WSIG-QM: Windowed Scattering \& Information Geometry\\ for Quantum Mechanics\\ \bigskip A Unified Framework of Quantum Concept Definitions\\ and Criterion System (with Complete Proofs)
Using the de Branges--Kreĭn (DBK) canonical system and weighted Mellin space as carriers, we explicitly incorporate the finite bandwidth/time window of real instruments into spectral measures, forming a windowed readout framework; we characterize ``commit (collapse/commit)'' through KL/Bregman information geometry; use scattering phase--spectral density--Wigner--Smith delay as energy scale; close non-asymptotic errors via Nyquist--Poisson--Euler--Maclaurin (three-term decomposition); ensure realizability and stability through variational optimization of frame/sampling density and window/kernel. The core unified formula is $ \,\varphi'(E)=-\pi\,\rho_{\mathrm{rel}(E)=1{2}trQ(E)\,}\quad(a.e.) $ unifying (single/multi-channel) scattering phase derivative, relative local density of states (LDOS) and Wigner--Smith delay; under information geometry we obtain Born probability = minimal-KL projection (I-projection), and ``pointer basis'' is spectral minimum of windowed readout operator. Above criteria consistent with Herglotz--Weyl, Birman--Kreĭn, Wigner--Smith, Ky Fan, Poisson/EM and other standard results, directly interchangeable and implementable.
Using the de Branges--Kreĭn (DBK) canonical system and weighted Mellin space as carriers, we explicitly incorporate the finite bandwidth/time window of real instruments into spectral measures, forming a windowed readout framework; we characterize commit (collapse/commit)'' through KL/Bregman information geometry; use scattering phase--spectral density--Wigner--Smith delay as energy scale; close non-asymptotic errors via Nyquist--Poisson--Euler--Maclaurin (three-term decomposition); ensure realizability and stability through variational optimization of frame/sampling density and window/kernel. The core unified formula is $ \,\varphi'(E)=-\pi\,\rho_{\mathrm{rel}(E)=1{2}trQ(E)\,}\quad(a.e.) $ unifying (single/multi-channel) scattering phase derivative, relative local density of states (LDOS) and Wigner--Smith delay; under information geometry we obtain Born probability = minimal-KL projection (I-projection), and pointer basis'' is spectral minimum of windowed readout operator. Above criteria consistent with Herglotz--Weyl, Birman--Kreĭn, Wigner--Smith, Ky Fan, Poisson/EM and other standard results, directly interchangeable and implementable.
This paper constructs and rigorizes WSIG-QFT (Windowed Scattering \& Information-Geometry Quantum Field Theory): under weighted Mellin--logarithmic model and de Branges--Kreĭn (DBK) canonical system, uses Weyl--Heisenberg kinematic scale for ``phase--scale'', connects scattering phase derivative with (relative) spectral density via Birman--Kreĭn (BK)--Wigner--Smith (WS) chain, realizes Born probability = relative entropy minimization through Csiszár-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--Kreĭn canonical system; Weyl--Heisenberg representation; Birman--Kreĭn; Wigner--Smith delay; I-projection; Wexler--Raz; Nyquist--Poisson--Euler--Maclaurin (NPE) error closure MSC: 81Txx; 47Bxx; 46E22; 42C15
Under WSIG-QM framework composed of de Branges--Kreĭn (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 non-asymptotically 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 $ \varphi'(E)=1{2}trQ(E),\qquad \rho_{rel}(E)=s_{\mathrm{BK}}{2\pi}trQ(E),\qquad \varphi(E)=s_{BK}\,\pi\,\xi(E)\pi, where Q(E)=-i\,S^\dagger(E)\, dS{dE}(E) is Wigner--Smith delay matrix, \rho_{rel}=\xi' spectral shift density, s_{BK} BK notation version parameter (this paper adopts s_{BK}=+1$); this given by Birman--Kreĭn 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-sum-exp 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--Kreĭn formula, Wigner--Smith delay, Poisson summation and Euler--Maclaurin formula, Nyquist--Shannon sampling, Wexler--Raz biorthogonality and ``painless'' expansion etc.
Within unified framework of mirror kernel--de Branges--Kreĭn canonical system--information geometry, this paper proposes and rigorously proves three main theorems: 1. Windowed Readout Theorem: Any realizable quantum measurement readout equivalent to weighting of (relative or absolute) local density of states (LDOS) by ``energy window w_R and frontend kernel h''; when adopting realistic discrete sampling--finite truncation procedure, error can be non-asymptotically closed by Nyquist (alias)--Poisson (sampling)--Euler--Maclaurin (EM, sum--integral difference) three terms, with alias term strictly zero under bandlimited + Nyquist conditions. Conclusion based on Herglotz property and boundary value dictionary (\Im m(E+i0)=\pi\rho(E)) of Weyl--Titchmarsh m-function and its equivalent formulation with canonical systems. 2. Born Probability = Minimal KL (Information Projection): When readout dictionary aligns with log-partition potential \Lambda(\rho)=\log\!\sum_j w_j e^{\langle\beta_j,\rho\rangle}, minimal energy projection with unit response equivalent to minimal Kullback--Leibler (KL) divergence under linear moment constraints; softmax probability precisely minimal-KL projection weights, converging via \Gamma-limit to hard projection (Hilbert orthogonal) as softening parameter \tau\!\downarrow\!0 (equivalently inverse temperature \kappa=1/\tau\!\uparrow\!\infty). Equivalently using Fenchel--Legendre duality / Bregman--KL identity / Csiszár I-projection. 3. Pointer Basis = Eigenbasis of Minimal Energy/Information Projection: Under finite dictionary, coefficient vector of minimal energy mollifier \beta^\star=G^{-1c}{c^\ast G^{-1}c}; in Gram spectral decomposition G=U\Lambda U^\ast, \beta^\star expanded along \{u_k\} weighted by \lambda_k^{-1}, thus direction contributing strongest to \beta^\star realized by $ \arg\max_k\ |\langle u_k,c\rangle|^2{\lambda_k}; small eigenvalue trend amplifies that direction, but whether dominates depends on simultaneously having sufficiently large projection |\langle u_k,c\rangle|. Soft version information Hessian \nabla^2\Lambda spectral basis isomorphic to this. On scattering side, via Birman--Kreĭn and Wigner--Smith standard construction, single-channel phase derivative and (relative) spectral density satisfy \varphi'(E)=\pi\,\rho_{rel}(E)=\pi\,\xi'(E),\qquad Q(E)=-i\,S(E)^\daggerdS{dE}, hence 1{2\pi}trQ(E)=\rho_{rel}(E)$. This interprets ``negative delay'' as result of reference choice and relative counting, not causality violation. Keywords: Windowed readout; Weyl--Titchmarsh; spectral shift function; Wigner--Smith time delay; de Branges space; BN--Bregman; minimal KL; PSWF; Nyquist--Poisson--EM; non-asymptotic error.
We construct a unified framework of Windowed Scattering--Information Geometry (WSIG) centered on the ``operator--measure--function'' triple: observation is operationalized through Toeplitz/Berezin compression Kw,h=ΠwMhΠw, reducing all readouts to linear functionals on spectral measures. The trinity scale identity φ′(E)/π=ρrel(E)=(2π)−1trQ(E)=−ξ′(E) unifies phase, relative density of states, and Wigner--Smith group delay as master scales. For observer filter chains {Kθ}θ∈Θ, we define an information action S and induce an information metric via the Hessian gμν=∂μ∂νS, interpreting connection and curvature as second-order non-commutative responses of readouts to observation parameters. Under the ``double-time separation'' postulate---where causal partial order is uniquely defined by frontier time t∗ while windowed group delay Tγ[w,h] serves only as operational scale---variation of the action A=(2κ)−1∫(R−2Λ)∣g∣dnθ+∫(Linfo+λμν[gμν−∂μ∂νSloc])∣g∣dnθ yields Einstein-type equations Gμν+Λgμν=κTμν. The right-hand stress-energy tensor Tμν arises from spectral responses of Kθ. Under band-limited and Nyquist--Poisson--finite-order Euler--Maclaurin (NPE) discipline, we achieve error closure with ``non-increasing singularity: poles = master scales''. Within the same geometric framework, we unify quantum measurement: superposition = convex superposition of compatible components in static blocks; collapse = anchor switching and π-semantic collapse; entanglement = inseparable information correlations in causal networks. Continuous monitoring satisfies GKSL dynamics and Spohn entropy production monotonicity, with Belavkin filtering providing conditional state updates. Core criteria and required external theorems are provided, along with one-dimensional potential scattering examples and complete proof chains.
Under unified framework of de Branges--Kreĭn (DBK) canonical system, scattering--functional equation dictionary and Bregman/information geometry, this paper establishes ``Trinity Theorem'' for windowed measurement. Conclusions in three tiers: (I) Born = Information Projection (iff): Under orthogonal projection measurement (and its generalization to POVM), optimal probability vector induced by windowed readout equivalent to I-projection (minimal KL/Bregman cost) over family of linear alignment constraints if and only if it equals Born probability. (II) Pointer = Spectral Minimum (iff): For any distinguishable window family, let Ky Fan partial sum of ``windowed trace quadratic form'' minimize over all orthonormal bases, then if and only if that basis is spectral eigenbasis of measured observable (modulo degeneracy). (III) Windows = Minimax Optimal: Under constraint of bandlimited even window with normalization w(0)=1, taking Nyquist--Poisson--Euler--Maclaurin ``alias + Bernoulli layer + truncation'' non-asymptotic error upper bound as adversary, optimal window exists and (under Hilbert strongly convex proxy) unique, satisfying frequency-domain projection KKT condition. Key bridge is Birman--Kreĭn formula and Wigner--Smith delay giving phase derivative = spectral density, precisely connecting windowed readout with relative state density (LDOS).
For multi-channel scattering with N(E) channels at energy E, establish unified framework connecting scattering phase derivative, relative spectral density, and Wigner--Smith delay. Core formula holding a.e. on absolutely continuous spectrum: $ \ \frac{1{N(E)}d{dE}\Arg\det S(E)=1{2\pi}\langle\tr Q(E)\rangle_N=\rho_{rel}(E)\ } where Q(E)=-iS^\dagger(E)\partial_E S(E) is Wigner--Smith delay matrix, \langle\cdot\rangle_N denotes per-channel average, \rho_{rel} relative spectral density from Birman--Kreĭn formula \det S=e^{-2\pi i\xi}$. Establish: (i) Channel covariance: formula invariant under unitary channel basis transformations; (ii) Threshold regularity: handle channel number jumps via proper boundary conditions; (iii) Windowed readout: finite-bandwidth measurement protocol with NPE error closure; (iv) Information geometry: Born probability as I-projection, pointer basis as Ky Fan minimum.