Nonlocal Uni ica ion Field Theo y and I s Quan um Op ics Limi : Obse e -Func ion
Fo mula ion
A. Chawla
REAL Ins i u e, Gu ug am, India
(Da ed: Oc obe 11, 2025)
We o mula e he Nonlocal Uni ica ion Field Theo y (NUFT) by ea ing he obse e unc ion
(∆I) as he p ima y ield gene a ing space ime geome y om in o ma ional di e en ials. This
aligns wi h he Nonlocal Cons uc (NC) o malism in which geome y is eme gen om smoo h,
bounded obse e mappings a he han p e-exis ing mani olds. The esul ing a ia ional ame-
wo k yields coupled ield equa ions ha uni y g a i a ional and in o ma ional dynamics. In he
weak-cu a u e limi , NUFT educes na u ally o quan um op ics, whe e he cu a u e o he ob-
se e unc ion gene a es op ical squeezing and nonlinea i y. This obse e - unc ion o mula ion
smoo hs he concep ual ansi ion be ween cosmological NU dynamics and labo a o y-scale op ical
phenomena.
I. INTRODUCTION
The Nonlocal Uni ica ion (NU) amewo k [1] posi s
ha space ime is no undamen al bu cons uc ed by
obse e -dependen mappings be ween in o ma ional di -
e en ials ∆Iand measu able in e als (∆x, ∆ ):
(∆x, ∆ ) = (∆I).(1)
He e :R→R1,3is a smoo h, mono onic, embodimen -
dependen obse e unc ion ha esides in a F ´eche -
opologized unc ion space O⊂C (M, R1,3), as es ab-
lished in [2]. The geome y pe cei ed by an obse e
a ises as he Cauchy limi ∞o a il a ion { n}, co e-
sponding o inc easing in o ma ional esolu ion.
Ou aim he e is o cons uc a co a ian ield heo y
whose dynamical a iable is (∆I), and o demons a e
ha in he small-cu a u e limi i ep oduces quan um-
op ical ield dynamics.
II. OBSERVER FUNCTION AS PRIMARY
FIELD
We de ine he eme gen me ic as a unc ional o he
obse e unc ion:
gµν ( ) = ∂µ (∆I)∂ν (∆I),(2)
which encodes he a e o in o ma ional sampling.
Smoo hness and boundedness o gua an ee he absence
o cu a u e singula i ies (No Singula i y Theo em).
Le ∇µdeno e he co a ian de i a i e compa ible
wi h gµν ( ). We in oduce he in o ma ional cu a u e
enso
Fµν =∇µ ν− ∇ν µ, µ≡ ∇µ . (3)
This enso measu es he nonlocal wis o obse e sam-
pling ac oss in o ma ion g adien s.
III. ACTION AND LAGRANGIAN DENSITY
We p opose he obse e - unc ion Lag angian
LNU =c3
16πG(R−2Λ) −λ
4Fµν Fµν −V( , ∂ , ∆I),
(4)
wi h ac ion
S[ ] = Zd4xp−g( )LNU.(5)
The po en ial V( , ∂ , ∆I) cap u es nonlinea depen-
dence on he obse e unc ion and i s g adien wi h e-
spec o ∆I. Va ia ion o Swi h espec o and gµν
yields he NU ield equa ions.
A. Field Equa ions
Va ia ion wi h espec o he me ic gi es
Rµν −1
2Rgµν + Λgµν =8πG
c4T( )
µν ,(6)
T( )
µν =λ(FµαFνα−1
4gµν FαβFαβ)−gµν V+δV
δgµν .
(7)
Va ia ion wi h espec o yields
∇µ(λFµν ) + ∂V
∂ ν
− ∇µ∂V
∂(∇µ ν)= 0.(8)
Equa ions (6)–(8) o m he In o ma ional Ein-
s ein–Maxwell sys em exp essed di ec ly in e ms o
(∆I).
IV. REDUCTION TO QUANTUM OPTICS
To eco e labo a o y-scale physics, we linea ize
a ound a la me ic:
gµν ( )→ηµν , (∆I) = α∆I+β(∆I)3+· · · .(9)
2
The linea e m ep oduces he s anda d Minkowski me -
ic, while cubic co ec ions gene a e op ical nonlinea i-
ies.
A. Linea (Cohe en ) Limi
Neglec ing βand highe -o de e ms,
∂µ∂µ +∂V
∂ = 0.(10)
Fo V( ) = 1
2ω2
0 2, his educes o he Klein–Go don
equa ion. Upon quan iza ion, exci a ions o co espond
o in o ma ional pho ons o in ons, iden ical in algeb a
o op ical ield quan a.
B. Nonlinea (Squeezed) Regime
Including β(∆I)3yields an in e ac ion Hamil onian
Hin ∝β(a†2+a2),(11)
p oducing he wo-pho on squeezing e m amilia in
quan um op ics. Thus, squeezing and Ke nonlinea i-
ies a ise om cu a u e in he obse e unc ion (∆I).
V. INTERPRETATION AND CONSISTENCY
a. Rela ion o NU Founda ions. The p esen ield
heo y is consis en wi h he appendices o he Conse-
quences o Nonlocal Uni ica ion o Limi Obse e s pa-
pe . In ha o malism, is he gene a o o he me ic,
and he en opy di e en ial ∆Iis he conse ed in o ma-
ional subs a e. Boundedness o ∂k ensu es he absence
o singula i ies, di ec ly embodying he No Singula i y
Theo em. The ac ion abo e co esponds o he a ia-
ional p inciple S[ ] = RL( , ∂ , ∆I)d4xde ined o e
he F ´eche space O.
b. Physical co espondence. The linea op ical limi
co esponds o small de ia ions o om i s mean ob-
se e con igu a ion, while cosmological beha io co e-
sponds o la ge-scale e olu ion o ac oss embeddings in
O. The same go e ning equa ions, in e p e ed a di e -
en scales, ep oduce bo h smoo h cosmic expansion and
cohe en op ical p opaga ion.
c. Expe imen al sec o . Quan um-op ical expe i-
men s p obe pe u ba ions o (∆I) a he labo a o y
scale, mani es ing as ield ampli ude luc ua ions. G a i-
a ional o cosmological obse a ions co espond o la ge-
scale a ia ions o cons ained by bounded en opy ∆I.
The sha ed o igin o bo h sec o s unde NUFT o e s a
uni ied epis emic ield heo y.
VI. CONCLUSION
The Nonlocal Uni ica ion Field Theo y in obse e -
unc ion o m es ablishes a con inuous b idge om in-
o ma ional geome y o quan um op ics. By ele a -
ing he obse e unc ion (∆I) o p ima y ield s a-
us, he o malism uni ies cu a u e, in o ma ion low,
and op ical ield dynamics wi hin a single a ia ional
amewo k. The weak- ield limi ep oduces quan um op-
ics; he s ong- ield egime yields cosmological egula i y
and he No Singula i y Theo em. This duali y posi ions
NUFT as bo h a concep ual and p ac ical uni ica ion o
in o ma ional and physical heo ies.
VII. ACKNOWLEDGMENT
Concep ual wo k is done by he au ho . An LLM was
used o w i e he de ails and edi he pape in collabo a-
ion wi h he au ho .
Appendix A: Ma hema ical Supplemen : F ´eche
S uc u e
Le O⊂C (M, R1,3), ≥2, deno e he space o ob-
se e unc ions. Equip Owi h semino ms ∥ ∥k,K =
supx∈K|Dk (x)| o compac K⊂M. (O, τO) is com-
ple e and locally con ex; e e y Cauchy il a ion { n}
con e ges o ∞∈Ogene a ing smoo h, singula i y- ee
me ics gµν ( ∞). This cons uc ion g ounds he ield he-
o y in he same ma hema ical domain desc ibed in he
NU appendices.
Appendix B: In o ma ional Regula i y and Ene gy
De ine he in o ma ional ene gy densi y E=1
2∥∇ ∥2+
V( , ∂ , ∆I). Unde bounded de i a i es, E(T)≤
E(0)eCT , ensu ing ini e cu a u e and ene gy h ough-
ou in o ma ional ime T.
Appendix C: Quan um Op ics Co espondence
Linea izing and iden i ying E↔elec ic ield
quad a u e yields he s anda d quan ized Hamil onian o
op ical modes. Cu a u e-induced de ia ions o map
o squeezing pa ame e s, connec ing labo a o y quan um
op ics di ec ly wi h NU obse e geome y.
3
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