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

WSIG-QFT: Axioms, Theo ems and P oo s
o Windowed Sca e ing and In o ma ion Geome y
in Quan um Field Theo y
Au ic (S-se ies / EBOC)
Ve sion 1.3
No embe 24, 2025
Abs ac
This pape cons uc s and igo izes WSIG-QFT (Windowed Sca e ing & In o ma ion-
Geome y Quan um Field Theo y): unde weigh ed Mellin–loga i hmic model and
de B anges–K e˘ın (DBK) canonical sys em, uses Weyl–Heisenbe g kinema ic scale o
“phase–scale”, connec s sca e ing phase de i a i e wi h ( ela i e) spec al densi y ia
Bi man–K e˘ın (BK)–Wigne –Smi h (WS) chain, ealizes Bo n p obabili y =
ela i e en opy minimiza ion h ough Csisz´a - ype I-p ojec ion, implemen s
poin e basis = spec al minimum o eadou quad a ic o m ia Ky-Fan
spec al minimum, p o ides non-asymp o ic e o closu e and bandlimi ed-sampling
c i e ion h ough Nyquis –Poisson–Eule –Maclau in (NPE). Fo mul i-channel es-
ablishes windowed BK iden i y and mul i-window ame–Wexle –Raz syne gy
condi ions, gi ing e i iable p emises, explici s a emen s and comple e p oo s (based
on ecognized c i e ia).
Keywo ds: Windowed eadou ; de B anges–K e˘ın canonical sys em; Weyl–Heisenbe g
ep esen a ion; Bi man–K e˘ın; Wigne –Smi h delay; I-p ojec ion; Wexle –Raz; Nyquis –
Poisson–Eule –Maclau in (NPE) e o closu e
MSC: 81Txx; 47Bxx; 46E22; 42C15
1 Se up and No a ion
1.1 Loga i hmic–Mellin Model and Mi o In olu ion
Take Ha=L2(R+, xa−1dx), le x=e hen isome ic wi h L2(R). De ine modula ion/scale
ac ion
(Uτ )(x) = xiτ (x),(Vσ )(x) = eσa/2 (eσx),
sa is ying VσUτ=eiτσUτVσ(Weyl ela ion). Mi o in olu ion (J )(x) = x−a (1/x)
uni a y, Mellin ans o m sa is ies Ma[J ](s) = Ma[ ](a−s). This symme y gi en by
s anda d Mellin iden i ies appea ing in handbooks and DLMF en ies.
1.2 DBK Canonical Sys em and He glo z–Weyl Dic iona y
Take hal -axis canonical sys em JY ′( , z) = zH( )Y( , z) (H⪰0 in eg able, J=0−1
1 0 ). I s
Weyl–Ti chma sh unc ion m(z) is He glo z, non- angen ial bounda y imagina y pa gi es
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spec al densi y ρ(E) = π−1ℑm(E+i0); e e y He glo z unc ion o igina es om some ace-
no med canonical sys em (de B anges heo em).
1.3 Sca e ing Da a and Phase–Delay Ma ix
Se sca e able pai (H0, H) sa is ying ace-class pe u ba ion p emise; S-ma ix S(E)’s
Wigne –Smi h delay ma ix Q(E) = −iS(E)∗∂ES(E) well-de ined, eigen alues a e “in insic
delay imes”.
2 WSIG-QFT Axioms
Axiom 2.1 (Weyl–Heisenbe g Co a iance and Mi o ).Physical obse able phase–scale ac-
ion ealized by p ojec i e uni a y ep esen a ion o (Uτ, Vσ), mi o J ealizes s7→ a−s
comple ion symme y (Mellin side).
Axiom 2.2 (Windowed Readou ).Any eal eadou equi alen o ene gy-side con olu ion–
weigh ed linea unc ional
R[F;ρ⋆]≡ZR
F(E)ρ⋆(E)dE, F := h∗wR,
whe e h on end ke nel, wRe en window, ρ⋆=ρo ela i e densi y ρ−ρ0.
Axiom 2.3 (Phase–Densi y Scale).Unde BK and WS chain, almos e e ywhe e
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2π Q(E) = ξ′(E) = ρ−ρ0(E),de S(E) = e2πi ξ(E),
whe e posi i e sign con en ion comple ely consis en wi h ξ′=1
2π Qand single-channel
φ′(E) = π ρ el(E)(S=e2iφ).
Axiom 2.4 (P obabili y–In o ma ion Consis ency).Solu ion minimizing KL-di e gence o e
linea momen cons ain amily equi alen o Bo n p obabili y; necessa y and su icien con-
di ions gi en by Csisz´a ’s I-p ojec ion geome y and Py hago ean iden i y.
Axiom 2.5 (NPE Non-Asymp o ic Closu e).Fo uni o m sampling/nume ical quad a u e o
F=h∗wR, e o decomposes as alias (Poisson) + EM Be noulli laye + ail h ee
e ms; i supp b
F⊂[−ΩF,ΩF]and ∆≤π/ΩF, alias e m is 0.
3 Kinema ics and Mi o Ke nel
Theo em 3.1 (CCR–Weyl Rela ion and Loga i hmic Rep esen a ion Equi alence).Le Uτ=
eiτA,Vσ=eiσB, whe e on co e D:= C∞
c(R+)
(A )(x) = (log x) (x),(B )(x) = −ix∂x+a
2 (x).
Then on common dense co e Dha e [A, B] = iI, a e closu e [A, B] = iI, exponen-
ial o ms gi e VσUτ=eiτσUτVσ. Via x=e isome y, uni a ily equi alen o modula ion–
ansla ion ep esen a ion o L2(R).
P oo . S one heo em gi es s ongly con inuous one-pa ame e g oups and gene a o s; di ec
calcula ion yields Weyl ela ion; isome ic map gi en by L2(R+, xa−1dx)≃L2(R) and Mellin–
Fou ie in e con e sion.
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Theo em 3.2 (Mi o Ke nel and Comple ed Func ion).I K(x) = x−aK(1/x)and K∈
L1(R+, xa−1dx), hen Mellin ans o m Φ(s) = R∞
0K(x)xs−1dx sa is ies Φ(s) = Φ(a−s).
Mul iplying by symme y ac o (s)gi es comple ed unc ion Ξ(s) = (s)Φ(s).
P oo . Di ec om de ini ion o Jand Ma[J ](s) = Ma[ ](a−s).
4 Dynamics: Phase–Densi y–Delay
Theo em 4.1 (Phase De i a i e = (Rela i e) Spec al Densi y).Se (H0, H)sel -adjoin
pai wi h H−H0∈S1. Deno e spec al shi unc ion ξand S-ma ix S(E). Then a.e. E
ha e
de S(E) = e2πi ξ(E), ξ′(E) = 1
2π Q(E) = (ρ−ρ0)(E),
single-channel S(E) = e2iφ(E)and φ′(E) = π ρ el(E).
P oo . Fi s o mula is Bi man–K e˘ın o mula; second om Q(E) = −iS∗∂ESand ∂Ea g de S(E) =
Q(E); equi alence o ξ′wi h ela i e local densi y o s a es ( ela i e LDOS) see spec al
shi – ace o mula (nex sec ion). Single-channel case subs i u e S=e2iφ p o es.
P oposi ion 4.2 (Th eshold and Phase C i ical Alignmen ).I a h eshold E0ha e ρ el(E0) =
0, hen φ′(E0)=0.
P oo . Di ec om Theo em 4.1 single-channel o mula φ′=π ρ el.
5 Windowed T ace Fo mula and Windowed BK Iden-
i y
Theo em 5.1 (Li shi s–K e˘ın T ace Fo mula–Windowed Ve sion).Se ∈OL(R)(ope a o
Lipschi z), ake p imi i e o = (h∗wR)such ha ′=F. Then
 (H)− (H0)=ZR
′(E)ξ(E)dE =ZR
F(E)ξ(E)dE.
P oo . Fo pai ed sel -adjoin ope a o s wi h H−H0∈S1, Li shi s–K e˘ın ace o mula holds
on OL class; se ing ′=Fyields “windowed ace”.
Theo em 5.2 (Windowed Bi man–K e˘ın Iden i y).Unde Theo em 5.1 p emises, in eg a ion
by pa s using de S(E) = e2πi ξ(E)gi es
ZR
F(E)ξ′(E)dE =−1
2πi ZR
F′(E) log de S(E)dE =−1
2πi ZRh′∗wR(E) log de S(E)dE.
P oo . In eg a ion by pa s and subs i u ing BK o mula.
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6 In o ma ion Geome y and Bo n P obabili y
Theo em 6.1 (Bo n P obabili y = I-P ojec ion).Fo linea momen cons ain C={p:
Pipiai=b}and e e ence q, minimal KL-di e gence
p⋆= a g min
p∈C DKL(p∥q)
has exponen ial amily o m p⋆
i∝qieλai. I Bo n weigh s wi=⟨ψ, Eiψ⟩a inely exp essible
in cons ain space, hen p⋆=w(Bo n p obabili y).
P oo . S ic con exi y o KL and Lag ange mul iplie s gi e exponen ial amily and unique-
ness; alignmen condi ion de i ed om exponen ial amily pa ame e iza ion. POVM case by
Naima k dila ion o PVM hen pushback.
7 Poin e Basis and Ky Fan Minimum
Theo em 7.1 (Poin e Basis = Spec al Minimum).Fo sel -adjoin window ope a o WR
and any m-dimensional o hogonal amily {ek},
m
X
k=1
⟨ek, WRek⟩ ≥
m
X
k=1
λ↑
k(WR),
equali y i and only i {ek}spans minimal eigensubspace o WR(Ky Fan minimum sum).
P oo . S anda d Ky Fan a ia ional p inciple (PNAS 1951).
8 Non-Asymp o ic E o Closu e: NPE Decomposi ion
Theo em 8.1 (Nyquis –Poisson–EM Th ee-Te m Decomposi ion).Fo ene gy-domain in e-
g al I=RRF(E)dE whe e F=wR·(h∗ρ⋆), unde :

Bandlimi ed: supp b
F⊂[−ΩF,ΩF];

Smoo hness: F∈C2M(R),F(2M)∈L1(R);

Sampling: s ep ∆>0, unca ion |n| ≤ N;
ha e disc e iza ion app oxima ion
I= ∆
N
X
n=−N
F(n∆) + εalias
|{z}
Poisson
+R2M
|{z}
EM
+ε ail
|{z}
unca ion
,
whe e alias e m εalias = 0 when ∆≤π/ΩF(Nyquis ), EM emainde |R2M| ≤ 2ζ(2M)
(2π)2MR|F(2M)|,
ail |ε ail| ≤ R|E|>N∆|F|.
P oo . Apply Poisson summa ion: o bandlimi ed F, eplicas a k= 0 all ou side suppo
when Nyquis sa is ied. Apply 2M-o de Eule –Maclau in o ini e sum, ob aining Be noulli
co ec ions and explici emainde . Tail om unca ion.
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9 Mul i-Window F ames and Wexle –Raz
Theo em 9.1 (Wexle –Raz Bio hogonali y o Mul i-Window).Fo Gabo ame wi h ime-
equency la ice (α, β)sa is ying αβ ≤1, window gand dual window egsa is y Wexle –Raz
bio hogonali y ela ion:
X
n∈Z
g( −nα)eg( −nα)e2πimβ =1
βδm,0,∀m∈Z,a.e. .
Equi alen ly in equency domain:
X
k∈Zbg(ξ−k/α)b
eg(ξ−k/α) = α, a.e. ξ.
P oo . S anda d esul om Gabo analysis (Daubechies–Landau–Landau 1995). Follows
om Poisson summa ion and ame ope a o p ope ies.
10 Discussion and Ou look
This wo k es ablishes igo ous ma hema ical ounda ions o WSIG-QFT:
1. Weyl–Heisenbe g kinema ic amewo k wi h mi o symme y
2. Phase–densi y–delay uni ica ion ia Bi man–K e˘ın and Wigne –Smi h
3. Bo n p obabili y as I-p ojec ion minimizing KL-di e gence
4. Poin e basis as Ky Fan spec al minimum
5. Non-asymp o ic e o closu e ia NPE decomposi ion
6. Mul i-window ame syne gy ia Wexle –Raz bio hogonali y
Key o mulas:

Scale iden i y: 1
2π Q=ξ′= (ρ−ρ0)

Windowed BK: RFξ′=−1
2πi RF′log de S

NPE e o : |ε| ≤ |εalias|+|R2M|+|ε ail|
Fu u e di ec ions:

Ex ension o quan um ield heo y and eno maliza ion

Connec ions o holog aphy and AdS/CFT

Nume ical implemen a ion and benchma king

Applica ions o quan um many-body sys ems
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