Quan um G a i a ional Field:
Uni ied Theo y ia Windowed Sca e ing
Phase–Delay–Spec al-Shi Measu e
Au ic (S-se ies / EBOC)
Ve sion 0.7, Oc obe 28, 2025
No embe 24, 2025
Abs ac
This pape p oposes quan um g a i a ional ield heo y comple ely scaled by ob-
se ables: o gi en space ime geome y gand e e ence geome y g0, wi h ixed-ene gy
sca e ing ma ix Sg(E), de ine co e Wigne –Smi h delay ope a o Qg(E) = −i Sg(E)†∂ESg(E),
de ining ela i e densi y o s a es ( DOS)
ρ el[g:g0](E) = 1
2πi S†
g∂ESg=1
2π Qg(E).
Unde uni a y sca e ing amewo k sa is ying Bi man–K e˘ın (BK) o mula de Sg(E) =
exp[−2πi ξg(E)], ha e ρ el[g:g0](E) = −ξ′
g(E), whe e ξgis K e˘ın spec al shi unc ion;
his uni ies phase–delay–spec al shi iple scale ela ion, consis en wi h F iedel/Smi h
ela ions. Wi h abso p ion (non-uni a y), use phase pa ial densi y o s a es
ρ el[g:g0](phase)(E) = 1
2π∂Ea g de Sg(E), cha ac e izing abso p ion in ensi y ia imagi-
na y pa o o al complex delay τ o .
Realize measu able eadou wi hin expe imen al esolu ion ia windowed obse a-
ion: choose window–dual ke nel pai (w, ˜w) sa is ying Wexle –Raz bio hogonali y and
Gabo ame necessa y densi y (∆E∆ /(2πℏ)≤1), de ining
Nw[g:g0;E0] = ZR
w(E−E0)ρ el[g:g0](E)dE,
gi ing windowed BK iden i y and non-asymp o ic e o h ee- e m decompo-
si ion (aliasing/Poisson + Be noulli laye /Eule –Maclau in + unca ion).
In geome ic sca e ing on asymp o ically la /hype bolic mani olds, s a iona y weak-
ield Shapi o g a i a ional ime delay, and non-uni a y sca e ing wi h abso p ion (e.g.,
black hole ex e io ), we p o e: (In a iance) in a ian unde di eomo phism/uni a y
equi alence; (Addi i i y) DOS addi i e o cascade sca e ing; (Semiclassical limi )
windowed DOS con olled by leng h spec um o pe iodic geodesic low, eco e ing classi-
cal dwell ime and Shapi o delay in low- equency limi .
Keywo ds: Wigne –Smi h delay; K e˘ın spec al shi ; Bi man–K e˘ın o mula; F iedel/Smi h
ela ion; windowed obse a ion; Gabo /Weyl–Heisenbe g amewo k; Landau sampling
densi y; mani old sca e ing; Shapi o delay
1 In oduc ion: Scaling by Obse ables
Fac ha sca e ing phase and ene gy de i a i e gi e DOS es ablished since Be h–Uhlenbeck
and F iedel; in mode n sca e ing heo y, igo ized by BK o mula as
1
de S(E) = e−2πi ξ(E), ξ′(E) = −1
2πi S†∂ES,
hus ρ el[g:g0](E) = 1
2πi (S†
g∂ESg) = −ξ′
g(E). Simul aneously equi alen o o al dwell
ime measu ed by Wigne –Smi h delay ope a o Qg=−iS†
g∂ESg.
Res ic ion: Abo e equi alence chain holds only when S(E) uni a y (S†S=I); wi h
abso p ion/leakage, use phase pa ial densi y o s a es ρ(phase)
el =1
2π∂Ea g de Sand o al complex
delay τ o =−i ∂Elog de S(see
§
5).
This pape ad oca es: quan um g a i a ional ield ope a ionally de ined as windowed
ela i e densi y o s a es, i.e., ρ el[g:g0](E) and i s eadou Nw[g:g0;E0] wi hin ins u-
men al esolu ion. De ini ion based on obse able sca e ing ma ix Sg(E), measu ed ia ene gy
de i a i e o a g de Sgo ace o Wigne –Smi h delay ope a o Qg, na u ally possessing: (i)
in a iance unde di eomo phism/uni a y equi alence; (ii) addi i i y o cascade sca e ing; (iii)
semiclassical limi and Poisson ela ion wi h wa e ace/geodesic spec um; (i ) complex delay
gene aliza ion o non-uni a y sca e ing (abso p ion).
2 Se up and No a ion
2.1 Geome y, Ope a o s and S anding Assump ions
Se (M, g) smoo h mani old wi h one o mo e non-compac ends, sa is ying asymp o ically Eu-
clidean (o asymp o ically hype bolic/long- ange) condi ions; le Hg=−∆g(o sel -adjoin
a ian wi h sui able sho /long- ange po en ial). Take e e ence geome y (M, g0) and Hg0.
S anding Assump ion (applies h oughou ): Assume pai (Hg, Hg0) sa is ies ela i e
ace class condi ion, i.e., exis s z∈ρ(Hg0) such ha
(Hg−Hg0)(Hg0−z)−1∈S1,
whe e S1 ace class ope a o ideal. Unde his condi ion, spec al shi unc ion ξg(E) and
ene gy-shell sca e ing ma ix Sg(E) well-de ined, BK o mula de Sg(E) = e−2πiξg(E)holds; he e
de Sgis pe u ba ion de e minan in BK sense (F edholm/de 1 ype). All BK o mu-
las, spec al shi unc ion iden i ies and ela i e ace exp essions in his pape
unde s ood unde his assump ion.
Re e ence geome y g0calib a ion and choice: Fo expe imen al/as onomical con-
nec ion, e e ence geome y g0should be chosen as known s anda d backg ound (such as
Minkowski la space ime, Schwa zschild solu ion, o s anda d asymp o ic cone o asymp o ically
la mani old). Key p inciples:
(i) Rela i e ace class gua an ee: di e ence be ween gand g0mus sa is y abo e ace
class condi ion;
(ii) Compa abili y: di e en obse a ions should use same g0 o same physical si ua ion,
ensu ing compa ison meaning o ρ el[g:g0];
(iii) Windowed calib a ion: bandwid h ∆Eand ime-domain wid h ∆ o window pai
(w, ˜w) should ma ch ins umen al esolu ion/obse a ion imescale;
(i ) Phase baseline: when pe o ming phase unw apping o a g de S, use phase a Emin as
baseline and ack cumula i ely, a oiding a bi a y 2πjumps.
Backg ound ansla ion iden i y:
ρ el[g:g0]−ρ el[g:g′
0] = ρ el[g′
0:g0],
2
whe e le side di e ence o DOS o a ge geome y g ela i e o wo di e en e e ences g0
and g′
0, igh side ixed backg ound di e ence e m, sys ema ically canceling when compa ing
di e en g.
3 Co e De ini ions
De ini ion 3.1 (Rela i e Densi y o S a es).Fo geome y gand e e ence g0sa is ying s anding
assump ion, ela i e densi y o s a es
ρ el[g:g0](E) := 1
2πi Sg(E)†∂ESg(E)=1
2π Qg(E),
whe e Qg(E) = −iSg(E)†∂ESg(E) is Wigne –Smi h delay ope a o .
Unde BK o mula de Sg=e−2πiξg, ha e ρ el[g:g0](E) = −ξ′
g(E) (a.e.).
De ini ion 3.2 (Windowed Readou ).Fo window wcen e ed a ene gy E0,windowed ela-
i e densi y
Nw[g:g0;E0] := ZR
w(E−E0)ρ el[g:g0](E)dE.
Window choice sa is ies: (i) Wexle –Raz bio hogonali y wi h dual ˜w; (ii) Gabo ame den-
si y ∆E∆ /(2πℏ)≤1; (iii) bandlimi ed o apid decay ensu ing NPE e o closu e.
4 Main Theo ems
Theo em 4.1 (In a iance Unde Di eomo phism/Uni a y Equi alence).Le ϕ:M→Mdi -
eomo phism, g′=ϕ∗gpullback me ic. Then
ρ el[g′:g0](E) = ρ el[g:g0](E).
Simila ly, i U:L2(M, g)→L2(M, g′)uni a y ope a o in e wining Hgand Hg′, hen DOS
p ese ed.
P oo . Di eomo phism in a iance ollows om spec al low and sca e ing ma ix ans o ma-
ion p ope ies. Uni a y equi alence p ese es ace and spec al shi unc ion.
Theo em 4.2 (Addi i i y o Cascade Sca e ing).Fo h ee geome ies g1, g2, g0wi h cascade
sca e ing Sg1→g2=Sg2Sg1, ha e
ρ el[g2:g0] + ρ el[g1:g2] = ρ el[g1:g0].
P oo . Follows om mul iplica i e p ope y o sca e ing ma ices and loga i hmic de i a i e
addi i i y. Spec al shi unc ion sa is ies ξg1:g0=ξg1:g2+ξg2:g0, di e en ia ing yields DOS
addi i i y.
Theo em 4.3 (Semiclassical Limi and Geodesic Leng h Spec um).In semiclassical limi ℏ→
0(o high-ene gy E→ ∞), windowed DOS con olled by leng h spec um o closed geodesics:
Nw[g:g0;E0]∼X
γ∈P bw(Lγ)Aγ(E0) + O(ℏ),
whe e Ppe iodic geodesics, Lγleng h, Aγampli ude ac o . Reco e s classical dwell ime
and Shapi o delay in app op ia e limi s.
3
P oo . S anda d ace o mula (Gu zwille , Duis e maa –Guillemin) connec s wa e ace o
geodesic leng h spec um. Windowing selec s ene gy ange, Fou ie ans o m gi es ime/leng h
dis ibu ion.
Theo em 4.4 (Non-Asymp o ic E o Closu e: NPE Decomposi ion).Fo disc e e sampling o
windowed eadou wi h s ep ∆Eand unca ion N,
E o =εalias
|{z}
Poisson
+εEM
|{z}
Eule –Maclau in
+ε ail
|{z}
unca ion
.
When window wbandlimi ed wi h bandwid h Ωwand ∆E≤π/Ωw(Nyquis ), alias e m
εalias = 0.
EM emainde |εEM| ≤ CM∆E2M o M- h o de co ec ion.
Tail con olled by window decay: |ε ail| ≤ R|E−E0|>N∆E|w(E−E0)| |ρ el|(E)dE.
P oo . Apply Poisson summa ion, Eule –Maclau in o mula, and unca ion analysis as in s an-
da d NPE heo y. Nyquis condi ion ensu es spec al eplicas don’ o e lap.
5 Non-Uni a y Sca e ing and Complex Delay
Fo non-uni a y sca e ing (wi h abso p ion/leakage), decompose
de Sg(E) = |de Sg(E)|eia g de Sg(E).
De ine:
Phase pa ial DOS:ρ(phase)
el [g:g0](E) = 1
2π∂Ea g de Sg(E)
To al complex delay:τ o (E) = −i ∂Elog de Sg(E)
Abso p ion a e: Γ(E) = −∂Elog |de Sg(E)|
Ha e ela ion:
τ o (E) = τphase(E)−iΓ(E),
whe e τphase =ℏρ(phase)
el ( es o ing ℏ).
6 Applica ions
6.1 Shapi o G a i a ional Time Delay
Fo weak g a i a ional ield wi h me ic pe u ba ion hµν, i s -o de Shapi o delay
∆τShapi o ≈ −2GM
c3log ou
in
,
eco e ed om windowed DOS in app op ia e low- equency, long-wa eleng h limi .
6.2 Black Hole Ex e io Sca e ing
Fo Schwa zschild geome y ex e io o e en ho izon, sca e ing ma ix exhibi s esonances
co esponding o pho on sphe e and quasi-no mal modes. Windowed DOS cap u es:
Resonance wid hs om complex poles
Abso p ion c oss-sec ion om non-uni a i y
Semiclassical co espondence wi h uns able null geodesics
4
7 Discussion and Ou look
This wo k es ablishes ope a ional de ini ion o quan um g a i a ional ield ia windowed sca -
e ing obse ables:
Key achie emen s:
1. Uni ied scale o mula ρ el =−(2π)−1 Qg=−ξ′
gconnec ing phase, delay, spec al shi
2. Windowed eadou amewo k wi h NPE non-asymp o ic e o closu e
3. Di eomo phism in a iance and cascade addi i i y
4. Semiclassical limi eco e ing classical dwell ime and Shapi o delay
5. Non-uni a y ex ension o abso p ion ia complex delay
Fu u e di ec ions:
Ex ension o ull dynamical space imes and cosmological se ings
Nume ical implemen a ion o ealis ic g a i a ional wa e scena ios
Connec ions o AdS/CFT and holog aphic en anglemen
Expe imen al p oposals o able- op quan um g a i y es s
In eg a ion wi h loop quan um g a i y and s ing heo y obse ables
Physical in e p e a ion: Quan um g a i a ional ield encoded in ela i e densi y o s a es,
measu able ia sca e ing phase/delay, p o iding b idge be ween quan um mechanics and gene al
ela i i y h ough ope a ional obse ables.
5
