Geome ic Ene gy Spec um and Redshi om Tempo al
Cu a u e:
A C3–Phase Fo malism App oach
Bo a Ak a¸s & Cha GPT (co-au ho )
Oc obe 2025
Abs ac
This wo k de elops a geome ic model in which empo al cu a u e di ec ly de e mines he
quan um ene gy spec um. Wi hin he C3–phase o malism, ime and ene gy a e ea ed as con-
juga e geome ic ope a o s ac ing on cu ed phase channels. The esul ing amewo k eplaces
he classical concep o andom unce ain y wi h a cu a u e-based necessi y, whe e edshi and
spec al de o ma ion eme ge as di ec consequences o ime cu a u e di e ences. The model
demons a es ha a cu ed empo al me ic ep oduces obse able edshi ela ions wi hou
in oking spa ial expansion, p o iding a uni ied geome ic in e p e a ion b idging quan um and
cosmological scales.
1. In oduc ion
S anda d quan um mechanics ea s ime as an ex e nal pa ame e and ene gy as i s conjuga e
obse able. Howe e , in cu ed phase geome ies, ime and ene gy mus be de ined on equal oo ing:
bo h a e di ec ions wi hin he same geome ic mani old. In he C3phase algeb a (j3=−1), he
iadic s uc u e allows he empo al axis o possess cu a u e, encoded in i s ope a o spec um.
The goal o his pape is o link ha cu a u e di ec ly o measu able quan i ies such as ene gy
le els and edshi .
This cons uc ion a ises na u ally om ea lie esul s in which he ime ope a o ˆ
Twas de ined
as C3–He mi ian, cyclic, and geome ically closed ( ˆ
T3=−τ3I). He e we ex end ha app oach by
coupling ˆ
Tand ˆ
H h ough a common cu a u e enso .
2. Ma hema ical F amewo k
In he C3–phase mani old, he ime and ene gy ope a o s belong o he same geome ic basis:
(ˆ
T, ˆ
H)∈ PC3={jpj2q|p, q ∈ {0,1,2} }.
The gene alized commu a ion ela ion eads
[ˆ
T, ˆ
H]3=iℏI+εˆ
Cj−j2,(1)
whe e ˆ
Cj−j2is he phase cu a u e enso , quan i ying he local wis be ween ime and ene gy
di ec ions.
The co esponding phase-space me ic is
ds2= (dT)2+ (dE)2+α dT dE, α =j−j2,(2)
in oducing a c oss e m dT dE ha ep esen s he in insic ime–ene gy coupling. In he la limi
(α= 0), his educes o he con en ional Sch ¨odinge me ic, while nonze o αencodes cu a u e-
induced mixing be ween he wo axes.
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The C3–Sch ¨odinge equa ion wi h empo al cu a u e hen eads
(jℏ∂ +j2γ∇2+µ+ Φ )ψ= 0,(3)
whe e Φ ep esen s he empo al po en ial associa ed wi h local cu a u e R .
3. Geome ic Ene gy Spec um
The ime ope a o ’s eigen alues,
λn=ℏ
µeiπ(2n+1)/3, n = 0,1,2,
de ine h ee dis inc empo al cu a u e modes co esponding o he C3phase channels. We now
pos ula e ha he local ene gy spec um couples o hese cu a u e modes:
En=E0(1+ε R(n)
),(4)
whe e R(n)
deno es he cu a u e eigen alue associa ed wi h each empo al channel. The expec a-
ion alue o he ene gy becomes
⟨E⟩=E01+εℜ⟨κj−j2⟩,(5)
wi h κj−j2 ep esen ing he p ojec ed cu a u e ope a o along he isible phase axis.
Thus, he ene gy spec um acqui es a geome ic co ec ion p opo ional o empo al cu a u e.
4. Redshi as Tempo al Cu a u e
In his amewo k, edshi is no due o spa ial ecession bu a ela i e di e ence in empo al
cu a u e. Le R(s)
and R(o)
deno e he sou ce and obse e ime cu a u es, espec i ely. Then
he edshi ela ion becomes
1+z=e∆R =e(R(s)
−R(o)
).(6)
The obse ed equency shi is he e o e a di ec measu e o empo al cu a u e con as . This
leads o a ein e p e a ion o cosmological edshi : galaxies appea eceding because hei local
ime po en ials di e , no because space i sel expands.
Equa ion (6) p edic s ha small cu a u e di e ences p oduce a linea edshi egime, while
la ge cu a u e g adien s yield exponen ial sa u a ion, consis en wi h high-zobse a ions.
5. Discussion
The cu a u e-dependen unce ain y ela ion
∆T∆H=ℏ
2eκ R +κxRx
implies ha ime cu a u e ac s as a scaling ac o o quan um unce ain y. When R →0, unce -
ain y app oaches i s classical limi ℏ/2, and bo h space and ime la en — he sys em classicalizes.
Con e sely, when R g ows, phase ajec o ies con ac , p oducing ene gy quan iza ion and
edshi simul aneously. This mechanism p o ides a geome ic link be ween quan um disc e eness
and cosmological expansion.
The enso ˆ
Cj−j2go e ns his coupling:
ˆ
Cj−j2=j ∂T−j2∂H,
binding isible and hidden channels o ime and ene gy e olu ion.
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6. Conclusion
We ha e shown ha :
1. Tempo al cu a u e di ec ly de e mines he quan um ene gy spec um.
2. Redshi a ises na u ally as an exponen ial o cu a u e di e ence.
3. The C3phase me ic p o ides a uni ied measu e o ime and ene gy, eplacing andomness
wi h geome y.
4. In he limi o anishing cu a u e, classical dynamics and la space ime a e eco e ed.
This app oach hus es ablishes a geome ic b idge be ween quan um mechanics and cosmology,
whe e ene gy quan iza ion and cosmological edshi become mani es a ions o he same empo al
cu a u e ield.
Appendix A: Tempo al Cu a u e Ope a o
Fo small cu a u e,
Φ =ℏ2
2µR +O(R2
),
and he associa ed Hamil onian co ec ion is
δˆ
H=εℏ2
2µR (j−j2).
This modi ies ene gy eigen alues by ∆En=ε E0R(n)
.
Appendix B: C3T ans o m and Phase P ojec ion
The p ojec ion ope a o s on isible and hidden channels a e
Π is =1
3(1+j−j2),Πhid =1
3(1+j+j2).
Expec a ion alues o obse ables decompose as
⟨ˆ
O⟩=⟨ˆ
O⟩ is +ε⟨ˆ
O⟩hid.
Appendix C: Obse a ional Implica ion
Using (6), a cu a u e g adien o ∆R ∼10−3p oduces a edshi o o de z∼10−3, ma ching
ypical nea by galaxies. Hence, cosmological edshi can be ein e p e ed as a di e en ial empo al
cu a u e e ec , consis en wi h he geome ic phase o he C3mani old.
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