D a s on he In es iga ion o he Na u e o Time in
Quan um Mechanics
Bo a Ak as¸
wi h con ibu ions by Cha GPT
Oc obe 2025
Abs ac
This documen does no aim o p esen a inal heo y, bu a he o p ese e he aces o
he in ellec ual pa hs ha lead owa d such a heo y. The sec ions collec ed he e con ain
no es o un inished ideas, and some imes e en opposing a emp s. Each o hem ep esen s
a kind o concep ual exe cise in unde s anding he na u e o ime wi hin he amewo k o
quan um mechanics — small labo a o ies whe e in ui ion is ansla ed in o equa ions and
equa ions back in o in ui ion.
The e o e, i would be mo e accu a e o ead his compila ion no as a cohe en whole,
bu as an e olu iona y sequence. Disconnec ions, epe i ions, and shi s o di ec ion may
be encoun e ed among he sec ions; ye hese belong o he na u al low o hough i sel .
The pu pose is o cons uc , s ep by s ep, an inne esea ch a las ha app oaches e e close
o he essence o ime a he quan um le el.
This ex should be ead less as a scien i ic a icle and mo e as a li ing no ebook o a
heo y in ma u a ion. I s leng h, i s looseness, and i s mul iplici y a e, o ha e y eason,
signs o i s hones y.
1 Cu a u e–Based In e p e a ion o Quan um Mechanics:
Cu en F amewo ks and Eme ging Di ec ions (2025)
1.1 In oduc ion: The Geome ic Hypo hesis
Mode n quan um mechanics desc ibes unce ain y as a s a is ical ea u e o wa e unc ion su-
pe posi ion. Howe e , i he space ime mani old on which he wa e unc ion e ol es is in insi-
cally cu ed, hen unce ain y may ins ead e lec he unde lying geome ic non–commu a i i y
1
o space and ime hemsel es. In his iew, he Planck cons an ep esen s no a uni e sal noise
loo bu he in a ian a ea o he minimal cu a u e cell in phase space:
eαR(3)+βK =h
2,(1)
whe e R(3) is he spa ial scala cu a u e, K he empo al (ex insic) cu a u e o spacelike
hype su aces, and α, β a e dimensionless coupling ac o s.
1.2 Es ablished Founda ions
Quan um Field Theo y in Cu ed Space. Quan um ields p opaga ing on a cu ed back-
g ound sa is y
(+ξR)ϕ= 0,(2)
wi h R he Ricci scala . This amewo k success ully explains Hawking adia ion, Un uh e -
ec s, and cosmological pa icle c ea ion. Ye , geome y emains an ex e nal pa ame e ; he
quan um s a e e ol es on cu a u e, no h ough cu a u e. Unce ain y ela ions s ay ixed a
¯h/2, independen o space ime s uc u e.
Loop Quan um G a i y (LQG). LQG quan izes he me ic i sel , yielding disc e e eigen-
alues o olume and a ea:
A= 8πγBa be o ℓ2
PX
iqji(ji+ 1).(3)
This in oduces in insic geome ic quan iza ion, implying ha classical space ime a ises om
spin–ne wo k a e ages. Howe e , he amewo k lacks a di ec Sch ¨
odinge o single–pa icle
limi ha links cu a u e luc ua ions o measu able unce ain y.
Causal Dynamical T iangula ions (CDT). In CDT, he pa h in eg al o e geome ies con-
s uc s space ime om disc e e simplices. Simula ions indica e dimensional educ ion (D→2)
a Planck scales—a geome ic signa u e o quan um unce ain y. S ill, he model is s a is ical
and lacks a di ec analy ic law connec ing cu a u e o ∆x∆p.
1.3 Eme gen and Geome ic Quan um Theo ies
Quan um Po en ial as Cu a u e (Shojai, Cas o, e al.). The Bohmian quan um po en ial,
Q=−¯h2
2m∇2A
A,(4)
2
can be ew i en as a cu a u e e m, Q∼¯h2
2mR, e ealing a one– o–one co espondence be-
ween quan um co ec ions and local space ime cu a u e. Hence, he unce ain y ampli ude is
in e p e able as geome ic bending o he p obabili y low.
Rela ional Quan um Geome y (Ro elli, Giacomini). He e, e e ence ames hemsel es
a e quan um objec s; space and ime coo dina es acqui e ela ional cu a u e. In his se ing,
he commu a o [ˆ
T, ˆ
H]=i¯h(1 + κ ) ep esen s he in ini esimal imp in o empo al cu a u e,
while spa ial non–commu a i i y encodes me ic shea . This amewo k di ec ly suppo s a
cu a u e–based in e p e a ion o quan um unce ain y.
1.4 The In insic Cu a u e Model
Co e Pos ula e. Quan um egime ⇒cu ed space ime; Classical egime ⇒ la space ime.
Unce ain y is he e o e no s ochas ic bu a geome ic necessi y:
∆x∆p= (R(3), K) = ¯h
2.(5)
He e ¯h/2is he in a ian measu e o he a e age mic oscopic cu a u e o space ime. Spa ial
and empo al cu a u es a e complemen a y p ojec ions o his in a ian :
•R(3) — spa ial conca i y ⇒wa e sp eading,
•K— empo al conca i y ⇒phase bending.
Physical In e p e a ion.
Regime Cu a u e Cha ac e Obse able Signa u e
R(3), K > 0Conca e ime, con ex space Phase wis ing, quan um pe sis ence
R(3), K < 0Con ex ime, conca e space Wa e sp eading, decohe ence
R(3) =K= 0 Fla mani old Classical de e minism
The quan um–classical ansi ion hus appea s as a geome ic la ening p ocess.
1.5 Empi ical P ospec s
1. A om In e e ome y: Tes phase d i s ∆ϕ=RΦ d induced by empo al cu a u e.
Nonlinea de ia ions om s anda d Ramsey inges would indica e cu a u e coupling.
2. Neu on & Elec on In e e ence: Beam sp ead depends on R(3); hype bolic cu a u e
enla ges ∆x, pa abolic comp esses i .
3
3. G a i a ional Wa e Backg ounds: Long–wa eleng h cu a u e luc ua ions should in-
duce co ela ed unce ain y modula ions ac oss in e e ome e s.
4. Quan um Re e ence F ames: F ame–dependen cu a u e co ec ions (κ , κx)measu -
able by synch onized a omic clocks in a ying po en ials.
1.6 Ou look: Towa d a Uni ied Quan um Geome y
No exis ing o mula ion ully me ges he quan um wa e unc ion wi h dynamic space ime cu -
a u e. The in insic–cu a u e model o e s a concep ual b idge:
• I es o es symme y be ween geome y and p obabili y.
• I p edic s measu able de o ma ions o he Heisenbe g bound in cu ed egions.
• I in e p e s Planck’s cons an as a geome ic in a ian , no me ely a p opo ionali y ac-
o .
Fu u e de elopmen s should de i e he cu a u e–unce ain y law om a a ia ional p inciple
on (M, gµν), es ∆x∆p∼eαR+βK expe imen ally ia p ecision in e e ome y, and ex end
he amewo k o ela i is ic wa e equa ions and ield quan iza ion.
Summa y.
Quan um = cu ed space + cu ed ime;
Classical = la space + la ime;
¯h/2= in a ian a ea o he minimal cu a u e cell.
Unce ain y hus ep esen s no measu emen noise, bu he isible p ojec ion o he uni e se’s
in insic space ime cu a u e.
2 The Expanding Role o CnAlgeb as in O e coming Hilbe -
Space Limi a ions
2.1 Abs ac
Classical quan um mechanics is con ined o a complex, single-ca ie Hilbe space HC, whe e
obse ables, inne p oduc s, and p obabili y measu es a e s ic ly C-linea . When empo-
al o spa ial cu a u e is in oduced, his one-dimensional imagina y ounda ion becomes
insu icien . Cnalgeb as (jn=−1) p o ide a na u al mul i-ca ie gene aliza ion ha ex-
ends Hilbe -space geome y, allowing ime, cu a u e, and mul i-phase obse ables o coexis
wi hin a uni ied algeb aic amewo k.
4
2.2 Mo i a ion
The pu pose o in oducing Cnis no o mal no el y bu necessi y: he geome ic s uc u e o
quan um space- ime canno be ep esen ed wi hin he single-imagina y complex ield. Cu ed
ime and cu ed space demand mul iple o hogonal phase di ec ions—each encoded by a ca ie
jksa is ying jn
k=−1. These ca ie s ac as independen geome ic channels, enabling he
desc ip ion o in e nal cu a u es ha con en ional C-based quan um mechanics supp esses.
2.3 Hilbe -Space Bo lenecks
Domain Limi a ion in C-Hilbe space Impac
Time ope a o [ˆ
T, ˆ
H]=i¯hcanno hold wi h sel -adjoin ˆ
TNo in insic empo al obse able
Dynamic cu a u e R(3), K en e only as ex e nal pa ame e s No eedback be ween geome y and s a e
Uni a i y U†U=I ied o single imagina y axis Phase-cu a u e coupling b eaks no m
These a e s uc u al a he han in e p e i e limi s—a ising di ec ly om he single-ca ie
na u e o C.
2.4 Cnas a Hilbe -Space Ex ension
When Cn eplaces Cas he scala ield,
⟨ψ|ϕ⟩ ∈ Cn,⟨ψ|ψ⟩=a+jb +j2c+. . . ,
he inne p oduc gains mul iple eal and hidden-phase componen s. The eal pa encodes
measu able p obabili y; highe -o de ca ie s s o e geome ic and empo al phase cu a u e.
No m p ese a ion gene alizes o a mul i-uni a y condi ion,
U⋆nU=I,
whe e ⋆ndeno es he Cn-adjoin , ensu ing o al ( isible + hidden) p obabili y conse a ion
e en unde cu a u e coupling.
2.5 Resolu ion o Exis ing Bo lenecks
P oblem C-space s a us Cn emedy
Time ope a o Unde ined sel -adjoin o m Tempo al ca ie j ac s as in insic imagina y uni
Dynamic cu a u e Ex e nal ield Nega i e-phase ca ie s encode local cu a u e →mu ual eedback gµν[ψ]
Uni a i y B oken unde cu ed phases Reco e ed h ough U⋆nU=Imul i-uni a i y
Measu emen no m Real-only Spli no m: isible = eal, hidden = phase cu a u e
5
Hence, Cndoes no bypass he bo leneck bu ex ends he geome y un il he bo leneck
dissol es.
2.6 Physical In e p e a ion
• The isible componen o ⟨ψ|ψ⟩yields s anda d p obabili ies.
• The hidden ca ie s con ain cu a u e ene gy and phase o sion.
• Thei in e e ence de ines he geome ic unce ain y:
∆x∆p=¯h
2eαR(3)+βK ,
whe e R(3) and Ka e encoded di ec ly in he Cnbasis elemen s.
Thus, he Planck scale eme ges as he in a ian cu a u e o he Cnmani old, no an ex e -
nally imposed cons an .
2.7 Conclusion
The Cnalgeb aic amewo k is no an auxilia y ool bu he na u al comple ion o he Hilbe
o malism when cu a u e, ime, and quan um geome y a e insepa able. The appa en “bo -
lenecks” o ime-ope a o de ini ion, cu a u e dynamics, and uni a i y conse a ion all s em
om he o e simpli ied scala ield o C. Once he inne p oduc , adjoin , and no m a e e o -
mula ed o e Cn, hese issues become in e nal ea u es a he han inconsis encies.
Cnis no a bypass – i is he ex ended ab ic on which cu ed-quan um geome y li es.
P elimina y S udies: On he Ope a o iza ion o Time
3 The Sea ch o a Time Ope a o
3.1 The Pauli Obs uc ion in Con inuous Quan um Mechanics
In he s anda d Sch ¨
odinge o mula ion,
i¯h∂
∂ ψ( ) = ˆ
H ψ( ),(6)
6
ime is me ely a con inuous ex e nal pa ame e , no an ope a o . A emp ing o in oduce a
He mi ian ope a o ˆ
Tsa is ying
[ˆ
T, ˆ
H]=i¯h(7)
leads o he Pauli no-go heo em: i ˆ
Hhas a lowe -bounded spec um, such a ˆ
Tcanno exis .
Thus, in o dina y quan um heo y, ime canno be an obse able.
3.2 Why C3Ci cum en s he Pauli Theo em
The algeb a C3de ines a ini e, cyclic phase space wi h a p imi i e cube oo o uni y ω=e2πi/3
and ope a o s (U, V )sa is ying
UV =ω V U, U3=V3=I. (8)
He e, ene gy and ime a e de ined modulo h ee: he spec um is cyclic, no bounded below,
and he Pauli a gumen no longe applies.
We de ine spec al p ojec o s
Pk=1
3
2
X
n=0
ω−knUn, k = 0,1,2,(9)
and he disc e e ime ope a o
ˆ
T=
2
X
k=0
τkPk, τk=k∆ , (10)
which is He mi ian wi h h ee eigen alues co esponding o he disc e e clock s a es. The
companion ope a o
V=
2
X
k=0
ωkPk= exp −i2π
3
ˆ
T
∆ !(11)
gene a es cyclic phase o a ions in ime space.
3.3 Disc e e Sch ¨
odinge Dynamics on C3
Time e olu ion now occu s in disc e e s eps:
i¯hψk+1 −ψk
∆ =ˆ
H ψk, k = 0,1,2 (mod 3),(12)
o equi alen ly,
ψk+1 =U ψk, U = exp−i
¯hˆ
H∆ , U3=I. (13)
7
Thus, a e h ee s eps, he wa e unc ion comple es a ull phase o a ion: ψk+3 =ψk. In he
small-s ep limi , he disc e e di e ence app oaches he con inuous de i a i e, eco e ing he
s anda d Sch ¨
odinge equa ion.
3.4 Physical In e p e a ion
Con inuous ime co esponds o an unbounded linea low o ene gy le els. In con as , he
C3clock ep esen s a closed phase ci cle: ime does no low, i o a es. The sys em e ol es
by phase cycling a he han ansla ion. This modula geome y pe mi s a genuine, He mi ian
ime ope a o and allows cu a u e—in oduced la e ia C5o C6ex ensions— o mani es as
he o igin o quan um beha iou .
3.5 Summa y
• In con inuous Hilbe spaces, ime canno be an ope a o due o he bounded spec um o
ˆ
H(Pauli’s heo em).
• In cyclic spaces such as C3, ene gy is pe iodic and he obs uc ion disappea s.
• The esul ing disc e e- ime Sch ¨
odinge equa ion desc ibes phase o a ion, no linea
low.
• Highe cyclici ies (C5,C6) will in oduce measu able conca i y ( ime cu a u e) in o his
baseline s uc u e.
4 T ansi ion om C3 o C6: Eme gen Time Cu a u e
4.1 F om Fla Cyclic Time o Cu ed Tempo al Geome y
In he C3model, he disc e e ime ope a o
ˆ
T=
2
X
k=0
τkPk, τk=k∆
de ines a la cyclic clock wi h uni o m eigen alue spacing ∆τ=τk+1 −τk= ∆ . The disc e e
cu a u e
κ (k)=τk+1 −2τk+τk−1
anishes iden ically, κ = 0. Quan um e olu ion is hen pu ely o a ional wi h no empo al
s ess: ψk+3 =ψk ep esen s a pe ec ly closed phase o bi .
8
Ex ending o C5o C6in oduces highe -o de cyclic oo s (ω5=e2πi/5,ω6=eiπ/3) and
he eby allows non-uni o m eigen alue ladde s,
τk=k∆ +δτk, δτk∝sin 2πk
N!,(14)
yielding a nonze o disc e e cu a u e κ = 0. The sign o κ de e mines he local conca i y o
he ime spec um:
κ <0⇒conca e(slowing ime), κ >0⇒con ex(accele a ing ime).
4.2 In e p e a ion as Tempo al Po en ial
The eme gence o κ can be iewed as he appea ance o a empo al po en ial ield Φ sa is ying
d2τ
dk2∼ −∂Φ
∂τ .(15)
In he la C3limi , Φ =cons . and ime is homogeneous. In he C6phase, Φ de elops
cu a u e, and his empo al s ess ac s analogously o a po en ial di e ence along he ime
axis. Quan um supe posi ion may hen be in e p e ed as a esponse o he g adien o Φ a he
han an in insic inde e minacy.
4.3 Modi ied Sch ¨
odinge Dynamics
Replacing he uni o m ime s ep ∆ by he locally de o med in e al ∆ k=τk+1 −τkleads o
a cu a u e-co ec ed disc e e Sch ¨
odinge equa ion:
i¯hψk+1 −ψk
∆ k
=ˆ
H ψk.(16)
To i s o de in cu a u e, his can be ew i en as
i¯h∂ψ
∂ = ˆ
H+¯h
2i
˙κ
κ !ψ, (17)
whe e he addi ional e m ep esen s he empo al po en ial’s eedback on he phase e olu ion.
4.4 Geome ic and Physical Consequences
•C3: la cyclic ime, ze o cu a u e, exac pe iodici y.
•C5/C6: cu ed cyclic ime, nonze o κ , phase s ess.
9
bedding ime in o a closed o a ional loop a he han a linea con inuum. Hence, C2⇒
spa ial–momen umgeome y( wo −phase),
C3⇒ empo alphase opology( h ee −phase).The C3ope a o amewo k hus a aches
“ o a ion” o ime, while space obse ables con inue o li e in he complex plane.
B.3 Ex ension o C4and Phase Space ime
The uni ica ion o space and ime wi hin he same algeb aic body equi es a ou -mode cyclic
sys em C4wi h k4=−1. A minimal co espondence can be p oposed as:
(k1, k2, k3)←→ (x, y, z), k4←→ .
He e he i s h ee modes ep esen spa ial o a ions, while he ou h ac s as a empo al
o a ion—a cyclic con inua ion o he C3clock. The esul ing s uc u e admi s a na u al Minkowski-
like phase me ic,
⟨Ψ|Ψ⟩C4=a2+b2+c2−d2,
whe e he nega i e empo al signa u e a ises om he ou h (cyclic) mode. In his sense, C4
p o ides a phase-space ime in which bo h space and ime a e in e nal coo dina es o a uni ied
algeb aic o a ion.
B.4 Con inuous Flow as Cyclic A e aging
The appa en con inui y o mac oscopic ime eme ges as a coa se-g ained limi o epea ed C3
cycles. Le ∆ deno e he elemen a y s ep o he h ee-phase clock. A e Ncycles comple e
o a ions, he accumula ed e ec i e ime is
Te = 3Ncycles ∆ .
Thus, linea low is eco e ed as he s a is ical a e age o cyclic phase u ns. Each o a ion
co esponds o a “uni phase e en ” o he unde lying quan um clock. Wha we pe cei e as he
o wa d passage o ime is he e o e a coun ing p ocess o e closed phase o a ions.
B.5 Physical In e p e a ion
The C3cyclic clock desc ibes ime as o a ion a he han ansla ion:
ψk+1 =U ψk, U3=I.
16
Successi e o a ions gene a e a pe iodic—bu non i ial—e olu ion, which becomes con inu-
ous when obse ed o e many cycles. Wi hin his model:
• The isible ime di ec ion a ises om p ojec ion o he cyclic phase on o he eal axis o
obse a ion.
• The a ow o ime co esponds o he o ien a ion o his p ojec ion.
• Quan um beha iou o igina es om cu a u e (nonlinea i y) o he ime phase, while
classical beha iou appea s when his cu a u e ends o ze o.
B.6 Ou look
The C3cons uc ion co ec s he imbalance be ween ime and o he obse ables wi hou dis-
u bing he complex o malism o s anda d quan um heo y. The nex algeb aic le els, C4and
C6, a e na u al ex ensions ha can encode spa ial cu a u e and empo al conca i y in a uni ied
me ic. Wi hin his b oade iew, he p og ession
C2→C3→C4→C6
ep esen s he e olu ion om he complex plane o a comple e phase-space ime geome y,
whe e ime is no longe ex e nal bu an ope a o -shaped cu a u e o he quan um phase i -
sel .
C Ma hema ical F amewo k: The C4Me ic and Tempo al
Cu a u e Tenso
C.1 Embedding he C3Clock in C4
Le C4= span{1, k, k2, k3}wi h k4=−1. A gene al elemen is w i en as
Ψ = a+k b +k2c+k3d, a, b, c, d ∈R. (44)
We in e p e (a, b, c)as spa ial ampli udes and das he empo al ampli ude. The C3 ime ope -
a o ˆ
Tde ined on {1, ω, ω2}is now embedded as he ou h cyclic mode:
ˆ
TC4=k3ˆ
TC3, wi h(k3)3=k9=−k.
Thus, ime is ealized as a highe o a ional componen wi hin he same algeb aic ame ha
al eady hos s spa ial di ec ions.
17
C.2 C4Phase Me ic
De ine he C4inne p oduc by
⟨Ψ1,Ψ2⟩C4=a1a2+b1b2+c1c2−d1d2,(45)
which mimics he Minkowski signa u e bu eme ges algeb aically om he sign e e sal o he
k4=−1mode. The co esponding me ic enso in componen o m is
gµν = diag(1,1,1,−1), µ, ν ∈ {1,2,3,4}.(46)
A di e en ial elemen in his phase-space ime eads
ds2=dx2+dy2+dz2−dτ2,
whe e dτ is he cyclic ime inc emen inhe i ed om he C3clock.
C.3 Tempo al Cu a u e Tenso
The cu a u e o he empo al sec o o igina es om he nonuni o m spacing o he cyclic ime
spec um {τk}. Le τk=k∆ +δτk, whe e δτkdeno es he de o ma ion induced by conca i y
(in e nal ime cu a u e). De ine he disc e e empo al cu a u e as
K (k) = τk+1 −2τk+τk−1.(47)
In he con inuous limi , his becomes
K =∂2τ
∂k2≈∂2
∂ϕ2,
in e p e ed as cu a u e o he ime-phase angle ϕ. Hence he empo al cu a u e enso
R44 =−K ,
ac s as a sou ce e m o quan um beha iou : nonze o K gene a es de ia ions om classical
de e minism.
18
C.4 Dynamics wi h Cu ed Time
Replacing he la ime de i a i e by a co a ian one,
∂
∂ −→ ∇ =∂
∂ + Γ
,
he Sch ¨
odinge equa ion in cu ed C4 ime eads
i¯h∇ Ψ = ˆ
HΨ,Γ
∝ K .(48)
The connec ion coe icien Γ
encodes he in e nal conca i y o he cyclic ime mani old. When
K = 0 ( la ime), s anda d linea e olu ion is eco e ed; o K = 0, phase accele a ion occu s,
mani es ing as quan iza ion o decohe ence.
C.5 Geome ic–Physical Co espondence
Quan i y Algeb aic o igin Physical meaning
k1, k2, k3spa ial modes (C4) momen um / posi ion axes
k4 empo al mode (C4) cyclic ime ope a o
g44 =−1k4=−1 empo al signa u e
K de o ma ion o τkspacing ime cu a u e (quan umness)
R44 =−K cu a u e enso componen sou ce o quan um beha iou
Γ
phase connec ion phase accele a ion / decohe ence a e
C.6 In e p e a ion
The C4 amewo k hus uni ies he spa ial and empo al o a ions o he s a e ec o . Whe eas
C3in oduced he no ion o cyclic ime, C4embeds i in a ull ou -mode phase geome y,
yielding:
• A consis en algeb aic o igin o he Lo en z signa u e.
• A na u al de ini ion o ime cu a u e, measu able as quan um de ia ion.
• The eme gence o classical physics in he la - ime limi K →0.
C.7 Towa ds C6: Conca i y and Hidden Phases
The nex s age, C6, in oduces addi ional hidden-phase channels, p oducing in insic conca i y
in bo h empo al and spa ial sec o s. The cu a u e hie a chy
C3( la clock)−→ C4(phasespace ime)−→ C6(conca e empo almani old)
19
p o ides a sys ema ic ou e om disc e e algeb aic cycles o con inuous cu ed dynamics, in
which he quan um- o-classical ansi ion is go e ned by he geome y o ime i sel .
D Quan um-Mechanical Consequences o a Cu ed C3Time
Ope a o
Se -up. Le he C3 ime ope a o be ˆ
T=P2
k=0 τkPkwi h τk=k∆ in he la case and
τk7→ τk+δτkwhen a small disc e e ime-cu a u e is p esen . We encode cu a u e by a
dimensionless pa ame e κ =O(δτ/∆ ), and w i e he Weyl pai as
U= exp−i
¯hˆ
H∆ , V = exp −i2π
3
ˆ
T
∆ !.(49)
De o med Weyl ela ion and e ec i e commu a o . Time cu a u e gene ically de o ms
he Weyl ela ion o
U V =ω eiϵ ˆ
KV U, ω =e2πi/3, ϵ =O(κ ),(50)
whe e ˆ
Kis a He mi ian ope a o suppo ed on hidden channels (P1,2). A BCH expansion yields
he e ec i e commu a o
[ˆ
T, ˆ
H]=i¯h(I+ϵˆ
C+O(ϵ2)),(51)
wi h ˆ
Ca bounded He mi ian unc ional o ˆ
Kand he p ojec o weigh s.
Modi ied ime–ene gy unce ain y. Fo any s a e ρ,
∆T∆H≥¯h
2|⟨I+ϵˆ
C⟩ρ|=¯h
21+ϵ⟨ˆ
C⟩ρ+O(ϵ2).(52)
Conca e (nega i e) cu a u e can igh en he bound depending on he hidden-channel popula-
ion.
Disc e e Sch ¨
odinge dynamics wi h cu a u e. The la C3e olu ion eads
i¯hψk+1 −ψk
∆ =ˆ
H ψk, k ∈Z3.(53)
Cu a u e eno malizes he ime-s ep and adds a small Hamil onian co ec ion (wi hou b eak-
ing uni a i y):
i¯hψk+1 −ψk
∆ e
=ˆ
H+δˆ
H[κ ]ψk,∆ e = ∆ (1 + α κ +···).(54)
20
Consequen ly, ansi ion equencies expe ience shi s δΩmn = (δEm−δEn)/¯hwi h δEn=
⟨n|δˆ
H|n⟩.
Quan um speed limi s (QSL). The Mandels am–Tamm bound de o ms o
τ≥a ccos |⟨ψ0|ψτ⟩|
(∆H/¯h) (1 + β κ +···).(55)
Conca e ime cu a u e inc eases he minimal e olu ion ime o ixed ene gy dispe sion.
Visible s hidden channels and measu emen . Measu emen as no m p ojec ion, ρ7→
PkPkρPk, eweigh s he isible channel (P0) agains hidden ones (P1,2). Cu a u e en e s
ia δˆ
H[κ ]and he p ojec o weigh s, leading o asymme ic in e e ence con as in h ee-pa h
Ramsey sequences.
Con inui y limi and ad ancing ime. Mac oscopic ime eme ges as cycle coun ing:
T=Ncycles (3 ∆ e ),(56)
so ha cu a u e p oduces a slow d i o he e ec i e clock a e, a genuine phase- opological
o igin o an “ad ancing” ime.
Expe imen al signa u es. (i) Th ee-phase Ramsey in e e ome y: cu a u e induces a sys-
ema ic d i o inge con as s. cycle numbe . (ii) Six-pa h pho onic ings: mul i-peak shi s
δΩmn calib a e κ . All e ec s a e uni a y and e e o he la C3p edic ions as κ →0.
E Sch ¨
odinge Dynamics on he C4Mani old
E.1 Co a ian E olu ion Equa ion
The wa e unc ion Ψ(xµ)on he C4mani old obeys a co a ian Sch ¨
odinge equa ion
i¯h g44 ∇4Ψ = ˆ
HΨ,∇4=∂
∂x4+ Γ4
44,(57)
whe e x4=τdeno es he cyclic ime coo dina e, g44 =−1, and Γ4
44 is he empo al connec ion
associa ed wi h he cu a u e K in oduced ea lie . Expanding he co a ian de i a i e gi es
i¯h ∂Ψ
∂τ + Γ4
44Ψ!=ˆ
HΨ.(58)
21
The addi ional e m Γ4
44Ψ ep esen s phase accele a ion caused by he conca i y o he ime
mani old.
E.2 Phase G adien and P obabili y Flow
Le Ψ = R eiϕ. Inse ion in o he co a ian equa ion yields wo coupled ela ions: ¯h∇4ϕ=
−E−¯hIm Γ4
44,
∇4R=−RRe Γ4
44.Thus he imagina y pa o he empo al connec ion al e s he local phase
g adien (ene gy shi ), while he eal pa changes he ampli ude e olu ion (no m low o de-
cohe ence).
De ine he p obabili y cu en in cu ed ime as
J4=R2∇4ϕ, (59)
so ha he con inui y equa ion becomes
∇4J4=−2R2Re Γ4
44.
Hence, nonze o ime cu a u e p oduces no m exchange be ween isible and hidden phase
channels.
E.3 Fla -Time Limi
Fo K = 0 ( la C3clock), he connec ion anishes: Γ4
44 = 0,g44 =−1, and he equa ion
educes o he s anda d o m
i¯h∂Ψ
∂ =ˆ
HΨ.
The e o e, classical quan um mechanics appea s as he la limi o he C4mani old.
E.4 Cu ed-Time Pe u ba ion
Fo weak conca i y, we may se Γ4
44 =ϵ (τ)wi h |ϵ| ≪ 1. To i s o de ,
i¯h∂Ψ
∂τ =ˆ
HΨ−i¯h ϵ (τ)Ψ,(60)
implying an e ec i e non-He mi ian co ec ion o he Hamil onian:
ˆ
He =ˆ
H−i¯h ϵ (τ).
22
The co esponding decay o ampli ica ion o |Ψ|2can be expe imen ally in e p e ed as a cu a u e-
induced quan um- o-classical ansi ion a e.
E.5 Classicaliza ion C i e ion
Le θ(τ)be he accumula ed phase due o Γ4
44:
θ(τ) = Zτ
0Im Γ4
44(s)ds.
When θ(τ)→0, phase alignmen occu s and he wa e unc ion beha es classically (s a iona y
phase condi ion); o θ(τ)= 0, phase dispe sion p oduces quan um in e e ence. Thus he
classicaliza ion c i e ion eads
|θ(τ)| ≪ 1⇐⇒ classical egime, |θ(τ)|1⇐⇒ quan um egime. (61)
E.6 In e p e a ion and Ou look
The Sch ¨
odinge equa ion on he C4mani old desc ibes a wa e unc ion p opaga ing no along
a linea ime axis bu wi hin a cyclic, cu ed phase mani old. The empo al connec ion en-
codes he cu a u e o his mani old, go e ning phase di usion and no m exchange. Quan um
beha iou hus eme ges om he geome y o ime:
• Fla ime (K = 0)⇒de e minis ic phase low.
• Cu ed ime (K = 0)⇒p obabilis ic in e e ence.
• Inc easing cu a u e ⇒decohe ence and classicaliza ion.
This o malism p o ides a b idge be ween disc e e cyclic clocks and con inuous quan um
dynamics, sugges ing ha he passage o ime is a geome ic e ec o epea ed phase o a ions
wi hin he algeb aic s uc u e o C4. [11p ,a4pape ]a icle
[T1] on enc [u 8]inpu enc lmode n amsma h,amssymb,ams hm ma h ools bm hype e
ema k Rema k
Cu a u e-Tigh ened Time–Ene gy Bound in a C3Time-Ope a o
F amewo k
De ini ion 1 (C3 ime ope a o and cu a u e) Le he C3 ime ope a o be
ˆ
T=
2
X
k=0
τkPk, τk=k∆ ,
23
wi h {Pk}2
k=0 he spec al p ojec o s o he C3clock. A small disc e e ime-cu a u e is encoded
by τk7→ τk+δτkand a dimensionless pa ame e κ , which induces he Weyl de o ma ion
U V =ω eiϵ ˆ
KV U, ω =e2πi/3, ϵ =χ κ +O(κ2
),
whe e ˆ
Kis He mi ian and suppo ed in he hidden channels, and χis a eal calib a ion con-
s an .
Okunus¸: C3saa inde zaman ¨
ozde˘
ge le i τkk¨
uc¸ ¨
uk bi e˘
g ilikle bozuluyo ; Weyl ilis¸kisi aza
eiϵ ˆ
Kd¨
uzel mesi ekliyo .
Fiziksel yo um: E˘
g ilik, gizli az kanalla ına hassas k¨
uc¸ ¨
uk bi az sapması ¨
u e i ; bu sapma
zaman–ene ji cebi ini e kili bic¸imde de o me ede .
P oposi ion 1 (Cu a u e- igh ened e ec i e commu a o and unce ain y) Unde he abo e
de o ma ion, a BCH expansion yields he e ec i e commu a o
[ˆ
T, ˆ
H]e =i¯h(I+ϵˆ
C)+O(ϵ2),
o some bounded He mi ian ˆ
C. Consequen ly, o any s a e ρ,
∆T∆H≥¯h
2|⟨I+ϵˆ
C⟩ρ|=¯h
21+ϵ⟨ˆ
C⟩ρ+O(ϵ2).
Okunus¸: E˘
g ilik, kom¨
u a ¨
o ¨
ui¯h’ın bi c¸a panı olacak s¸ekilde de˘
gis¸ i i e beli sizlik sını ı
¯h/2’nin yakınında do˘
g usal bi d¨
uzel me alı .
Fiziksel yo um: ˙
Ic¸b¨
ukey (nega i ) e˘
g ilik e, uygun du um n¨
u usla ıyla ⟨ˆ
C⟩ρ<0sec¸ile ek
zaman–ene ji beli sizli˘
gi sıkılas¸ ı ılabili (da al ılabili ).
Co olla y 1 (Global ope a o -no m bound and c i icali y) Using |⟨ˆ
C⟩ρ| ≤ ∥ˆ
C∥, one ob-
ains he uni o m bound
∆T∆H≥¯h
21−|ϵ|∥ˆ
C∥+O(ϵ2).
Hence he p oduc canno anish in he pe u ba i e egime |ϵ|∥ˆ
C∥<1. A o mal ze o equi es
|1+ϵ⟨ˆ
C⟩ρ|= 0 =⇒ϵc i =1
|⟨ˆ
C⟩ρ|,
which lies beyond he small-cu a u e alidi y unless ∥ˆ
C∥is unbounded (i is no ).
Okunus¸: E ensel al sını , ∥ˆ
C∥ile ¨
olc¸ ¨
ul¨
u e |ϵ|∥ˆ
C∥<1iken sı ı a inmez.
Fiziksel yo um: Beli sizli˘
gi c¸ok da al mak m¨
umk¨
und¨
u ; aka sı ı lamak, pe ¨
u ba i e
¨
uni e yen ejimin dıs¸ındaki bi “k i ik e˘
g ilik” ge ek i i .
24
[Sch ¨
odinge dynamics wi h cu a u e] The disc e e C3Sch ¨
odinge equa ion becomes
i¯hψk+1 −ψk
∆ e
=ˆ
H+δˆ
H[κ ]ψk,∆ e = ∆ (1 + α κ +···),
wi h a small He mi ian co ec ion δˆ
Hsuppo ed by hidden channels. T ansi ion equencies
acqui e shi s δΩmn = (δEm−δEn)/¯h.
Okunus¸: E˘
g ilik, e kin zaman adımını e Hamil onyeni k¨
uc¸ ¨
uk ¨
olc¸ ¨
ude de˘
gis¸ i i .
Fiziksel yo um: ¨
Uni a lık ko unu ; ancak az e imi e gec¸is¸ ekansla ında ¨
olc¸ ¨
ulebili kay-
mala olus¸u .
Me hods: Ex ac ing ϵepsilon and Maximizing Tigh ening
Th ee-phase Ramsey calib a ion. P epa e a h ee-phase sequence add essing he C3chan-
nels P0, P1, P2wi h popula ions w= (w0, w1, w2),Pkwk= 1. Measu e he in e e ence
con as C(n)a e ncycles and i he e ec i e s ep and equency shi s:
C(n)≈C0exp( −γn) cos(Ωe n+ϕ0),Ωe = Ω0+X
m<n
δΩmn.
F om Ωe and he d i in ∆ e in e ϵ=χ κ and de e mine
⟨ˆ
C⟩w=T [ρ(w)ˆ
C], ρ(w) =
2
X
k=0
wkPk.
Okunus¸: Ramsey e isinden e kin ekans e zaman-adımı sapmala ını i ede ek ϵ e ⟨ˆ
C⟩
c¸ıka ılı .
Fiziksel yo um: Kon as –c¸e im ilis¸kisi, gizli/g¨
o ¨
un¨
u kanalla daki n¨
u us dengesinin e˘
g ili˘
ge
e di˘
gi yanı ı sayısallas¸ ı ı .
Popula ion op imiza ion ( igh ening maximiza ion). Fo ixed ϵ, minimize he bound o e
w:
B(w) = ¯h
2|1+ϵ⟨ˆ
C⟩w|s. . wk≥0,X
k
wk= 1.
Choose w o make ⟨ˆ
C⟩was nega i e as allowed by he expe imen . The global loo is
Bmin ≥¯h
2(1 −|ϵ|∥ˆ
C∥).
Okunus¸: N¨
u us ka ıs¸ımı, al sını ı minimize e mek ic¸in aya lanı .
Fiziksel yo um: En iyi da alma, ⟨ˆ
C⟩’yi en nega i yapan ka ıs¸ımda elde edili ; yine de ∥ˆ
C∥
ile sını lıdı .
25
In a ian unce ain y wi h sec o budge . We keep he in a ian bound
∆T∆H≥¯h
2,(80)
and esol e i in o ime/space con ibu ions ia posi i e ope a o s Q (κ )and Qx(κx),
¯h
2=¯h
2⟨Q ⟩+¯h
2⟨Qx⟩, Q +Qx=I. (81)
Dual C3cons uc ion. Le C( )
3gene a e he clock wi h p ojec o s P( )
0,1,2and C(s)
3gene -
a e spa ial phases wi h P(s)
0,1,2, mu ually commu ing. Cu a u es κ , κxa e calib a ed by hid-
den/ isible popula ions.
Disc e e dynamics. E olu ion o e one clock s ep eads
i¯hψk+1 −ψk
∆ e
=ˆ
H+δˆ
H[κ , κx]ψk,∆ e = ∆ (1 + α κ +αxκx).(82)
Calib a ion condi ions. Ramsey-3 da a yield κ and ⟨Q ⟩; posi ion–momen um in e e om-
e y yields κxand ⟨Qx⟩. Consis ency equi es (1+κ )(1 + κx) = 1 wi hin e o ba s.
K Spa io–Tempo al P obabili y and Cu en s in he Dual C3
Model
K.1 Se -up and Cu a u e Pa ame e s
We conside small cu a u es κx, κ ha de o m he spa ial and empo al sec o s, espec i ely.
In 1D o cla i y: [ ˆ
X, ˆ
P] = i¯h(1 + κx),[ˆ
T, ˆ
H]=i¯h(1+κ ),
(1 + κ )(1 + κx) = 1 (phase–cu a u econse a ion).The spa ial Sch ¨
odinge equa ion
(ha monic ap V=1
2mω2x2) becomes
−¯h2
2m(1+κx)∂2
xψ+1
2mω2x2ψ=E ψ, (83)
which is equi alen o a ha monic oscilla o o e ec i e equency
Ω(κx) = ω
√1+κx
, En(κx)=¯hω√1+κx(n+ 12).(84)
Tempo al cu a u e escales he phase a e (uni a ily):
e = (1 + κ ), U( ) = exp−i¯hˆ
H e .(85)
32
K.2 Analy ic Spa ial Solu ions (HO)
Le α(κx)=mΩ(κx)/¯h. The no malized g ound and i s exci ed eigen unc ions a e ψ0(x;κx) =
απ1/4e−αx2/2,
ψ1(x;κx) = √2α x ψ0(x;κx).Hence
P(0)
x(x) = |ψ0|2=√απ e−αx2, P(1)
x(x) = |ψ1|2= 2αx2P(0)
x(x),(86)
wi h α(κx) = mω
¯h
1
√1+κx. Fo κx>0 he densi y b oadens (la ge spa ial a iance), o κx<0
i igh ens.
K.3 Tempo al Phase and In e e ence
Fo a supe posi ion Ψ(0) = 1
√2(0 + 1), he ime-e ol ed wa e unc ion is
Ψ(x, ) = 1
√2hψ0(x;κx)e−iE0 e /¯h+ψ1(x;κx)e−iE1 e /¯hi,(87)
so he local p obabili y densi y is P(x, )=—Ψ(x, )|2= 12P(0)
x+P(1)
x+ℜhψ∗
0ψ1e−i∆E e /¯hi,
∆E=E1(κx)−E0(κx) = ¯hω√1+κx.Using he explici HO o ms,
ψ∗
0ψ1=√2α x P(0)
x(x), P(x, ) = 12P(0)
x+P(1)
x+√2α x P(0)
x(x) cos∆E¯h e .(88)
Thus empo al cu a u e simply escales he in e e ence equency:
ωin (κx, κ ) = ∆E¯h(1+κ ) = ω√1+κx(1+κ )(89)
subjec o (1+κ )(1 + κx)=1, which keeps global phase cu a u e conse ed.
K.4 In eg a ed Signals (De ec o Windows)
Fo a de ec ion window x∈[a, b], S( )=
Rb
aP(x, )dx = 12Rb
a(P(0)
x+P(1)
x)dx+cos∆E¯h e Rb
a√2α x P(0)
x(x)dx.
Fo symme ic windows [−L, L] he in e e ence in eg al anishes by odd pa i y; an asymme -
ic window (e.g. [0, L]) yields a nonze o oscilla ion ampli ude (p opo ional o RL
0xe−αx2dx).
K.5 P obabili y Cu en s
Spa ial cu en (s anda d o m wi h cu a u e- eno malized Laplacian):
Jx(x, ) = ¯h
m(1+κx)ℑ(Ψ∗∂xΨ).(90)
33
Tempo al “phase” cu en (Heisenbe g con inui y coun e pa ) o he pai (ˆ
T, ˆ
H):
∂ P(x, )+∂xJx(x, ) = 1
i¯hΨ∗(ˆ
H−ˆ
H†)Ψ uni a y
= 0,(91)
and he cu a u e en e s h ough e in he phase e olu ion and (1 + κx)in Jx. Equi alen ly, in
expec a ion o m one has
d
d ⟨ˆ
T⟩=1
i¯h⟨[ˆ
T, ˆ
H]⟩= (1 + κ ),d
d ⟨ˆ
X⟩=1
m(1+κx)⟨ˆ
P⟩.(92)
K.6 Summa y (Ope a ional P edic ions)
• Spa ial cu a u e κxde o ms he HO wid h ia α(κx), di ec ly isible in Px(x).
• Tempo al cu a u e κ escales he in e e ence equency h ough e .
• The p oduc law (1+κ )(1+κx) = 1 couples bo h, keeping a conse ed phase–cu a u e
measu e.
• Asymme ic de ec o s e eal he empo al oscilla ions S( )∝cos[ωin (κx, κ ) ].
L Sch ¨
odinge Dynamics wi h an In insic Space–Time Cu -
a u e Tenso
L.1 Pos ula es and Geome ic Da a
We model quan um kinema ics on a (3 + 1) decomposi ion o a phase-geome y (M, gµν)wi h
lapse Nand shi Ni, and spa ial me ic γij (µ= 0,1,2,3,i= 1,2,3). P obabili y ampli udes
a e sec ions o a complex line bundle wi h a quan um connec ion Aµ(U(1) gauge ield) and
spa ial Le i–Ci i a connec ion ∇io γij. We de ine he co a ian de i a i es
D := ∂ +iA , Di:= ∇i+iAi.(93)
L.2 Cu ed Sch ¨
odinge Equa ion (Canonical Fo m)
The in insic-cu a u e Sch ¨
odinge equa ion eads
i¯h D ψ=h−¯h2
2m
1
√γDi
(√γ γijDj)+V(x) + Φ + Ξ R(3)iψ, (94)
34
whe e γ= de γij,R(3) is he scala cu a u e o (Σ , γij),Ξis a dimensionless coupling
ha encodes he s eng h o geome ic back eac ion on quan um sp ead, and Φ isa empo al
cu a u e po en ial ex ac ed om he 3+1 spli o gµν (see below).
Tempo al cu a u e po en ial. Le gµν admi a 3+1 spli wi h lapse Nand ex insic cu a-
u e Kij o he slices Σ . We pa ame ize he ime-cu a u e imp in on phase by
Φ =¯h
2Θwi h Θ := ∂ ln(N√γ)−N K , K := γijKij.(95)
In he la limi (N= 1,Kij = 0,γij =δij) one has Φ = 0 and Eq. eq:cu edSE educes o he
s anda d Sch ¨
odinge equa ion.
L.3 P obabili y Conse a ion
De ine he co a ian densi y ϱ:= |ψ|2and cu en
Ji:= ¯h
mℑ(ψ∗γijDjψ).(96)
Then Eq. eq:cu edSE implies he con inui y law
∂
(√γ ϱ)+∂i
(√γ Ji) = 0,(97)
p o ided Φ is eal and Ξ∈R, gua an eeing uni a i y.
L.4 Unce ain y and Cu a u e
Fo He mi ian ˆ
Xand he co a ian momen um ˆ
Pi:= −i¯hDi, he canonical commu a o is
geome ically de o med by he connec ion and me ic:
[ˆ
Xi,ˆ
Pj]=i¯h δij,[ˆ
Pi,ˆ
Pj]=i¯hFij,Fµν := ∂µAν−∂νAµ.(98)
The in a ian Heisenbe g bound holds in cu ed space wi h he usual o m, while sa u abili y
depends on he local geome y h ough γij and Fij ia he co a iance e m in he Robe son
inequali y.
L.5 Weak-Cu a u e Expansion (Linea Response)
W i e γij =δij +hij wi h |hij|≪1,N= 1 + νwi h |ν| ≪ 1, and K=O(∂h). To i s o de ,
δˆ
H=−¯h2
2mhhij∂i∂j+1
2(∂ihij)∂j+1
2h∆i+¯h
2Θ+ΞR(3) + ¯hA ,(99)
35
whe e h=δijhij and ∆is he la Laplacian. Fo an unpe u bed eigens a e |n⟩wi h H0|n⟩=
En|n⟩, he leading ene gy shi is
δEn=⟨n|δˆ
H|n⟩.(100)
L.6 Fla Limi and Consis ency
In he la limi hij →0,ν→0,Kij →0,Aµ→0, one has
i¯h ∂ ψ=h−¯h2
2m∆+Viψ, ∂ Z|ψ|2d3x= 0.(101)
Thus he cons uc ion is no m-p ese ing and educes o s anda d quan um mechanics.
M Geome ic O igin o Quan um Unce ain y
M.1 Cu a u e–Unce ain y Co espondence
We p opose ha quan um unce ain y is no me ely epis emic bu geome ic, a ising om he
in insic cu a u e o space– ime. Fo mally,
∆x∆p=¯h
2exp(α R(3) +β K),(102)
whe e R(3) deno es he spa ial scala cu a u e, K he ex insic ( empo al) cu a u e o he 3+1
olia ion, and α, β a e phenomenological esponse coe icien s cap u ing he local sensi i i y o
he phase geome y.
Physical in e p e a ion.
•R(3) >0(spa ially conca e) ⇒wa e packe s con ac , ∆xdec eases, ∆pinc eases; he
sys em ends owa d classical de e minism.
•R(3) <0(spa ially con ex) ⇒wa e packe s sp ead, ∆xinc eases, unce ain y b oadens,
quan um cohe ence s eng hens.
•K >0(conca e ime) ⇒phase e olu ion cu es non-linea ly, supp essing collapse, main-
aining supe posi ion.
•K <0(con ex ime) ⇒phase low s aigh ens, a ou ing classicalisa ion.
36
M.2 Cu ed-Phase In e p e a ion
Time cu a u e Φ ∝Kmodi ies he e ec i e phase eloci y,
ϕ( ) = 1
¯hZ(E+ Φ )d ⇒dϕ
d =E
¯h(1+κ ),(103)
so he phase ajec o y in Hilbe space becomes a geodesic on a cu ed mani old o phases.
Spa ial cu a u e, in u n, eshapes he me ic o con igu a ion space, al e ing he canonical
Fou ie duali y be ween posi ion and momen um. Hence, unce ain y is he p ojec ion o a
single geome ic cons ain on wo complemen a y submani olds:
Gspace(R(3))⊕ G ime(K)⇒∆x∆p= ¯h2 (R(3), K).
M.3 Expe imen al P obes
•A om in e e ome y: Phase d i in Ramsey o Mach–Zehnde sequences e eals K.
•Wa e-packe dynamics: Expansion o con ac ion a es o apped a oms o elec on
beams measu e R(3).
•Op ical analogues: Cu ed-index me ama e ials simula e (R(3), K)and isualise he
de o ma ion o he unce ain y ellipse.
M.4 Fla -Limi Reco e y
Fo anishing cu a u e R(3), K →0, Eq. eq:geomunce educes o hecanonicalHeisenbe g o m∆x∆p=
¯h
2, husensu ingconsis encywi ho dina yquan ummechanics.
M.5 Concep ual Implica ion
Unce ain y ceases o be a measu emen limi a ion; i becomes a s a emen abou he local
geome y o space– ime. Planck’s cons an ¯h he eby quan i ies he minimal cu a u e–a ea
elemen in phase space, linking quan um inde e minacy o he mic oscopic geome y o eali y.
N Cu a u e in Non ela i is ic Quan um Mechanics: F om
DeWi o Pa ke and Beyond
This sec ion e iews how spa ial and empo al cu a u e en e non ela i is ic quan um me-
chanics, om DeWi ’s cu ed-space Sch ¨
odinge equa ion o Pa ke ’s non ela i is ic limi o a
cu ed Klein–Go don ield, and places ou in insic phase-geome y o mula ion in his con ex .
37
N.1 Fla Sch ¨
odinge and he Absence o Cu a u e
In s anda d quan um mechanics on la R3,
i¯h ∂ ψ= −¯h2
2m∆+V!ψ, ∆=δij∂i∂j,(104)
he Euclidean me ic δij is ixed in he backg ound and no explici cu a u e enso appea s.
N.2 DeWi ’s Cu ed-Space Sch ¨
odinge (Laplace–Bel ami)
On a Riemannian 3-mani old (Σ, γij)wi h Le i–Ci i a connec ion ∇i, he kine ic ope a o is
p omo ed o he Laplace–Bel ami ope a o
∆γψ=1
√γ∂i√γ γij∂jψ, γ = de γij,(105)
so ha he cu ed-space Sch ¨
odinge equa ion eads [?]
i¯h ∂ ψ="−¯h2
2m∆γ+V(x)#ψ. (106)
Equa ion eq:dewi SE mani es s cu a u e implici ly ia γij, bu con ains no explici scala
cu a u e R(3) e m. Impo an ly, he p obabili y cu en and con inui y law e ain a co a ian
o m
ρ:= |ψ|2, Ji:= ¯h
mℑ(ψ∗γij∂jψ), ∂ (√γ ρ)+∂i(√γ Ji)=0,(107)
ensu ing uni a i y unde s anda d (He mi ian) bounda y condi ions.
Ope a o o de ing ambigui y. In cu ed space, p omo ing pipj→ −¯h2∇i∇jis no unique
because ∇iac s on scala s and on √γwi h di e en weigh s. DeWi ’s choice leading o
eq:dewi SE is consis en wi h minimal-coupling and he co a ian p obabili y conse a ion
eq:cu ed-con inui y. Al e na i e o de ings may di e by e ms p opo ional o R(3) (see be-
low).
N.3 Nonminimal Coupling om Rela i is ic O igin: Pa ke ’s Resul
Fo a scala ield in a cu ed (3+1)-dimensional space ime (M, gµν), he Klein–Go don equa-
ion wi h nonminimal coupling eads
(g+ξR) Φ = 0,g:= gµν∇µ∇ν, R := gµνRµν.(108)
38
In he non ela i is ic (NR) limi Φ(x, )=e−imc2 /¯hψ(x, )and weak- ield, Pa ke showed ha
an explici cu a u e e m su i es in he Sch ¨
odinge Hamil onian [?,?]:
i¯h ∂ ψ="−¯h2
2m∆γ+Ve (x) + ¯h2
2mζ R(3)(x)#ψ, ζ =ζ(ξ),(109)
whe e R(3) is he scala cu a u e o spa ial slices and ζdepends on he ela i is ic coupling ξ
(and on de ails o he NR educ ion; e.g. o speci ic o de ings one inds ζ= 1/12). Equa ion
eq:Pa ke SE ealizes a di ec cu a u e con ibu ion o he NR Hamil onian.
Gauge/connec ion s uc u e. I aU(1) connec ion Aµcouples o he phase (elec omagne ic
o geome ic), he co a ian de i a i es Di:= ∇i+iAiand D := ∂ +iA en e , wi h ield
s eng h Fµν =∂µAν−∂νAµ. Commu a o s become [ˆ
Xi,ˆ
Pj]=i¯hδijand [ˆ
Pi,ˆ
Pj] = i¯hFij.
N.4 Tempo al Geome y in a 3+1 Spli and Ou Ex ension
Le he space ime me ic be decomposed as gµν →(N, Ni, γij)wi h ex insic cu a u e Kij o
he spa ial slices. A empo al cu a u e po en ial can be pa ame ized a he NR le el by
Φ =¯h
2Θ,Θ := ∂ ln(N√γ)−NK, K := γijKij,(110)
so ha he in insic-cu a u e Sch ¨
odinge equa ion eads
i¯h D ψ=h−¯h22m1√γDi(√γ γijDj)+V+ Φ + Ξ R(3)iψ, (111)
wi h eal Ξ. In he la limi (N=1,Kij =0,γij =δij,Aµ=0), eq:ou SE educes o eq: la SE.
P obabili y conse a ion. De ining ρ=|ψ|2and Ji= (¯h/m)ℑ(ψ∗γijDjψ), one ob ains
∂ (√γ ρ)+∂i(√γ Ji)=0,(112)
so uni a i y is p ese ed p o ided Φ is eal and bounda y condi ions a e He mi ian.
N.5 O de ing, Equi alence P inciple, and Physical In e p e abili y
The explici R(3) e m in eq:Pa ke SE depends on he ela i is ic pa en heo y and he ope a o
o de ing chosen in he NR educ ion. Di e en choices lead o di e en ζ, e lec ing an ambi-
gui y ha is cons ained by (i) p obabili y conse a ion, (ii) he la limi , and (iii) expe imen al
consis ency. Ou pa ame iza ion eq:ou SE isola es wo independen geome ic imp in s: spa-
ial cu a u e ia R(3) and a empo al imp in ia Φ , bo h es able in p inciple.
39
N.6 Linea Response and Measu able Shi s
Fo weak cu a u e, w i e γij =δij +hij (|hij| ≪ 1), N=1+ν,K=O(∂h). Expanding
eq:ou SE yields a Hamil onian co ec ion
δˆ
H=−¯h2
2mhij∂i∂j+1
2(∂ihij)∂j+1
2h∆+¯h
2Θ+ΞR(3) + ¯hA ,(113)
and he leading ene gy shi o an eigens a e |n⟩is δEn=⟨n|δˆ
H|n⟩. Ramsey in e e ome y
p obes Θand A , while spec al shi s p obe R(3) and hij.
N.7 Open P oblems and Expe imen al P obes
•Tempo al geome y: A i s -p inciples de i a ion o Φ om a ela i is ic pa en heo y
emains open; ou Φ is a es able NR pa ame iza ion.
•O de ing and ζ:De e mining ζnonpe u ba i ely and isola ing i expe imen ally (e.g.
om apped-a om spec a in enginee ed cu a u e analogs) is an open p og am.
•Gauge/geome y sepa a ion: Disen angling Aµ(gauge) om geome ic con ibu ions
in p ecision phase measu emen s (Ramsey/echo) is c ucial.
•In e e ome ic signa u es: Mul i-a m in e e ome e s wi h spa ially a ying γij and
ime-modula ed N( )can sepa a ely cons ain (Ξ, ζ)and Φ .
N.8 Li e a u e Con ex
See DeWi o he cu ed Sch ¨
odinge ope a o [?]; Pa ke o NR limi s wi h cu a u e e ms
[?]; and he s anda d QFT-in-cu ed-space ime ea men s [?,?]. Ou o mula ion eq:ou SE
ex ends his line by in oducing an explici empo al cu a u e po en ial Φ a he NR le el
alongside a spa ial R(3) e m and co a ian U(1) coupling.
O In insic Cu a u e In e p e a ion o Planck’s Cons an
O.1 Fundamen al Pos ula e
In he con en ional iew, he unce ain y ela ion
∆x∆p=¯h
2
40
is ea ed as a s a is ical o epis emic limi . He e we ein e p e i geome ically: he quan um
domain i sel possesses an in insic space– ime cu a u e, and he cons an ¯h/2is he isible
p ojec ion o his cu a u e.
O.2 Cu a u e Equi alence Rela ion
We pos ula e ha quan um unce ain y a ises no om measu emen limi s, bu om he unde -
lying cu a u e o space and ime a mic oscopic scales. Hence,
eαR(3)+βK =¯h
2,
whe e R(3) deno es he spa ial scala cu a u e o he local hype su ace, K he ex insic ( em-
po al) cu a u e o he slice, and α, β a e dimensionless coupling ac o s con e ing geome ic
cu a u e o phase cu a u e.
In e p e a ion. - The exponen ial encapsula es he join con ibu ion o spa ial (R(3)) and
empo al (K) cu a u es. - The igh -hand side, ¯h/2, is no a cons an inse ed by hand: i is
he in a ian geome ic a ea o he minimal phase-space cell co esponding o one quan um
o in insic cu a u e. - Thus, Planck’s cons an measu es he a e age mic o-cu a u e o he
uni e se.
O.3 Physical Pic u e
Quan um mechanics co esponds o a egime o cu ed ime and cu ed space:
Quan um egime :(R(3), K)= 0.
The isible mani es a ion o his cu a u e is he unce ain y bound. Con e sely, when bo h
cu a u e enso s la en,
R(3) →0, K →0,
he exponen ial e m ends o uni y, and he phase-space cell becomes la :
eαR(3)+βK →1,⇒∆x∆p→0,
eco e ing classical de e minism.
41
