Windowed Path Integrals:\\ Spectral ``Window--Kernel'' Formulation\\ and Rigorous Equivalence to Propagators

Windowed Pa h In eg als:
Spec al “Window–Ke nel” Fo mula ion
and Rigo ous Equi alence o P opaga o s
Au ic (S-se ies / EBOC F amewo k)
Ve sion 0.8.1
·
Oc obe 28, 2025
No embe 24, 2025
Abs ac
Unde WSIG-QM amewo k composed o de B anges–K e˘ın (DBK) canon-
ical sys em and Weyl–Heisenbe g (including loga i hmic/Mellin) ep esen a-
ion, his pape akes spec al heo em + analy ic Fou ie duali y as main h ead,
gi ing igo ous ma hema ical cha ac e iza ion o pa h in eg al = p opaga o ke -
nel, p o ing windowed pa h in eg al heo em: any ealizable pa h in eg al- ype
obse a ion equi alen o “window–ke nel–densi y” con olu ion in ene gy domain; ime
domain p ecisely p opaga o ime ace (o s a e-weigh ed ke nel) Fou ie dual un-
de same window/ke nel. Fo nume ical implemen a ion, disc e iza ion e o non-
asymp o ically closes as “alias (Poisson) + Be noulli laye (Eule –Maclau in)
+ unca ion” h ee- e m decomposi ion; unde bandlimi ed + Nyquis condi ions
alias e m s ic ly ze o. Fo phase scale, on absolu ely con inuous spec um almos e -
e ywhe e holds
φ′(E) = 1
2 Q(E), ρ el(E) = sBK
2π Q(E), φ(E) = sBK π ξ(E) (mod π),
whe e Q(E) = −i S†(E)dS
dE (E) is Wigne –Smi h delay ma ix, ρ el =ξ′spec al
shi densi y, sBK BK no a ion e sion pa ame e ( his pape adop s sBK = +1);
his gi en by Bi man–K e˘ın o mula and ela i e sca e ing delay uni ica ion, closing
pa h weigh ac ion phase wi h measu able ene gy scale uni ied. On in o ma ion
geome y side, Bo n p obabili y = minimal-KL (I-p ojec ion) gi es log-sum-
exp so po en ial con ex dual seman ics; single-window and mul i-window syne gy o
window/ke nel exp essible as s ongly con ex/spa se op imiza ion in e acing wi h
ame–dual window heo y. All abo e ancho s anda d c i e ia: spec al heo em and
S one heo em, Bi man–K e˘ın o mula, Wigne –Smi h delay, Poisson summa ion and
Eule –Maclau in o mula, Nyquis –Shannon sampling, Wexle –Raz bio hogonali y and
“painless” expansion e c.
1 No a ion and Con en ions
1.1 Fou ie Con en ion
Take
1
b
(ξ) = ZR
(x)e−ixξ dx, (x) = 1
2πZRb
(ξ)eixξ dξ,
using Pa se al (ze o- equency equali y and Planche el join ly): Z g =1
2πZb
bg.
Quick e e ence ca d: Unde his con en ion,
e+iE 0(ξ) = 2πδ(ξ− 0),
e−iE 0(ξ) =
2πδ(ξ+ 0); scaling wR(E) = w(E/R) gi es bwR(ξ) = Rbw(Rξ) (ampli ude ac o R, suppo
sh inks o 1/R imes). Angula equency Ω co esponds o ime bandwid h Ω ( his pape
uni o mly akes his con en ion, di e en om some li e a u e’s 2πplacemen ).
1.2 Dimensions and Cons an s
Uni o mly ake ℏ= 1; when eco e ing subs i u e 7→ /ℏ.
1.3 Spec um and P opaga o
Hsel -adjoin ope a o , EHi s spec al measu e. Fo any ace class ope a o ρ∈S1(H)
(whe e s a e weigh means ρ≥0, obse able weigh means sign- ini e ace class ope a o
wi h T ρ= 0), de ine
Kρ( ) := T ρ e−iH =ZR
e−iE dνρ(E), νρ(B) := T ρ EH(B).
Unde his assump ion, Kρ( ) well-de ined and is con inuous bounded unc ion.
I absolu ely con inuous pa o νρhas densi y ρabs(E), i s con ibu ion sa is ies (dis ibu-
ional sense) dρabs( ) = RRe−iE ρabs(E)dE. Gene ally, Kρ( ) = dρabs( ) + dνsing( ); i and only
i νρpu ely absolu ely con inuous, ha e Kρ=dρabs. This om spec al heo em and S one
heo em cha ac e iza ion o e−i H .
1.4 Window and Ke nel
Take e en window wR(E) = w(E/R), whe e w∈PWe en
Ω(Paley–Wiene e en unc ion class
o bandwid h Ω), hen cwR(ξ) = Rbw(Rξ) also e en unc ion suppo ed on [−Ω/R, Ω/R].
Tes ke nel h∈W2M,1(R)∩L1(R) (no e enness equi emen , bandlimi ed i necessa y),
ensu ing con olu ion and eo de ing.
1.5 Phase–Densi y–Delay Scale
Se sca e ing ma ix ela i e o e e ence H0as S(E) (single/mul i-channel). This pape
ixes Bi man–K e˘ın no a ion
de S(E) = e+2πi ξ(E)(a.e. E),
in oducing Wigne –Smi h delay ma ix. Dimension and ℏuni ica ion: De ine
Qℏ(E) := −iℏS†(E)∂ES(E),Q(E) := 1
ℏQℏ(E) = −i S†(E)∂ES(E).
Then o any a.e. di e en iable sca e ing ene gy E,
Qℏ(E)=2ℏφ′(E) = 2πℏξ′(E), ρ el(E) = ξ′(E) = 1
2πℏ Qℏ(E).
Th oughou ex ake ℏ= 1, de aul ing Q=Qℏ/ℏ, hus
2
ξ′(E) = 1
2π Q(E), ρ el(E) := ξ′(E) = 1
2π Q(E) (spec al shi densi y).
De ine o al phase φ(E) := 1
2a g de S(E), choosing con inuous b anch consis en wi h
BK no a ion, no malizing ξ o anish a e e ence ene gy egion, making absolu e alue o
ξ(E) physically measu able. Then
φ′(E) = 1
2 Q(E), φ(E) = sBK π ξ(E) (mod π),
whe e sBK = +1 co esponds o his pape ’s e sion I no a ion (de S=e+2πiξ). Thus
ρ el(E) = ξ′(E) = sBK
2π Q(E).
2 Pa h In eg als and Spec al Window/Ke nel Dic io-
na y
P opaga o ke nel in posi ion eigenbasis
K(x , ;xi,0) = ⟨x |e−iH |xi⟩=ZR
e−iE dµx ,xi(E),
whe e µx ,xico esponding spec al S iel jes measu e. Fo mal Feynman pa h in eg al
p ecisely ano he ep esen a ion o his ke nel (consis en wi h ke nel in igo ous amewo k).
The e o e, choosing “window” wR(E) = e−iE 0and “ke nel” h=δ(gene alized unc ion
sense), ime p opaga o K(x , 0;xi,0) special case o ene gy-side windowed eadou ; h=δ
co esponds o ene gy smoo hing, ime domain mul iplying by b
h.
In WSIG-QM con ex , his equi alen o: all measu able pa h in eg al- ype obse -
a ions = ene gy-side “window–ke nel–densi y” eadou s; ime side p opaga o ime
ace/ke nel Fou ie dual unde same window/ke nel.
3 Windowed Pa h In eg al Theo em: Ene gy–Time Dual
Rep esen a ion
Assump ion 3.1 (Reo de ing and In eg abili y P emise).To make Theo em 3.2 Fou ie
duali y and eo de ing igo ously alid, assume:
(A1) Spec al densi y egula i y: ρ⋆ ini e signed Bo el measu e;
(A2) Window unc ion egula i y: wR∈L∞(R)∩C2M(R)e en unc ion, Paley–Wiene
class PWe en
Ω;
(A3) Ke nel unc ion egula i y: h∈W2M,1(R)∩L1(R), ensu ing h∗ρ⋆well-de ined
dis ibu ionally;
(A4) Fubini/Tonelli in e changeabili y: Unde abo e condi ions, h∗ρ⋆∈L1(R)and
wR·(h∗ρ⋆)∈L1(R);
(A5) S iel jes/dis ibu ional duali y: When ρ⋆=νρspec al measu e, Kρ⋆( ) = T (ρe−iH )
gua an eed con inuous bounded by S one heo em;
3
(A6) Time-side EM smoo hness (op ional): Fo 2M-o de Eule –Maclau in co ec ion
ime-side, equi e G ∈C2M([−T, T]).
Theo em 3.2 (Windowed Pa h In eg al Duali y).Unde Assump ion 3.1, o sel -adjoin
H, spec al measu e EH, spec al densi y ρ⋆, window wR∈PWe en
Ω, ke nel h∈W2M,1∩L1,
ha e ene gy– ime dual iden i ies:
Ene gy-domain iden i y:
ZR
wR(E) [h∗ρ⋆](E)dE =ZR
wR(E)ZR
h(E−E′)ρ⋆(E′)dE′dE
Time-domain Fou ie dual:
=1
2πZRcwR(− )b
h( )Kρ⋆( )d ,
whe e Kρ⋆( ) = RRe−iE ρ⋆(E)dE p opaga o ime ace/ke nel.
When ρ⋆=νρ om ace class ρ, ha e Kρ⋆( ) = T (ρe−iH ).
P oo . By spec al heo em, S one heo em and Pa se al iden i y. De ine G(E) := wR(E) [h∗
ρ⋆](E). Unde assump ions ha e G∈L1(R). Apply Fou ie ans o m:
b
G( ) = ZR
wR(E) [h∗ρ⋆](E)e−iE dE.
By con olu ion heo em
h∗ρ⋆=b
h·bρ⋆. By p oduc -con olu ion duali y:
b
G( ) = 1
2πcwR∗(b
h·bρ⋆)( ) = 1
2πZRcwR( −s)b
h(s)bρ⋆(s)ds.
Change a iable s→ −sand use wRe enness (cwRe en), ge ime-domain iden i y.
4 Phase Scale Uni ica ion
Theo em 4.1 (Sca e ing Phase–Densi y–Delay Scale Iden i y).Unde sca e ing egula i y
( ela i e ace class o Hilbe –Schmid , making S(E)a.e. di e en iable and BK o mula
applicable), on absolu ely con inuous spec um a.e. ha e:
φ′(E) = 1
2 Q(E), ξ′(E) = sBK
2π Q(E), ρ el(E) = ξ′(E),
whe e Q(E) = −i S†(E)∂ES(E)Wigne –Smi h delay ma ix, sBK ∈ {+1,−1}BK no a-
ion e sion pa ame e , ρ el spec al shi densi y.
Fo BK e sion I (de S=e+2πiξ,sBK = +1), ha e unc ion-le el equali y:
φ(E) = π ξ(E), ρ el(E) = 1
2π Q(E).
P oo . F om Bi man–K e˘ın o mula de S(E) = esBK·2πiξ(E), aking loga i hmic de i a i e:
d
dE ln de S(E) = (S−1∂ES) = (S†∂ES) = sBK ·2πi ξ′(E).
By de ini ion Q=−iS†∂ES, hus Q=i (S†∂ES) = sBK ·2π ξ′(E).
Fo o al phase φ=1
2a g de S=sBK ·πξ (mod π), di e en ia ing gi es φ′=1
2 Q.
Spec al shi densi y de ini ion ρ el := ξ′comple es chain.
4
5 Non-Asymp o ic E o Closu e
Theo em 5.1 (Poisson–EM–Tail Th ee-Te m Decomposi ion).Fo ene gy-domain in eg al
I=RRF(E)dE whe e F=wR·(h∗ρ⋆), unde :

Bandlimi ed: supp b
F⊂[−ΩF,ΩF]whe e ΩF= Ωw/R + Ωh;

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 wi h e o decomposi ion:
I= ∆
N
X
n=−N
F(n∆) + εalias
|{z}
Poisson
+R2M
|{z}
EM emainde
+ε ail
|{z}
unca ion
,
whe e:
1. Alias e m: εalias = 0 when ∆≤π/ΩF(Nyquis );
2. EM emainde : |R2M| ≤ 2ζ(2M)
(2π)2MRR|F(2M)(x)|dx;
3. Tail e m: |ε ail| ≤ R|E|>N∆|F(E)|dE.
P oo . Apply Poisson summa ion o mula: o Fbandlimi ed wi h supp b
F⊂[−ΩF,ΩF],
X
n∈Z
F(n∆) = 2π
∆X
k∈Zb
F2πk
∆.
When ∆ ≤π/ΩF, eplicas a k= 0 all ou side suppo o b
F, hus alias anishes. Apply
2M-o de Eule –Maclau in o ini e sum P|n|≤N, ob aining Be noulli co ec ion e ms and
explici emainde bound. Tail e m om unca ion a ±N.
6 Discussion and Ou look
This wo k es ablishes:
1. Rigo ous equi alence be ween pa h in eg als and windowed spec al eadou s ia ene gy–
ime Fou ie duali y
2. Phase–densi y–delay uni ica ion h ough Bi man–K e˘ın o mula
3. Non-asymp o ic e o closu e ia Poisson–EM– ail h ee- e m decomposi ion
4. Nyquis sampling c i e ion o alias elimina ion
Key o mulas:

Ene gy– ime duali y: RwR(E)[h∗ρ⋆](E)dE =1
2πRcwR(− )b
h( )Kρ⋆( )d

Phase scale: φ′=1
2 Q,ρ el =sBK
2π Q

E o bound: |ε| ≤ |εalias|+|R2M|+|ε ail|
5

Fu u e di ec ions:

Ex ension o non-He mi ian sca e ing and dissipa i e sys ems

Nume ical implemen a ion and benchma king

Applica ions o quan um ield heo y and g a i a ional sys ems

Connec ion wi h quan um in o ma ion and en anglemen measu es
6