Establish unified framework connecting scattering phase derivative, spectral density, and windowed readout. Core formula holding a.e.: wφ′(E)π=ρrel(E)=12π\trQ(E)hereφscatteringphase,ρrelrelativespectraldensityfromBirman−−Kreı˘ndetS=e−2πiξwithρrel=−ξ′,Q=−iS†∂ESWigner−−Smithdelaymatrix.WindowedreadoutRw=∫w(E)[h∗ρrel](E)dE with NPE three-term error decomposition. Applications: quantum metrology, scattering theory, condensed matter.
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.
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.
Under de Branges--Kreĭn canonical systems and multi-channel scattering theory, taking scattering phase derivative, relative state density, and Wigner--Smith group delay trace as three equivalent formulations of the same scale, we construct a ``universal measure coordinate'' and prove its change-of-variables consistency under windowed readouts and non-asymptotic error closure of Nyquist--Poisson--Euler--Maclaurin (NPE) three-fold decomposition. The core unification formula (almost everywhere on absolutely continuous spectrum) is equation d\mu_\varphi(E)=\varphi'(E){\pi}\,dE=1{2\pi}tr\mathsf Q(E)\,dE=\rho_{rel}(E)\,dE, equation where \mathsf Q(E)=-i\,S(E)^\dagger S'(E) is the Wigner--Smith delay matrix, \rho_{rel}=-\xi' is the spectral shift density relative to reference operator H_0, and \det S(E)=\exp(-2\pi i\,\xi(E)) is the standard gauge of Birman--Kreĭn formula. The trinity chain is derived consistently from Birman--Kreĭn formula and Wigner--Smith delay, applicable to multi-channel unitary scattering; sub-unitary systems admit compatible extension via generalized (complex) delay. The paper provides: (i) windowed change-of-variables consistency theorem (energy \leftrightarrow phase/delay/density coordinates); (ii) maximum entropy--minimum delay duality theorem; (iii) causal monotonicity \Leftrightarrow phase monotonicity; and formulates sampling--frame thresholds, Wexler--Raz biorthogonality, and Balian--Low impossibility as invariant expressions in phase coordinates, while establishing KKT and \Gamma-limit stability for multi-window/multi-kernel optimization. All conclusions align term-by-term with BK/SSF, Wigner--Smith, Herglotz--Weyl, and Carleson--frame theory criteria, directly verifiable 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.