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En opy as In o ma ion Cu a u e: Uni ying Ma e , Gauge Fields, and G a i a ion
h ough En opic Geome y
Bo os A ne h, Philipps Uni e si y Ma bu g, Jus us Liebig Uni e si y Giessen, Ge many,
[email p o ec ed]
Abs ac
We p esen a uni ied heo e ical amewo k in which ma e , gauge ields, and g a i a ion
eme ge as mani es a ions o an unde lying en opic in o ma ion geome y. The
o malism is cons uc ed on a diag am–Hilbe space, whe e opological p ojec ions
encode co ela ions be ween quan um deg ees o eedom, and en opy ac s as he
gene a ing unc ional o ield dynamics. Wi hin his se ing, he Fishe in o ma ion
me ic de ines an en opic cu a u e enso whose ex emiza ion yields he e ec i e
Eins ein and Yang–Mills equa ions as equilib ium condi ions o in o ma ional en opy.
G a i a ional coupling eme ges om he opological en opy densi y associa ed wi h
in o ma ion low be ween ma e sec o s, while gauge in e ac ions co espond o
localized cu a u e de ec s in he en opic mani old. The Bekens ein–Hawking en opy
and holog aphic bounds appea as limi ing con igu a ions o he same in o ma ion
po en ial, p o iding a con inuous ansi ion om black hole he modynamics o he
s anda d-model egime. The amewo k na u ally accoun s o mass gene a ion h ough
e ec i e en opic p ojec ions wi hin he diag am–Hilbe space, linking he s uc u e o
he S anda d Model o opological in a ian s o he en opic geome y. This app oach
es ablishes a sel -consis en and eno malizable uni ica ion o g a i y and gauge ields,
g ounded in he in o ma ional and he modynamic o igin o physical law.
1. In oduc ion
The deep in e play be ween en opy, in o ma ion, and space ime has p o oundly eshaped
he concep ual ounda ions o physics. Bekens ein demons a ed ha black holes possess
an en opy p opo ional o he ho izon a ea [1], and Hawking e ealed ha quan um
e ec s lead o black-hole adia ion [2]. Jacobson la e de i ed he Eins ein ield
equa ions om he Clausius ela ion, showing ha space ime dynamics ollow om
he modynamic consis ency [3]. Subsequen de elopmen s by Padmanabhan [4,5]
and Ve linde [4] ad anced he iew ha g a i y i sel may be an en opic o cea ising
om in o ma ion- heo e ic g adien s.
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Pa allel p og ess in in o ma ion geome y has shown ha he Fishe me ic de ines a
na u al cu a u e on s a is ical mani olds [6, 7-11], while Ca icha’s en opic dynamics
[7] e o mula e quan um heo y as in e ence o e p obabili y dis ibu ions. These insigh s
hin ha bo h g a i a ion and gauge in e ac ions could eme ge om an
unde lying in o ma ion mani old go e ned by en opy maximiza ion.
He e, we ex end hese ideas o a comple e en opic uni ica ion amewo k, in which
ma e , gauge ields, and g a i a ion a ise om an en opy unc ional de ined on
a diag am–Hilbe space. This ope a o mani old cap u es quan um co ela ions h ough
opological p ojec ions, linking in o ma ional cu a u e o e ec i e ield dynamics.
En opy—no ac ion—plays he cen al a ia ional ole. By ex emizing he en opy
unc ional, one ob ains he Eins ein–Yang–Mills equa ions as equilib ium condi ions in
he in o ma ion mani old. The S anda d-Model couplings and mass hie a chy appea as
consequences o en opic p ojec ions be ween subsys ems o he diag am–Hilbe space.
2. En opic In o ma ion Geome y and he Diag am–Hilbe Space
The diag am–Hilbe space ℋ! gene alizes he con en ional Hilbe space by encoding
bo h algeb aic and opological ela ions among s a es. Each node o he diag am
ep esen s a sub-Hilbe space associa ed wi h a local gauge o ma e sec o , and he
mo phisms be ween nodes de ine p ojec ion ope a o s 𝑃"#:ℋ"→ℋ#.
An en opy unc ional is de ined o e ampli ude dis ibu ions 𝜓∈ℋ!:
𝑆[𝜓]=−𝑘$∫𝜓%ln𝜓 𝑑Ω
whe e 𝑑Ω ep esen s he measu e on he in o ma ional mani old.
Va ia ions 𝛿𝑆=0 unde no maliza ion cons ain s yield equilib ium con igu a ions. The
esul ing Eule –Lag ange ela ions ep oduce bo h g a i a ional and gauge dynamics
when in e p e ed geome ically.
The Fishe in o ma ion me ic
𝑔"# =5(∂"ln𝑝):∂#ln𝑝; 𝑝 𝑑𝑥
induces a cu a u e enso 𝑅&'"# ha quan i ies how dis inguishabili y be ween
p obabili y ampli udes changes ac oss pa ame e space. This cu a u e co esponds
physically o ene gy–momen um low and gauge ield s eng h:
𝑅"# ⟷𝐹"# +𝐺"#
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whe e 𝐹"# deno es Yang–Mills cu a u e and 𝐺"# he geome ic cu a u e o space ime.
3. En opy Func ional and Eme gen Field Equa ions
Ex emizing he en opy unc ional unde he cons ain o conse ed p obabili y and
in o ma ion lux gi es
𝛿:𝑆[𝜓]−𝜆∫𝜓%𝜓 𝑑Ω;=0
leading o an en opic balance equa ion
∇":𝑔"#∇#ln𝑝;=0
which, when exp essed in geome ical a iables, ep oduces he Eins ein equa ion in
en opic o m [3, 5]:
𝑅() −1
2𝑅𝑔() =8𝜋𝐺 𝑇()
(+,-)
whe e 𝑇()
(+,-) is he in o ma ional ene gy–momen um enso de i ed om en opy
g adien s.
Simila ly, p ojec ing a ia ions wi hin a subsys em ℋ"⊂ℋ! yields an e ec i e Yang–
Mills equa ion:
∇(𝐹/
() =𝐽/
)
whe e he cu en 𝐽/
) co esponds o he en opic lux associa ed wi h gauge deg ees o
eedom.
Thus, he g a i a ional and gauge ields a ise simul aneously as ex emal esponses o he
same en opy unc ional—a di ec in o ma ion- heo e ic uni ica ion.
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4. En opic Uni ica ion o Gauge Fields and G a i a ion
In his iew, bo h gauge and g a i a ional couplings s em om a sha ed en opic
cu a u e enso ℛ01 de ined on he diag am–Hilbe mani old.
P ojec ing ℛ01 on o space ime coo dina es gi es he Eins ein enso 𝐺(), while
p ojec ion on o in e nal symme y coo dina es yields he ield s eng hs 𝐹()
/.
The uni ied en opic ela ion
∇0ℛ01 =0
encodes he conse a ion o o al in o ma ion cu a u e.
Gauge in a iance co esponds o he in a iance o he en opy unc ional unde local
ans o ma ions 𝜓→𝑈(𝑥)𝜓 ha p ese e o al in o ma ional measu e.
In he limi whe e en opic cu a u e localizes, he heo y educes o classical gene al
ela i i y and Yang–Mills ields. In he high-cu a u e (quan um) egime, en opic non-
locali y induces coupling uni ica ion and po en ial quan um-g a i y co ec ions.
5. Mass Gene a ion and Symme y B eaking ia En opic P ojec ions
Masses a ise when in o ma ional symme y is educed h ough en opic p ojec ion.
Le Π:ℋ!→ℋ+22 deno e he p ojec ion om ull in o ma ional space o an obse able
submani old. The associa ed en opy change
Δ𝑆=−𝑘$ T (𝜌lnR𝜌−Π𝜌lnRΠ𝜌)
co esponds o e ec i e mass gene a ion. The mass pa ame e 𝑚 can be in e p e ed as a
Lag ange mul iplie en o cing in o ma ional equilib ium be ween p ojec ed and ull
dis ibu ions:
∂𝑆
∂Π∼𝑚𝑐3
This connec s mass o he en opic cos o p ojec ion, analogous o how he Higgs
mechanism in oduces mass h ough symme y educ ion.
The hie a chy o pa icle masses hen e lec s he opology o p ojec ion pa hways wi hin
he diag am–Hilbe space—an in o ma ional pa allel o he s uc u e o he S anda d
Model.
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6. Black-Hole and Cosmological En opy
When he en opic cu a u e enso localizes on a wo-su ace ∂Σ, he en opy unc ional
ep oduces he Bekens ein–Hawking ela ion [1, 2]:
𝑆$4 =𝑘$𝐴
4ℓ5
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In his limi , he diag am–Hilbe en opy densi y equals he geome ic a ea densi y,
indica ing ha black-hole en opy ep esen s he bounda y p ojec ion o in o ma ional
cu a u e.
A cosmological scales, en opy low be ween subsys ems d i es space ime expansion.
Following Padmanabhan’s holog aphic equipa i ion [5, 12], he en opic imbalance
be ween bulk and bounda y deg ees o eedom yields an eme gen accele a ion
consis en wi h he F iedmann equa ions.
En opy p oduc ion in ea ly-uni e se symme y b eaking na u ally links o ba yogenesis
and he obse ed ma e –an ima e asymme y.
7. Discussion and Ou look
The en opic in o ma ion-geome y amewo k es ablishes a common in o ma ional
o igin o all undamen al in e ac ions. G a i y and gauge o ces a e no sepa a e ields
bu mani es a ions o a uni e sal en opy unc ional de ined on a opological in o ma ion
mani old.
This pe spec i e uni ies:
• he modynamic g a i y [3–5, 12,13],
• in o ma ion geome y [6, 7],
• holog aphic and quan um-in o ma ion models o space ime [14–20], and
• mass gene a ion h ough in o ma ional p ojec ion.
Fu u e wo k will explo e eno maliza ion wi hin his en opic pic u e and iden i y
expe imen al signa u es—such as en opic co ec ions o coupling uni ica ion o ho izon
he modynamics— ha could be p obed ia cosmological obse a ions o quan um-
in o ma ion analog sys ems.
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Re e ences
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