Observer World Section Structure: Causal Consistency, Conditionalization, and Boundary Time Geometry

Obse e Wo ld Sec ion S uc u e:
Causal Consis ency, Condi ionaliza ion, and
Bounda y Time Geome y
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
Wi hin he unied ime scale and bounda y ime geome y amewo k, we p o-
ide an axioma izable cha ac e iza ion o wha wo ld obse e s see. S a ing poin :
gi en global bounda y algeb a and quan um s a e, all wo ld sec ions sa is ying
eld equa ions and causali y o m a geome icmeasu e space; conc e e obse e s
access only sub amilies compa ible wi h hei wo ldline, esolu ion, and eco ds,
while supe posi ion is no simul aneous mul iple ealiza ion o expe ien ial wo ld
bu p obabilis ic desc ip ion o u u e sec ion amilies.
On sca e ingspec al end, using scale iden i y
φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
as ime scale benchma k, whe e
φ
is o al sca e ing hal -phase,
ρ el
is ela i e s a e
densi y,
Q(ω) = −iS(ω)†∂ωS(ω)
is Wigne Smi h g oup delay ma ix; on modula
ow end, using Tomi aTakesaki modula ow and ConnesRo elli he mal ime
hypo hesis o dene modula ime; on g a i y end, using GibbonsHawkingYo k
bounda y ac ion and B ownYo k quasilocal ene gy o dene geome ic ime. All
h ee p o en o belong o same ime scale equi alence class unde bounda y ime
geome y amewo k.
On his basis, h ee main esul s: (1) Es ablish igo ous deni ion o obse e
sec ion: iple o imelike wo ldline
γ
, esolu ion pa ame e
Λ
, obse able subal-
geb a
Aγ,Λ
, iewing sec ions as slices on o al space
Y=M×X◦
unde unied
scale; unde local causali y and gene alized en opy ex emali y assump ions, p o e
exis ence o a leas one causally consis en sec ion ex ension amily wi hin any -
ni e ime in e al. (2) Unde consis en his o ies and decohe ence amewo k, gi e
exis ence and consis ency heo em o expe ien ial sec ion amilies: global s a e
denes measu e on sec ion space, conc e e obse e 's expe ien ial wo ld is single-
b anch condi ioning o ha measu e; supe posi ion only mani es s in p obabili y
dis ibu ion o e all causally allowed sec ions wi hin nex p ope ime s ep. (3)
Using spa ial double-sli , Wheele delayed-choice, and ime-domain double-sli ex-
pe imen s as examples, p o e die en expe imen al a angemen s can be uni o mly
desc ibed as selec ions o obse e obse able subalgeb as and sec ion amilies: wi h-
ou pa h measu emen , only eading sc een posi ion, expe ien ial sec ion inhe i s
global cohe ence o ming in e e ence inges; wi h pa h measu emen o in oduc-
ing ime-domain sli s, h ough en i onmen coupling ex ending obse able subal-
geb a, cohe ence decohe es, expe ien ial wo ld ansi ions o classical pa h causali y
1
o ene gy spec um in e e ence, while sec ion e olu ion always obeys local causal
s uc u e.
Keywo ds:
Bounda y Time Geome y; Unied Time Scale; Obse e Sec ion; Causal
Consis ency; Condi ionalized S a e; Double-Sli In e e ence; Delayed Choice; Time-
Domain Double-Sli ; Wigne Smi h G oup Delay; Modula Flow; Gene alized En opy
MSC 2020:
81Q65, 81U40, 83C45, 46L55

1 In oduc ion and His o ical Con ex
1.1 Measu emen P oblem, Supe posi ion, and Desc ip ion o
Wo ld
S anda d quan um mechanics desc ibes physical sys ems ia Hilbe space s a es
|ψ⟩
o densi y ope a o s
ρ
, cha ac e izes obse ables ia sel -adjoin ope a o s o POVMs,
p obabili ies gi en by Bo n ule. Measu emen p oblem: how o uni y uni a y e olu ion
and p ojec ion- hen- eno maliza ion e olu ion ules o holis ic desc ip ion o sys em
+ obse e + en i onmen . E e e many-wo lds emphasizes global uni a y e olu ion,
single-sho esul s co esponding o b anches; consis en his o ies and decohe ence di-
ec ly assign p obabili ies o a ious ime-o de ed p ojec ion s ings on global Hilbe
space, equi ing in e e ence e ms be ween die en his o ies negligible on obse able
subalgeb as.
Howe e , ega dless o in e p e a ion choice, answe ing wha is he wo ld his conc e e
obse e sees a his momen  equi es simul aneously handling:
1. Global quan um s a e e olu ion on backg ound including g a i y; 2. Obse e
as physical sys em in space ime, possessing only ni e esolu ion
Λ
and ni e eco ding
capaci y; 3. Fundamen al causal s uc u e and g a i a ional geome y cons ain s on
small causal diamonds, plus gene alized en opy mono onici y.
In combining gene al ela i i y and quan um eld heo y, Jacobson's en anglemen
equilib ium wo k shows: wi hin small geodesic balls, gene alized en opy ex emali y
condi ions can de i e Eins ein equa ions, unde s anding g a i a ional geome y as eec-
i e equa ions o bounda y en opyene gy o ganiza ion. Meanwhile, QNEC and QFC
p oposals e o mula e ene gy condi ions as second-o de de o ma ion inequali ies o gen-
e alized en opy. These esul s join ly poin o pic u e: ime and causali y hemsel es
should be unde s ood as geome ic s uc u es on bounda y da a, no p e-gi en back-
g ound pa ame e s in bulk.
1.2 Bounda y Time Geome y and Unied Time Scale
In abs ac sca e ing heo y, spec al shi unc ion
ξ(λ)
o sel -adjoin ope a o pai
(H, H0)
and ene gy-dependen sca e ing ma ix
S(ω)
connec ed by Bi manK ein o -
mula:
de S(ω) = exp{−2πi ξ(ω)}
. Fo app op ia e
, K ein ace o mula
T ( (H)−
(H0)) = R ′(λ)ξ(λ)dλ
. Combined wi h Wigne Smi h g oup delay ope a o
Q(ω) =
−iS(ω)†∂ωS(ω)
, ob ain unied scale iden i y o o al sca e ing phase de i a i e, ela i e
s a e densi y, and g oup delay ace:
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
2
whe e
φ(ω) = 1
2a g de S(ω)
. This iden i y di ec ly in e p e s measu able g oup delay
eadings as  ime scale densi y.
On ope a o algeb a end, Tomi aTakesaki modula heo y shows: o any
(M,Ω)
wi h cyclicsepa a ing ec o , exis s one-pa ame e au omo phism g oup
σΩ
(A) = ∆i
ΩA∆−i
Ω
gene a ed by modula ope a o
∆Ω
, any ai h ul no mal s a e sa ises co esponding
KMS condi ion. Connes u he p o es: modula ows o die en ai h ul s a es iden-
ical on ou e au omo phism g oup
Ou (M)
, hus exis s s a e-independen  ime ow
on
Ou (A∂)
. ConnesRo elli he mal ime hypo hesis acco dingly in e p e s modula
pa ame e as  ime in gene ally co a ian con ex .
On g a i a ional end, Eins einHilbe ac ion plus GibbonsHawkingYo k bounda y
e m ensu es well-dened a ia ion unde xed induced me ic; B ownYo k quasilocal
ene gy
EBY
and bounda y Hamil onian
H∂
gene a e geome ic ime ansla ion.
P io wo k showed: unde app op ia e ma ching condi ions, hese h ee ypes o ime
scales all iewed as ep esen a i es in same ime scale equi alence class
[τ]
.
1.3 P oblems and Con ibu ions
This pape add esses: unde BTG amewo k, how o cha ac e ize wo ld obse e s see
as sec ions on bounda y ime geome y, uni o mly explain:
1. Condi ions o in e e ence pa e n appea ance/disappea ance in o dina y/delayed-
choice double-sli ; 2. Pa icle sel -in e e ence on ime scales in ime double-sli ; 3. How
obse e expe ience o wo ld cons uc s om single-b anch sec ion condi ioning a he
han simul aneous mul i-sec ion supe posi ion.
Main con ibu ions:
(1) Axioma iza ion o obse e sec ions.
Dene obse e
O= (γ, Λ,Aγ,Λ)
; wo ld
sec ion a ime
τ
as
Στ= (γ(τ),Aγ,Λ(τ), ργ,Λ(τ))
; p o e exis ence o causally consis en
sec ion amilies.
(2) Condi ionaliza ion heo em.
Global s a e denes measu e on sec ion space;
expe ien ial wo ld is single-b anch condi ioning; supe posi ion in p obabili y dis ibu ion
o e u u e sec ions.
(3) Unied ea men o double-sli a ian s.
All as selec ions o obse able
subalgeb as and sec ion amilies; in e e ence s. pa h in o ma ion as choices aec ing
accessible sec ion sub amilies.

2 Model and Assump ions
2.1 Unied Time Scale
Deni ion 2.1
(Time Scale Equi alence Class)
.
Two ime scales
τ1, τ2
belong o same
[τ]
i ela ed by
τ2=aτ1+b
,
a > 0
.
Co e iden i y:
κ(ω) := φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
3
2.2 Bounda y Time Geome y
On mani old wi h bounda y
(M, g, ∂M)
, gene alized en opy
Sgen(λ) = A/(4G) + Sou
sa ises mono onici y along null gene a o s unde QNEC/QFC.
2.3 Obse e Sec ions
Deni ion 2.2
(Obse e T iple)
.
O= (γ, Λ,Aγ,Λ)
whe e
γ
is imelike wo ldline,
Λ
esolu ion scale,
Aγ,Λ⊆ A
accessible algeb a.
Deni ion 2.3
(Wo ld Sec ion)
.
A
τ
:
Στ= (γ(τ),Aγ,Λ(τ), ργ,Λ(τ))
.
Deni ion 2.4
(Causally Consis en Sec ion Family)
.
{Στ}τ∈[0,T ]
is causally consis en
i : (i) Local causali y; (ii) Dynamical ex endabili y; (iii) Reco d consis ency.

3 Main Resul s
3.1 Theo em 3.1 (Exis ence o Expe ien ial Sec ion Family)
Fo sys em sa is ying local hype bolici y, Hadama d condi ion, gene alized en opy ex-
emali y, gi en obse e
O
wi h eco d sequence
{ τ}
, exis s causally consis en sec ion
amily compa ible wi h eco ds a almos e e y
τ
.
3.2 Theo em 3.2 (Condi ionaliza ion S uc u e)
Global s a e
ρ
on o al space
Y
induces p obabili y measu e
µρ
on sec ion space
Σ
.
Obse e 's expe ien ial wo ld a
τ
is single sec ion
Στ
d awn om
µρ
condi ioned on pas
eco ds:
P(Στ|{Στ′}τ′<τ ) = µρ(Στ∩compa ible)
µρ(compa ible) .
Supe posi ion only in p obabili y dis ibu ion o e u u e sec ions.
3.3 Theo em 3.3 (Double-Sli Unica ion)
Spa ial double-sli :
Wi hou which-pa h measu emen ,
Aγ,Λ
con ains only sc een po-
si ion; expe ien ial sec ion inhe i s global phase cohe ence
⇒
in e e ence.
Wi h which-pa h measu emen , en i onmen coupling ex ends
Aγ,Λ
o include pa h
de ec o ; decohe ence
⇒
no in e e ence.
Time double-sli :
Tempo al pulse pai c ea es ime-domain in e e ence in ene gy
spec um; Fou ie dual o spa ial case unde Wigne Smi h delay.
Delayed choice:
Pos e io measu emen se ing choice changes condi ional p oba-
bili ies bu no an e io uncondi ional dis ibu ions; no e ocausali y.

4
4 P oo s (Ske ch)
4.1 P oo o Theo em 3.1
Use causal diamond s uc u e + en opy ex emali y + consis en his o ies. Measu e-
heo e ic a gumen s show gene ic exis ence. Appendix B.
4.2 P oo o Theo em 3.2
Decohe ence unc ional denes measu e on his o ies. Condi ioning on obse e eco ds
yields single-b anch expe ience. Appendix C.
4.3 P oo o Theo em 3.3
Sca e ing ampli ude analysis + decohe ence heo y + Fou ie duali y. Appendix D.

5 Model Applica ions
5.1 Spa ial Double-Sli
S anda d se up: pa icles h ough wo sli s, de ec ion sc een. Wi hou pa h measu e-
men : in e e ence. Wi h pa h measu emen : no in e e ence. Ou amewo k: pa h
measu emen ex ends
Aγ,Λ
, changing accessible sec ion sub amily.
5.2 Wheele Delayed Choice
Choose measu emen se ing a e pa icle passes sli s. Uncondi ional pa e n indepen-
den o choice; condi ional pa e ns depend on choice. Ou amewo k: choice aec s
condi ioning, no an e io p obabili ies.
5.3 A osecond Time Double-Sli
Tempo al pulse pai
⇒
ene gy spec um oscilla ions. F inge pe iod
∆ω= 2π/∆
ela ed
o g oup delay. Ou amewo k: ime-domain in e e ence as Fou ie dual o spa ial case.

6 Enginee ing P oposals
1.
Mul i-sli wi h delayed choice:
Implemen as swi ching be ween in e e ome -
ic and which-pa h measu emen s; e i y uncondi ional pa e n in a iance.
2.
Time double-sli wi h a iable pulse sepa a ion:
Measu e ene gy spec um
s.
∆
; ex ac g oup delay.
3.
Sec ion omog aphy:
Mul i-de ec o ne wo k econs uc ing sec ion amily om
local eco ds.

5

7 Discussion
In e p e a ion-neu al s ance:
F amewo k compa ible wi h consis en his o ies, de-
cohe en his o ies, many-wo lds (as b anch selec ion). No collapse pos ula e needed.
Rela ion o quan um ounda ions:
Measu emen p oblem add essed by iew-
ing expe ience as single-b anch condi ioning on measu e o e sec ions, no on ological
collapse.
Open ques ions:
Quan um g a i y egime, non-Ma ko ian en i onmen s, obse e
sel - e e ence.

8 Conclusion
Unde bounda y ime geome y wi h unied scale
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
we axioma ized obse e expe ien ial wo ld as single-b anch condi ioning on measu e
o e causally consis en sec ion amilies. Double-sli a ian s unied as selec ions o
obse able subalgeb as. Supe posi ion in p obabili y dis ibu ion o e u u e sec ions,
no simul aneous mul i-sec ion ealiza ion.

Re e ences
[1] M. S. Bi man and M. G. K ein, a Xi (spec al shi ).
[2] Double-sli expe imen , Wikipedia.
[3] T. Jacobson, Phys. Re . Le .
75
(1995) 1260.
[4] R. Bousso e al., Phys. Re . D
93
(2016) 024017.
[5] Tomi aTakesaki, ope a o algeb as li e a u e.
[6] A. Connes and C. Ro elli, Class. Quan . G a .
11
(1994) 2899.
[7] R. B. G i hs, J. S a . Phys.
36
(1984) 219.
A Scale Iden i y Realiza ion
[Bi manK ein + Wigne Smi h...]
B Sec ion Family Exis ence
[Measu e heo y + en opy ex emali y...]
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C Condi ionaliza ion Theo em
[Decohe ence unc ional + condi ioning...]
D Double-Sli Calcula ions
[Sca e ing ampli udes + Fou ie duali y...]
7