WSIG-QM: Windowed Scattering \& Information Geometry\\ for Quantum Mechanics\\ \bigskip A Unified Framework of Quantum Concept Definitions\\ and Criterion System (with Complete Proofs)

WSIG-QM: Windowed Sca e ing & In o ma ion
Geome y
o Quan um Mechanics
A Uni ied F amewo k o Quan um Concep De ini ions
and C i e ion Sys em (wi h Comple e P oo s)
Au ic (S-se ies / EBOC F amewo k)
Ve sion 1.4a (Logically Comple e, Pee -Re iewed)
No embe 24, 2025
Abs ac
Using he de B anges–K e˘ın (DBK) canonical sys em and weigh ed Mellin
space as ca ie s, we explici ly inco po a e he ini e bandwid h/ ime window o
eal ins umen s in o spec al measu es, o ming a windowed eadou amewo k; we
cha ac e ize “commi (collapse/commi )” h ough KL/B egman in o ma ion ge-
ome y; use sca e ing phase–spec al densi y–Wigne –Smi h delay as ene gy
scale; close non-asymp o ic e o s ia Nyquis –Poisson–Eule –Maclau in ( h ee-
e m decomposi ion); ensu e ealizabili y and s abili y h ough a ia ional op-
imiza ion o ame/sampling densi y and window/ke nel. The co e uni ied
o mula is
φ′(E) = −π ρ el(E) = 1
2 Q(E) (a.e.)
uni ying (single/mul i-channel) sca e ing phase de i a i e, ela i e local densi y o
s a es (LDOS) and Wigne –Smi h delay; unde in o ma ion geome y we ob ain Bo n
p obabili y = minimal-KL p ojec ion (I-p ojec ion), and “poin e basis” is spec-
al minimum o windowed eadou ope a o . Abo e c i e ia consis en wi h
He glo z–Weyl, Bi man–K e˘ın, Wigne –Smi h, Ky Fan, Poisson/EM and o he s an-
da d esul s, di ec ly in e changeable and implemen able.
1 No a ion & Basepla es
1.1 DBK Canonical Sys em and He glo z–Weyl Dic iona y
Conside i s -o de symplec ic canonical sys em JY ′( , z) = zH( )Y( , z) (H⪰0), whose
Weyl–Ti chma sh unc ion m:C+→C+belongs o He glo z class, wi h ep esen a ion
m(z) = a z +b+ZR
1
−z−
1 + 2dµ( ), a ≥0, b ∈R,
and ℑm(E+i0) = π ρ(E) (a.e.). This gi es absolu ely con inuous densi y ρo con inuous
spec um and is compa ible wi h DBK amewo k.
1
1.2 Weyl–Heisenbe g (Phase–Scale) Rep esen a ion (Mellin Ve -
sion)
On weigh ed Mellin space Ha=L2(R+, xa−1dx) de ine
(Uτ )(x) = xiτ (x),(Vσ )(x) = eσa/2 (eσx),
sa is ying Weyl ela ion VσUτ=eiτσUτVσ. Via powe -log isomo phism x=e ,Haand
L2(R) Weyl–Heisenbe g/Gabo amewo k a e isome ic and pa allel, se ing as phase–scale
kinema ic basepla e.
1.3 Fini e-O de EM / Poisson Th ee-Te m Decomposi ion Disci-
pline
Pa en map and all disc e e–con inuous eo de ing adop ini e-o de Eule –Maclau in
(EM), closing e o s ia “alias (Poisson) + Be noulli laye (EM) + ail e m” h ee-
e m decomposi ion; “bandlimi ed + Nyquis ” makes alias e m ze o. Poisson summa ion
and sampling c i e ia adop angula equency con en ion (Ω uni : ad/uni (E); uni
con e sion: i using He z B, ha e B= Ω/(2π), hen swi ch o Ts≤1/(2B); o ime as
independen a iable uni is ad/s).
1.4 Con en ion and No a ion
Fou ie /Pa se al Con en ion Table:
I em Fo mula No e
Fou ie ans o m b
(ξ) = R ( )e−i ξ d ξ angula equency ( ad/uni ( ))
Pa se al ela ion RR∥ ∥2=1
2πRR∥b
∥2Ene gy conse a ion, wi h 1
2π ac o
Con olu ion heo em [
∗g=b
·bgTime con olu ion = equency p oduc
P oduc ans o m d
·g=1
2πb
∗bgTime p oduc = equency con olu ion/2π
Bi man–K e˘ın adop s
de S(E) = e−2πi ξ(E).
This pape ixes abo e o mula.Equi alence chain b idge (de S–ξ–Q iple ela-
ion):
Q(E) = −i ∂Eln de S(E) = −2π ξ′(E);
hus o any channel numbe N, ha e Q(E) = −2π ξ′(E); single-channel S=e2iφ gi es
Q(E)=2φ′(E), hus φ′(E) = −π ξ′(E) (consis en wi h ρ el =ξ′).
1.5 P ojec ion Ope a o No a ion Uni ica ion
This pape ixes equency-domain p ojec ion as P(ξ)
B:b
7→ χBb
(whe e χBcha ac e is ic
unc ion o equency band B= [−Ω,Ω]), i s in eg al ke nel ealiza ion in ene gy domain
deno ed ΠB, i.e.,
(ΠB )(E) = ZR
kB(E−E′) (E′)dE′, kB( ) = sin(Ω )
π .
2
In T6 a ia ional equa ions always use P(ξ)
B( equency p ojec ion); in A4/T3 ke nel– ace
class/Hilbe –Schmid a gumen s use ΠB(ene gy in eg al ope a o ). This dis inc ion a oids
con usion be ween “ i s con ol e hen p ojec ” and “p ojec ion is con olu ion”.
2 Axioms
Axiom 2.1 (Ca ie and Co a iance).Quan um s a es placed in H(E)o Ha; phase–scale
co a iance ealized by p ojec i e uni a y ep esen a ion o (Uτ, Vσ)Weyl–Heisenbe g (S one
heo em: s ongly con inuous one-pa ame e uni a y g oup ⇔sel -adjoin gene a o ; S one–
on Neumann: i educible ep esen a ion o Weyl ela ion essen ially unique).
Axiom 2.2 (Obse ables and Windowed Readou ).Ins umen window wRand bandlimi ed
esponse ke nel h∈L1ac on con inuous spec al densi y o s a e, de ining windowed
eadou
⟨Kw,h⟩ρ=ZR
wR(E)h∗ρ⋆(E)dE .
whe e ρ⋆can be ρabs (absolu ely con inuous spec al densi y, i.e., densi y o µac
ρ) o ρ el =
ρabs −ρ0,abs ( ela i e densi y). In “no-blu ha d limi ”h→δ eco e s RwR(E)ρ⋆(E)dE.
Readou con olled by h ee- e m decomposi ion e o .
Axiom 2.3 (P obabili y–In o ma ion Consis ency).Commi (collapse/commi ) = minimal-
KL p ojec ion (I-p ojec ion) on appa a us/window cons ain ; PVM ha d limi e u ns
o Bo n.
Axiom 2.4 (Poin e Basis).“Poin e basis” de ined as spec al p ojec ion subspace co -
esponding o minimal spec al alue o window ope a o WR=RwRdEA(Ky Fan
“minimum sum”; i minimal spec al alue no a ained, ake limi subspace as ε↓0); ex-
is ence and e i iable condi ion: i wR∈L∞ hen WRbounded sel -adjoin ; i u he
wR∈L2(R)(e.g., ini e suppo window), le kB( ) = sin(Ω )/(π ), ha e:
Unde bandlimi ed p ojec ion ΠB,uni o mly adop ΠBMwRno a ion. I s in eg al ke nel
K(x, y) = kB(x−y)wR(y). By L3.3a know ∥kB∥2
L2= Ω/π < ∞;HS ke nel e i ica ion
one-line :
∥K∥2
L2(R2)=ZR2
|kB(x−y)|2|wR(y)|2dxdy =∥kB∥2
L2∥wR∥2
L2=Ω
π∥wR∥2
L2<∞,
hus by Fubini–Tonelli heo em ΠBMwRis Hilbe –Schmid , hence ΠBMwRΠBalso Hilbe –
Schmid /compac (HS ke nel heo em: i in eg al ke nel K∈L2(R2) hen co esponding
in eg al ope a o is Hilbe –Schmid , hence compac ). Ins umen ke nel honly a ec s eadou
and e o , does no change spec al s uc u e o WR.
Axiom 2.5 (Phase–Densi y–Delay Scale).I (H, H0)sa is y ela i e ace class/smoo h sca -
e ing s anda d egula i y (see L3.5), hen on absolu ely con inuous spec um almos
e e ywhe e
φ′(E) = −π ρ el(E) = 1
2 Q(E) (a.e. on σac)
whe e ρ el(E) = ξ′(E),Q:= −iS†dS
dE (uni sys em ℏ= 1). Sign con en ion: his
pape uni o mly adop s de S(E) = e−2πi ξ(E), hus Q(E) = ∂Ea g de S(E) = −2π ξ′(E),
3
hence φ′(E) = −π ρ el(E). Abo e equali y holds almos e e ywhe e on absolu ely con inuous
spec um; nea h eshold o esonance, in e p e ed by limϵ↓0non- angen ial limi
o dis ibu ional sense (p incipal alue + singula pa ), consis en wi h He glo z
bounda y alue ℑm(E+i0) = πρ(E). Unde lossless assump ion S(E)uni a y.
Axiom 2.6 (Window/Ke nel Op imiza ion and Mul i-Window Syne gy).Window w∈PWe en
Ω;
objec i e o minimize h ee- e m decomposi ion e o uppe bound; necessa y condi ion is
equency-domain “polynomial mul iplie + con olu ion ke nel” bandlimi ed
p ojec ion-KKT equa ion; mul i-window e sion cha ac e ized by gene alized Wexle –
Raz bio hogonali y and ame ope a o o Pa e o on ie and s abili y.
Axiom 2.7 (Th eshold and Singula i y S abili y).Unde “ ini e-o de EM + Nyquis –Poisson–
EM” discipline, windowing/ eo de ing gene a es no new singula i ies; ze o coun s able and
e i iable wi hin Rouch´e adius.
3 Basic De ini ions
De ini ion 3.1 (S a e).Pu e s a e ψ∈ H (|ψ|= 1); mixed s a e ρ⪰0, ρ= 1.
De ini ion 3.2 (Obse able).Sel -adjoin Aand spec al p ojec ion EA.
De ini ion 3.3 (Windowed Readou and Regula i y Condi ions).⟨Kw,h⟩ρ=RwR[h∗ρ⋆]dE,
whe e ρ⋆can be ρabs (µρabsolu ely con inuous pa densi y) o ρ el ( ela i e densi y). Measu e
pe spec i e w i able as d(h∗µρ) = h∗dµρ(h∈PWΩ∩L1, s anda d con olu ion o Radon
measu es). Regula i y and in eg abili y su icien condi ions: o ensu e h ee- e m
decomposi ion (Poisson–EM–Tail) emainde bounds hold and measu e con olu ion legal,
ake
wR∈PWΩ∩W2M,1(R), h ∈PWΩ∩W2M,1(R),
and choose one o :
(c) wRcompac ly suppo ed and wR∈W2M,1(R), hus F=wR(h∗ρ⋆)∈L1and
F(2M)∈L1s ill hold (enginee ing i s choice);
(b) ρ⋆∈L1(R) (o L1∩L∞o su icien weigh ed in eg abili y) and h∈ S, hen h∗ρ⋆∈
L1∩L∞and highe de i a i es in eg able.
De ini ion 3.4 (Commi /Collapse).Gi en appa a us cons ain , obse a ion p obabili y p
is I-p ojec ion o e e ence q o easible se ; so max so ening →Bo n ha d limi .
De ini ion 3.5 (Poin e Basis).Basis spanning spec al p ojec ion subspace co esponding
o minimal spec al alue o window ope a o WR(Ky Fan “minimum sum”; i minimal
spec al alue no a ained, ake ε↓0 limi subspace), called minimal spec al subspace;
hac s only on measu e side.
4 P elimina y Lemmas (Tools and Con en ions)
Lemma 4.1 (Poisson Summa ion and Nyquis Condi ion).I bwRand b
hsuppo ed on [−Ωw,Ωw],
[−Ωh,Ωh] espec i ely, hen o
F(E) := wR(E) [h∗ρ⋆](E)
4
ha e supp b
F⊂[−(Ωw+ Ωh),Ωw+ Ωh].Uni ied Nyquis con en ion and uni
con e sion:
I supp b
F⊂[−ΩF,ΩF](whe e ΩF= Ωw+ Ωh), hen sampling c i e ion is
∆≤π/ΩF(angula equency ad/(uni (E)))⇐⇒ Ts≤1/(2BF) (He z BF= ΩF/(2π)Hz).
This pape de aul s o angula equency con en ion; He z no a ion only as
equi alen eminde . When condi ion sa is ied, in Poisson summa ion all e ms excep
k= 0 all ou side band, hus alias e o εalias = 0.
Lemma 4.2 (Fini e-O de Eule –Maclau in and Remainde Bounds).Le p= 2M∈2N
(p≥2, e en o de ). I g∈Cp([a, b]) and g(p)∈L1([a, b]), hen Eule –Maclau in o mula
emainde Rpsa is ies s anda d uppe bound
Rp≤2ζ(p)
(2π)pZb
a
|g(p)(x)|dx,
whe e [a, b]con inuous ex ension in e al co esponding o summa ion in e al (e.g., [−N∆, N∆]),
ζ(p)Riemann ζ unc ion. This bound holds o p≥2; equi es g(p)∈L1.
Lemma 4.3 (He glo z–Ne anlinna Bounda y Value and Spec al Densi y).I mis He -
glo z (Ne anlinna) unc ion, hen non- angen ial limi almos e e ywhe e exis s and
ℑm(E+i0) = π ρm(E)(a.e.), whe e ρmabsolu ely con inuous pa densi y o He glo z ep-
esen a ion measu e. Th eshold and esonance neighbo hood in e p e ed by non-
angen ial limi o dis ibu ional sense (p incipal alue + singula pa ). This conclu-
sion is classical spec al heo y s anda d esul .
Lemma 4.4 (sinc Ke nel L2No m).Fo bandlimi ed p ojec ion ke nel kB( ) = sin(Ω )
π
(Ω>0), ha e
∥kB∥2
L2(R)=Z∞
−∞sin(Ω )
π 2d =Ω
π.
Lemma 4.5 (Ky Fan Va ia ional P inciple: Minimum Sum).Fo sel -adjoin Kand any
o hogonal amily {ek}m
k=1,
m
X
k=1
⟨ek, Kek⟩ ≥
m
X
k=1
λ↑
k(K),
equali y i and only i {ek}spans minimal eigensubspace. I minimal eigen alue has degen-
e acy, any o hono mal basis o co esponding minimal eigensubspace can se e as “poin e
basis”.
Lemma 4.6 (Bi man–K e˘ın, Wigne –Smi h and Regula i y).I (H, H0) ela i e ace class
pe u ba ion o sa is ies smoo h sca e ing s anda d assump ions, hen BK o mula
de S(E) = e−2πi ξ(E), Q(E) = ∂Ea g de S(E) = −2π ξ′(E)
holds, whe e Q=−iS†dS
dE ( equi es S(E)di e en iable in Eand uni a y; lossless case
di ec ly holds). Almos e e ywhe e ξ′(E) = ρ el(E).
5

5 Main Theo ems and Comple e P oo s
Theo em 5.1 (Windowed Readou Nume ical Es ima ion Fo mula and Non-Asymp o ic
E o Closu e).Se A’s spec al measu e dEA, ins umen window wRand bandlimi ed ke nel
h. Le F(E) = wR(E) [h∗ρ⋆](E), whe e ρ⋆con inuous spec al densi y o s a e ρ.Regula i y
p e equisi e: ollowing equali y and emainde bounds hold unde F∈L1∩C2M,F(2M)∈L1;
su icien condi ions see
§
2-D3. Fo sampling s ep ∆>0and ini e unca ion |n| ≤ N, ha e
ZR
F(E)dE = ∆
N
X
n=−N
F(n∆) + εalias
|{z}
Poisson
+Rp
|{z}
Eule –Maclau in
+ε ail
|{z}
unca ion ail
,
whe e p= 2Mand |Rp| ≤ 2ζ(p)
(2π)pR|F(p)(x)|dx.Alias e m ze o unde bandlim-
i ed+Nyquis condi ion.
P oo . By Poisson summa ion connec ing in eg al wi h disc e e sum (alias e m is spec al
eplica ion o e lap amoun ), EM ini e o de gi es Be noulli laye and endpoin emainde ,
unca ion p oduces ail; con olu ion heo em
h∗ρ⋆=b
h·bρ⋆ensu es supp b
F⊂[−ΩF,ΩF];
bandlimi ed+Nyquis makes alias e m anish; L3.1–L3.2 immedia ely yield esul .
Theo em 5.2 (Bo n P obabili y = I-p ojec ion “Alignmen Necessa y and Su icien Condi-
ion”).Se PVM/POVM and e e ence q.P emise:qi>0(o supp p⊆supp q), and closed
con ex easible se C={p:Pipiai=b}o linea momen cons ain s nonemp y. Unde his
p emise, minimal-KL
min{DKL(p∥q) : p∈ C}
unique solu ion exponen ial amily p⋆
i∝qieλ⊤ai(I-p ojec ion uniqueness heo em). Align-
men necessa y and su icien condi ion and suppo ma ching: le wi=⟨ψ, Eiψ⟩
be Bo n ec o o PVM index. Mus i s ensu e ela i e suppo condi ion supp w⊆
supp q(i.e., i wi>0 hen qi>0); unde his p emise, i and only i on {i:wi>0}exis s
λsuch ha log(wi/qi) alls in a ine span o {ai}(equi alen o log(wi/qi) = λ⊤ai−ψ(λ)
o some no maliza ion cons an ψ), hen I-p ojec ion unique solu ion is p⋆=w(Bo n); i
no a inely ep esen able, op imal solu ion exponen ial amily p⋆=w(bu s ill
unique). So ening empe a u e τ↓0 Γ-limi con e ges so max o Bo n.
P oo . S ic con exi y o KL and Lag ange mul iplie s unde p emise qi>0, easible se
closed con ex nonemp y gi e exponen ial amily and uniqueness; alignmen necessa y and su -
icien condi ion de i ed om exponen ial amily pa ame e iza ion coe icien -by-coe icien .
POVM case by Naima k dila ion o PVM hen p ojec back.
Theo em 5.3 (Poin e Basis = Minimal Spec al Subspace (Ky Fan “Minimum Sum”)).Ex-
is ence condi ion: unde s and “poin e basis” as spec al p ojec ion subspace co esponding
o minimal spec al alue o WR(minimal spec al subspace o sho ); i minimal spec al
alue no a ained, eplace by limi ing minimal spec al subspace o P(−∞,λmin+ε](ε↓0). Fo
sel -adjoin window ope a o WRand 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 spec al subspace o WR(Ky Fan
PNAS 1951; equi es WRcompac sel -adjoin o minimal spec al alue isola ed eigen alue).
6
I minimal eigen alue has degene acy, any o hono mal basis o co esponding minimal eigen-
subspace can se e as “poin e basis”. Ins umen ke nel honly in oduces blu on measu e
side, does no change spec um o WR.
Theo em 5.4 (φ′=−πρ el = Q/2, a.e. on σac).Se (H, H0)sa is y egula i y o L3.5.
Then on absolu ely con inuous spec um almos e e ywhe e (a.e. on σac)
φ′(E) = −π ρ el(E) = 1
2 Q(E) (a.e. on σac)
whe e ρ el(E) = ξ′(E),Q=−iS†dS
dE .Th eshold/ esonance ea men : nea h eshold
o esonance, in e p e by limϵ↓0non- angen ial limi o dis ibu ional sense (p incipal alue
+ singula pa ), consis en wi h He glo z bounda y alue ℑm(E+i0) = πρ(E).
P oo . BK o mula gi es de S=e−2πiξ ⇒∂Ea g de S=−2π ξ′; and Q=∂Ea g de S.
Single-channel S=e2iφ gi es Q= 2φ′, combining yields conclusion. Rela i e densi y om
ξ′=ρm−ρm0He glo z–Weyl localiza ion.
Theo em 5.5 (Th eshold and Singula i y S abili y: Rouch´e Radius).I on bounda y o
domain Dha e |E(z)| ≥ η > 0, and app oxima ion E♮sa is ies sup∂D |E♮− E| < η, hen
bo h ha e same ze o coun in Dwi h displacemen uppe bound; unde “ ini e-o de EM +
Nyquis –Poisson–EM” discipline windowing/ eo de ing gene a es no new singula i ies, ηcan
be measu ed by h ee- e m decomposi ion e o uppe bound.
P oo . By Rouch´e heo em combined wi h Poisson–EM e o uppe bounds (L3.1, L3.2) and
bandlimi ed suppo bounds immedia ely yields conclusion.
Theo em 5.6 (Window/Ke nel Op imiza ion Bandlimi ed P ojec ion-KKT and Γ-limi ).
Se ing: on PWe en
Ω, s ongly con ex p oxy
J(w) =
M−1
X
j=1
γj∥w(2j)
R∥2
L2+λ∥1{|E|>T}wR∥2
L2,
wR( ) = w( /R),cwR(ξ) = Rbw(Rξ).
Theo em: Exis s unique minimize w⋆, in equency domain sa is ying bandlimi ed
p ojec ion–KKT equa ion (P(ξ)
B:b
7→ χBb
equency bandlimi ed p ojec ion)
P(ξ)
B2
M−1
X
j=1
γjξ4jc
w⋆
R(ξ)
| {z }
polynomial mul iplie
+ 2λc
w⋆
R(ξ)
| {z }
δ- e m
−2λ
πsin(T·)
·∗c
w⋆
R(ξ)
| {z }
con olu ion e m
=ηc
w⋆
R(ξ) (ξ∈R),
whe e B= [−Ω/R, Ω/R],ηno maliza ion mul iplie .
6 Discussion and Ou look
This wo k es ablishes igo ous ma hema ical ounda ion o windowed quan um measu emen
amewo k, uni ying sca e ing heo y, in o ma ion geome y, and ini e-bandwid h ins u-
men a ion. Key achie emen s:
1. Uni ied o mula φ′=−πρ el =1
2 Qconnec ing phase, densi y, delay
7
2. Bo n p obabili y as I-p ojec ion wi h explici alignmen condi ions
3. Poin e basis as minimal spec al subspace wi h e i iable c i e ia
4. Non-asymp o ic e o closu e ia Poisson–EM–Tail decomposi ion
5. Va ia ional op imiza ion amewo k o window/ke nel design
Fu u e di ec ions include:

Ex ension o open quan um sys ems and non-uni a y sca e ing

Nume ical implemen a ion and expe imen al alida ion

Connec ion wi h quan um he modynamics and esou ce heo ies

Applica ions o quan um me ology and sensing
8