowed Scattering and Information Geometry\\ in Quantum Field Theory

WSIG-QFT: Axioms, Theorems and Proofs for Windowed Scattering and Information Geometry in Quantum Field Theory Auric (S-series / EBOC) Version 1.3 November 24, 2025 Abstract This paper constructs and rigorizes WSIG-QFT (Windowed Scattering & InformationGeometry Quantum Field Theory): under weighted Mellin–logarithmic model and de Branges–Kre˘ın (DBK) canonical system, uses Weyl–Heisenberg kinematic scale for “phase–scale”, connects scattering phase derivative with (relative) spectral density via Birman–Kre˘ın (BK)–Wigner–Smith (WS) chain, realizes Born probability = relative entropy minimization through Csisz´ar-type I-projection, implements pointer basis = spectral minimum of readout quadratic form via Ky-Fan spectral minimum, provides non-asymptotic error closure and bandlimited-sampling criterion through Nyquist–Poisson–Euler–Maclaurin (NPE). For multi-channel establishes windowed BK identity and multi-window frame–Wexler–Raz synergy conditions, giving verifiable premises, explicit statements and complete proofs (based on recognized criteria). Keywords: Windowed readout; de Branges–Kre˘ın canonical system; Weyl–Heisenberg representation; Birman–Kre˘ın; Wigner–Smith delay; I-projection; Wexler–Raz; Nyquist– Poisson–Euler–Maclaurin (NPE) error closure MSC: 81Txx; 47Bxx; 46E22; 42C15 1 Setup and Notation 1.1 Logarithmic–Mellin Model and Mirror Involution Take Ha=L2(R+, xa−1dx), let x=etthen isometric with L2(R). Define modulation/scale action (Uτf)(x) = xiτ f(x),(Vσf)(x) = eσa/2f(eσx), satisfying VσUτ=eiτσUτVσ(Weyl relation). Mirror involution (Jf)(x) = x−af(1/x) unitary, Mellin transform satisfies Ma[Jf](s) = Ma[f](a−s). This symmetry given by standard Mellin identities appearing in handbooks and DLMF entries. 1.2 DBK Canonical System and Herglotz–Weyl Dictionary Take half-axis canonical system JY ′(t, z) = zH(t)Y(t, z) (H⪰0 integrable, J=0−1 1 0 ). Its Weyl–Titchmarsh function m(z) is Herglotz, non-tangential boundary imaginary part gives 1 spectral density ρ(E) = π−1ℑm(E+i0); every Herglotz function originates from some tracenormed canonical system (de Branges theorem). 1.3 Scattering Data and Phase–Delay Matrix Set scatterable pair (H0, H) satisfying trace-class perturbation premise; S-matrix S(E)’s Wigner–Smith delay matrix Q(E) = −iS(E)∗∂ES(E) well-defined, eigenvalues are “intrinsic delay times”. 2 WSIG-QFT Axioms Axiom 2.1 (Weyl–Heisenberg Covariance and Mirror).Physical observable phase–scale action realized by projective unitary representation of (Uτ, Vσ), mirror Jrealizes s7→ a−s completion symmetry (Mellin side). Axiom 2.2 (Windowed Readout).Any real readout equivalent to energy-side convolution– weighted linear functional R[F;ρ⋆]≡ZR F(E)ρ⋆(E)dE, F := h∗wR, where hfrontend kernel, wReven window, ρ⋆=ρor relative density ρ−ρ0. Axiom 2.3 (Phase–Density Scale).Under BK and WS chain, almost everywhere 1 2πtr Q(E) = ξ′(E) = tr ρ−ρ0(E),det S(E) = e2πi ξ(E), where positive sign convention completely consistent with ξ′=1 2πtr Qand single-channel φ′(E) = π ρrel(E)(S=e2iφ). Axiom 2.4 (Probability–Information Consistency).Solution minimizing KL-divergence over linear moment constraint family equivalent to Born probability; necessary and sufficient conditions given by Csisz´ar’s I-projection geometry and Pythagorean identity. Axiom 2.5 (NPE Non-Asymptotic Closure).For uniform sampling/numerical quadrature of F=h∗wR, error decomposes as alias (Poisson) + EM Bernoulli layer + tail three terms; if supp b F⊂[−ΩF,ΩF]and ∆≤π/ΩF, alias term is 0. 3 Kinematics and Mirror Kernel Theorem 3.1 (CCR–Weyl Relation and Logarithmic Representation Equivalence).Let Uτ= eiτA,Vσ=eiσB, where on core D:= C∞ c(R+) (Af)(x) = (log x)f(x),(Bf)(x) = −ix∂x+a 2f(x). Then on common dense core Dhave [A, B] = iI, after closure [A, B] = iI, exponential forms give VσUτ=eiτσUτVσ. Via x=etisometry, unitarily equivalent to modulation– translation representation of L2(R). Proof. Stone theorem gives strongly continuous one-parameter groups and generators; direct calculation yields Weyl relation; isometric map given by L2(R+, xa−1dx)≃L2(R) and Mellin– Fourier interconversion. 2 Theorem 3.2 (Mirror Kernel and Completed Function).If K(x) = x−aK(1/x)and K∈ L1(R+, xa−1dx), then Mellin transform Φ(s) = R∞ 0K(x)xs−1dx satisfies Φ(s) = Φ(a−s). Multiplying by symmetry factor r(s)gives completed function Ξ(s) = r(s)Φ(s). Proof. Direct from definition of Jand Ma[Jf](s) = Ma[f](a−s). 4 Dynamics: Phase–Density–Delay Theorem 4.1 (Phase Derivative = (Relative) Spectral Density).Set (H0, H)self-adjoint pair with H−H0∈S1. Denote spectral shift function ξand S-matrix S(E). Then a.e. E have det S(E) = e2πi ξ(E), ξ′(E) = 1 2πtr Q(E) = tr(ρ−ρ0)(E), single-channel S(E) = e2iφ(E)and φ′(E) = π ρrel(E). Proof. First formula is Birman–Kre˘ın formula; second from Q(E) = −iS∗∂ESand ∂Earg det S(E) = tr Q(E); equivalence of ξ′with relative local density of states (relative LDOS) see spectral shift–trace formula (next section). Single-channel case substitute S=e2iφ proves. Proposition 4.2 (Threshold and Phase Critical Alignment).If at threshold E0have ρrel(E0) = 0, then φ′(E0)=0. Proof. Direct from Theorem 4.1 single-channel formula φ′=π ρrel. 5 Windowed Trace Formula and Windowed BK Identity Theorem 5.1 (Lifshits–Kre˘ın Trace Formula–Windowed Version).Set f∈OL(R)(operator Lipschitz), take primitive of f= (h∗wR)such that f′=F. Then trf(H)−f(H0)=ZR f′(E)ξ(E)dE =ZR F(E)ξ(E)dE. Proof. For paired self-adjoint operators with H−H0∈S1, Lifshits–Kre˘ın trace formula holds on OL class; setting f′=Fyields “windowed trace”. Theorem 5.2 (Windowed Birman–Kre˘ın Identity).Under Theorem 5.1 premises, integration by parts using det S(E) = e2πi ξ(E)gives ZR F(E)ξ′(E)dE =−1 2πi ZR F′(E) log det S(E)dE =−1 2πi ZRh′∗wR(E) log det S(E)dE. Proof. Integration by parts and substituting BK formula. 3 6 Information Geometry and Born Probability Theorem 6.1 (Born Probability = I-Projection).For linear moment constraint C={p: Pipiai=b}and reference q, minimal KL-divergence p⋆= arg min p∈C DKL(p∥q) has exponential family form p⋆ i∝qieλai. If Born weights wi=⟨ψ, Eiψ⟩affinely expressible in constraint space, then p⋆=w(Born probability). Proof. Strict convexity of KL and Lagrange multipliers give exponential family and uniqueness; alignment condition derived from exponential family parameterization. POVM case by Naimark dilation to PVM then pushback. 7 Pointer Basis and Ky Fan Minimum Theorem 7.1 (Pointer Basis = Spectral Minimum).For self-adjoint window operator WR and any m-dimensional orthogonal family {ek}, m X k=1 ⟨ek, WRek⟩ ≥ m X k=1 λ↑ k(WR), equality if and only if {ek}spans minimal eigensubspace of WR(Ky Fan minimum sum). Proof. Standard Ky Fan variational principle (PNAS 1951). 8 Non-Asymptotic Error Closure: NPE Decomposition Theorem 8.1 (Nyquist–Poisson–EM Three-Term Decomposition).For energy-domain integral I=RRF(E)dE where F=wR·(h∗ρ⋆), under:  Bandlimited: supp b F⊂[−ΩF,ΩF];  Smoothness: F∈C2M(R),F(2M)∈L1(R);  Sampling: step ∆>0, truncation |n| ≤ N; have discretization approximation I= ∆ N X n=−N F(n∆) + εalias |{z} Poisson +R2M |{z} EM +εtail |{z} truncation , where alias term εalias = 0 when ∆≤π/ΩF(Nyquist), EM remainder |R2M| ≤ 2ζ(2M) (2π)2MR|F(2M)|, tail |εtail| ≤ R|E|>N∆|F|. Proof. Apply Poisson summation: for bandlimited F, replicas at k= 0 fall outside support when Nyquist satisfied. Apply 2M-order Euler–Maclaurin to finite sum, obtaining Bernoulli corrections and explicit remainder. Tail from truncation. 4 9 Multi-Window Frames and Wexler–Raz Theorem 9.1 (Wexler–Raz Biorthogonality for Multi-Window).For Gabor frame with timefrequency lattice (α, β)satisfying αβ ≤1, window gand dual window egsatisfy Wexler–Raz biorthogonality relation: X n∈Z g(t−nα)eg(t−nα)e2πimβt =1 βδm,0,∀m∈Z,a.e. t. Equivalently in frequency domain: X k∈Zbg(ξ−k/α)b eg(ξ−k/α) = α, a.e. ξ. Proof. Standard result from Gabor analysis (Daubechies–Landau–Landau 1995). Follows from Poisson summation and frame operator properties. 10 Discussion and Outlook This work establishes rigorous mathematical foundations for WSIG-QFT: 1. Weyl–Heisenberg kinematic framework with mirror symmetry 2. Phase–density–delay unification via Birman–Kre˘ın and Wigner–Smith 3. Born probability as I-projection minimizing KL-divergence 4. Pointer basis as Ky Fan spectral minimum 5. Non-asymptotic error closure via NPE decomposition 6. Multi-window frame synergy via Wexler–Raz biorthogonality Key formulas:  Scale identity: 1 2πtr Q=ξ′= tr(ρ−ρ0)  Windowed BK: RFξ′=−1 2πi RF′log det S  NPE error: |ε| ≤ |εalias|+|R2M|+|εtail| Future directions:  Extension to quantum field theory and renormalization  Connections to holography and AdS/CFT  Numerical implementation and benchmarking  Applications to quantum many-body systems 5