Boundary as Unified Stage: From Time Translation Operator, Null--Modular Double Cover to GHY Boundary Term

Bounda y as Unied S age:
F om Time T ansla ion Ope a o , NullModula
Double Co e
o GHY Bounda y Te m
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
Cons uc unied amewo k wi h bounda y as sole undamen al s age, gluing
h ee ma u e bu usually sepa a e s uc u es as h ee p ojec ions o same objec :
(i) Based on Bi manK einF iedelWigne Smi h amewo k,  ime ansla ion
ope a o  cha ac e ized by sca e ing phase, spec al shi unc ion, Wigne Smi h
ime delay ma ix;
(ii) Based on Tomi aTakesaki modula heo y and ecen esul s on causal dia-
mond/null su ace modula Hamil onians, Ma ko p ope y, QNEC, NullModula
double co e and o e lapping causal diamond chains;
(iii) Rep esen ed by GibbonsHawkingYo k (GHY) bounda y e m and i s gen-
e aliza ion wi h null shee s and join s, g a i a ional bounda y ac ion and B own
Yo k quasilocal ene gy.
Unde clea ly s a ed applicabili y domains and assump ions, gi e ou main e-
sul s:
(1) In sca e ing sys ems sa is ying ace-class pe u ba ion condi ions, sca e -
ing hal -phase de i a i e, K ein spec al shi densi y, Wigne Smi h delay ma ix
ace cons i u e same bounda y ime measu e;
(2) In ela i is ic quan um eld heo y unde s anda d assump ions, modula
Hamil onians on causal diamonds and null su aces localizable as weigh ed in eg als
o s essene gy ow on NullModula double co e , sa is ying Ma ko inclusion-
exclusion law o o e lapping diamond amilies;
(3) On piecewise space ime bounda ies (wi h null shee s and join s) in gen-
e al ela i i y, a e adding GHY- ype bounda y/join e ms, ac ion a ia ion o
xed induced geome y well-dened; B ownYo k bounda y s ess enso gene a es
bounda y ime ansla ion as hi d ype o bounda y ime;
(4) Wi h app op ia e ma ching maps, hese h ee bounda y imes uniable as
die en ealiza ions o same one-pa ame e au omo phism g oup, hus es a ing
ime, algeb a, geome y simul aneously as die en aspec s o bounda y da a.
P o ide unied scale examples in black hole he modynamics, AdS/CFT, sca -
e ing ne wo k expe imen s; p opose se e al enginee ing schemes es able on mesoscale
expe imen al pla o ms.
1
Keywo ds:
Bounda y Physics; Time T ansla ion Ope a o ; Bi manK ein Spec al
Shi ; Wigne Smi h Time Delay; Tomi aTakesaki Modula Theo y; NullModula Dou-
ble Co e ; Ma ko P ope y; GibbonsHawkingYo k Bounda y Te m; B ownYo k Quasilo-
cal Ene gy; Holog aphy

1 In oduc ion and His o ical Con ex
1.1 Pa adigm Shi om Bulk o Bounda y
De elopmen o ela i i y, quan um eld heo y, quan um g a i y shows clea bulk- o-
bounda y end. Quan um sca e ing heo y cen e ed on
S
-ma ix di ec ly encodes dy-
namical in o ma ion on space ime asymp o ic adia ion bounda y; Bi manK ein iden i y
connec s sca e ing de e minan phase wi h spec al shi unc ion, making bulk spec al
changes eadable om bounda y sca e ing da a.
In algeb aic quan um eld heo y, Tomi aTakesaki modula heo y shows: gi en
local algeb a and ai h ul s a e, modula ow
σω
is canonical one-pa ame e au omo -
phism g oup o algeb a; geome ic ealiza ion gi en clea cha ac e iza ion by Bisognano
Wichmann heo em, sphe ical egion and null su ace modula Hamil onian local exp es-
sions.
In gene al ela i i y, Eins einHilbe bulk e m alone canno gi e well-dened a ia-
ional p inciple o mani olds wi h bounda y; mus add GibbonsHawkingYo k bounda y
e m; when null shee s and join s p esen , mus in oduce co esponding null bounda y
and co ne e ms o ensu e ac ion die en iabili y and Hamil onJacobi s uc u e.
These ad ances join ly sugges : uly compu able physical objec s o en concen-
a ed on bounda y, while bulk mo e like econs uc ion o e olu ion esul o bounda y
da a.
1.2 Th ee Seemingly Die en Theo e ical Pa adigms
Th ee specic pa adigms his pape ocuses on:
(1) Sca e ing end: Time as bounda y ansla ion scale
Le
(H0, H0+V)
be sel -adjoin ope a o pai ,
V
in app op ia e ace class making
wa e ope a o s and sca e ing ma ix
S(ω)
well-dened sa is ying Bi manK ein condi-
ions. Then spec al shi unc ion
ξ(ω)
exis s sa is ying
de S(ω) = exp(−2πiξ(ω)),
whose de i a i e
ξ′(ω)
has exac ace o mula ela ion wi h sca e ing phase de i a i e
and Wigne Smi h ime delay ma ix
Q(ω) = −iS(ω)†∂ωS(ω)
ace.
This makes
φ′(ω)
,
ξ′(ω)
,
Q(ω)
join ly cha ac e ize bounda y ime measu e, iew-
ing ime as ansla ion pa ame e gene a ed by bounda y spec al da a.
(2) Algeb aicgeome ic end: NullModula double co e and o e lapping
causal diamond chains
In Minkowski o AdS bounda y CFT, o acuum s a e es ic ed o wedge egion,
sphe ical diamond, o null su ace egion, modula Hamil onian w i able as local in eg al
o s essene gy enso on bounda y, o ming inni e-dimensional Lie algeb a and Ma ko
p ope y on null su aces.
2
Fo causal diamond
D
, in oduce NullModula double co e composed o u u e/pas
null bounda ies
(E+, E−)
; modula Hamil onian
KD
w i able as
KD= 2πX
σ=±ZEσ
gσ(λ, x⊥)Tσσ(λ, x⊥)dλ dd−2x,
whe e
Tσσ
a e null di ec ion componen s. Modula Hamil onians o o e lapping diamond
chains sa is y inclusion-exclusion and Ma ko s i ching p ope ies.
(3) G a i y end: GHY bounda y e m, null bounda ies, B ownYo k
quasilocal ene gy
On space ime mani old
M
wi h bounda y, a e adding GHY bounda y e m
S=SEH +SGHY
a ia ion o xed induced me ic well-dened; when bounda y includes null shee s and
join s, mus supplemen null bounda y and co ne e ms.
Hamil onJacobi analysis o egions wi h bounda y yields B ownYo k bounda y
s ess enso
Tab
BY =2
√−h
δS
δhab
,
whose ime componen gi es quasilocal ene gy and Hamil onian gene a ing bounda y
ime ansla ion.
These h ee ou es espec i ely highligh spec alsca e ing, modula algeb aic, geome ic
g a i a ional aspec s, bu all depend on bounda y: sca e ing dened by asymp o ic
bounda y, modula ow localized on egion bounda y, g a i a ional ac ion die en iabil-
i y de e mined by bounda y e ms.
1.3 Goals and Main Th ead
Goal: P o ide ma hema ically sel -consis en amewo k wi h clea applicabili y domain,
iewing abo e h ee pa adigms as h ee p ojec ions o same bounda y s uc u e.
Co e idea: 1. Take app op ia e bounda y da a iple
(∂M,A∂, µ∂)
as undamen al objec :
∂M
geome ic bounda y,
A∂
obse able algeb a o sca e ing
algeb a on i ,
µ∂
ime scale measu e om spec al shi o ene gy ow;
2. Res a e ime delay in sca e ing heo y, modula ow in modula heo y, B own
Yo k bounda y Hamil onian in g a i y all as die en ep esen a ions o one-pa ame e
au omo phism g oup on his bounda y s uc u e;
3. Unde s ic ly limi ed condi ions, p o e hese ep esen a ions mu ually equi alen ,
comp essing  imealgeb ageome y h ee h eads on o unied bounda y s age.

2 Model and Assump ions
Cons uc abs ac model simul aneously accommoda ing sca e ing sys ems, algeb aic
QFT, g a i a ional bounda ies; cla i y applicabili y domains o all esul s.
3
2.1 Abs ac Bounda y T iple
Deni ion 2.1
(Bounda y T iple)
.
Bounda y iple is da a
(∂M,A∂, ω∂)
whe e:
1.
∂M
is piecewise smoo h h ee-dimensional mani old, decomposable in o imelike,
spacelike, null shee s and hei join s
C
;
2.
A∂
is on Neumann algeb a ac ing on Hilbe space
H
, con aining bounda y
obse ables (sca e ing channels, bounda y elds, quasilocal ene gy ope a o s, e c.);
3.
ω∂
is ai h ul no mal s a e on
A∂
; GNS iple deno ed
(πω,Hω,Ωω)
.
Pos ula e 1
(Bounda y Comple eness)
.
Physical con en o bulk egion
M
comple ely e-
cons uc ible om some bounda y iple
(∂M,A∂, ω∂)
(wi hin gi en heo y's applicabili y
ange); ime e olu ion and esponse ope a o s all de e mined by bounda y one-pa ame e
au omo phism g oup and s a e e olu ion.
This pos ula e has die en conc e e ealiza ions in die en con ex s: wa e ope a o s
and
S
-ma ix in sca e ing heo y, bounda y CFT and bulk geome y in AdS/CFT, bulk
solu ion econs uc ion om bounda y da a in Hamil onJacobi pe spec i e.
2.2 Sca e ing End Assump ions
Assump ion 1
(S.1: T ace-class pe u ba ion and BK condi ions)
.
On Hilbe space
Hsca
,
H0
and
H=H0+V
a e sel -adjoin ope a o s,
V
in ace-class o s onge ideal
making sca e ing ma ix
S(ω)
exis o almos all ene gies
ω
wi h
S(ω)−1∈S1
. Unde
hese assump ions, K ein spec al shi unc ion
ξ(ω)
and Bi manK ein iden i y
de S(ω) = exp(−2πiξ(ω))
exis .
Assump ion 2
(S.2: Time delay ma ix and ace o mula)
.
Wigne Smi h ime delay
ope a o dened as
Q(ω) = −iS(ω)†∂ωS(ω),
Q(ω)
is ace-class ope a o sa is ying
Q(ω) = 2πξ′(ω)
in app op ia e sense.
Unde hese condi ions, dene bounda y ime measu e
dµsca
∂(ω) := 1
2π Q(ω)dω.
4
2.3 Modula Theo y and NullModula Double Co e Assump-
ions
Assump ion 3
(M.1: S anda d modula s uc u e)
.
A∂⊂ B(H)
is local algeb a o
causal egion
O
,
ω
is acuum o KMS s a e making
Ωω
cyclic and sepa a ing ec o o
A∂
, hus Tomi aTakesaki modula ope a o
∆
and one-pa ame e modula ow
σω
(A)=∆i A∆−i
exis .
Assump ion 4
(M.2: Geome ic modula ow)
.
Fo wedge egion, sphe ical causal di-
amond, o null su ace egion
O
in acuum s a e, modula ow's geome ic ac ion is
co esponding egion's Lo en z ans o ma ion o con o mal Killing ow; modula Hamil-
onian
KO=−log ∆
w i able as local in eg al o s essene gy enso .
Assump ion 5
(M.3: Ma ko p ope y and inclusion-exclusion)
.
Fo egion amilies
on null su ace, modula Hamil onians sa is y Ma ko p ope y p o ed by CasiniTes e
To oba: o nes ed o o e lapping egions along null line, condi ional mu ual in o ma ion
sa u a es s ong subaddi i i y, co esponding o modula Hamil onian inclusion-exclusion
iden i ies.
Based on his, dene NullModula double co e : o causal diamond
D
, bounda y
null hype su aces decompose as
E+∪E−
; dene weigh ed ene gy ow in eg al ep esen-
a ion modula Hamil onians on bo h shee s.
2.4 G a i a ional Bounda y Assump ions
Assump ion 6
(G.1: Ac ion wi h bounda ies/null bounda ies)
.
On ou -dimensional
Lo en zian mani old
M
, g a i a ional ac ion akes s anda d o m
S=1
16πG ZM
√−g(R−2Λ) d4x+S∂,
whe e
S∂=S l/sp
GHY +Snull
N+Sco ne
C
a e GHY bounda y e ms on imelike/spacelike shee s, imp o ed e ms on null shee s,
co ne e ms a join s.
Assump ion 7
(G.2: Va ia ion well-denedness and B ownYo k s ess enso )
.
On
a ia ion amily xing bounda y induced me ic (and app op ia e equi alen da a o
null bounda ies),
δS
con ains only bulk e ms, yielding Eins ein equa ions. Fo gi en
imelike bounda y h ee-shee
3B
, ac ion a ia ion wi h espec o bounda y me ic de-
nes B ownYo k bounda y s ess enso
Tab
BY =2
√−h
δS
δhab
,
whose con ac ion wi h bounda y Killing ec o
ξa
gi es quasilocal ene gy and Hamil o-
nian gene a ing bounda y ime ansla ion.
Unde hese assump ions, iew B ownYo k Hamil onian as hi d ype o bounda y
ime gene a o .

5

3 Main Resul s (Theo ems and Alignmen s)
S a e ou main esul s es ablishing co espondences among h ee heo e ical h eads.
Theo em 3.1
(A: BKWigne Smi hTime Scale Iden i y)
.
Unde Assump ions S.1
S.2, dene:
•
To al sca e ing phase
Φ(ω) = a g de S(ω)
, hal -phase
φ(ω) = 1
2Φ(ω)
;
•
K ein spec al shi unc ion
ξ(ω)
sa is ying
de S(ω) = exp(−2πiξ(ω))
;
•
Wigne Smi h
ime delay ope a o
Q(ω) = −iS(ω)†∂ωS(ω)
.
Then almos e e ywhe e ni e de i a i es
ξ′(ω)
and
φ′(ω)
exis such ha
φ′(ω)
π=ξ′(ω) = 1
2π Q(ω)
holds almos e e ywhe e.
Thus measu e
dµsca
∂(ω) := 1
2π Q(ω)dω
equi alen o spec al shi measu e
dξ(ω)
and sca e ing phase scale
π−1dφ(ω)
, iewable
as unied bounda y ime scale.
Deni ion 3.2
(NullModula Double Co e )
.
In
d
-dimensional Minkowski space ime,
conside causal diamond wi h e ices
p, q
:
D(p, q) = J+(p)∩J−(q).
Bounda y composed o u u e null hype su ace
N+
and pas null hype su ace
N−
.
Dene NullModula double co e
e
ED:= E+⊔E−,
whe e
E±
a e wo smoo h lea es o
N±
a e emo ing join poin s; in oduce ane
pa ame e
λ
and ans e se coo dina es
x⊥
on each lea .
Theo em 3.3
(B: Modula Hamil onian Null Measu e Localiza ion)
.
Unde Assump ions
M.1M.2, o Minkowski acuum s a e es ic ed o causal diamond
D
local algeb a
A(D)
,
modula Hamil onian w i able as
KD= 2πX
σ=±ZEσ
gσ(λ, x⊥)Tσσ(λ, x⊥)dλ dd−2x ,
whe e: 1.
T++ =T
,
T−− =Tuu
a e s essene gy enso componen s along wo null
di ec ions; 2. Weigh unc ions
gσ(λ, x⊥)
de e mined solely by causal diamond geome ic
da a, linea ly degene a ing a endpoin s.
Mo eo e , o o e lapping causal diamond amily
{Dj}
on same null su ace, modula
Hamil onians sa is y inclusion-exclusion iden i y
K∪jDj=X
k≥1
(−1)k−1X
j1<···<jk
KDj1∩···∩Djk,
and Minkowski acuum s a e es ic ed o hese egions sa ises Ma ko p ope y: o
app op ia ely nes ed
A, B, C
, condi ional mu ual in o ma ion
I(A:C|B)=0
equi alen o abo e inclusion-exclusion iden i y.
6
Theo em B shows
: Modula Hamil onian comple ely localizable on null measu e
bounda y; NullModula double co e p o ides pu ely bounda y geome ic ealiza ion o
modula ow.
Theo em 3.4
(C: GHYB ownYo k Bounda y Hamil onian)
.
Unde Assump ions G.1
G.2, conside space ime egion
M
wi h imelike/spacelike/null shee s and join s; o al
ac ion
S=1
16πG ZM
√−g(R−2Λ) d4x+S l/sp
GHY +Snull
N+Sco ne
C
sa ises:
1. Fo all me ic a ia ions
δgµν
xing induced geome y (and equi alen da a on null
bounda ies) on bounda y, o al a ia ion
δS =1
16πG ZM
√−g Gµν δgµν d4x,
yielding Eins ein equa ions;
2. Fo gi en imelike bounda y
3B
wi h imelike Killing ec o
ξa
, B ownYo k bound-
a y s ess enso
Tab
BY =2
√−h
δS
δhab
denes bounda y Hamil onian
Hg a
∂[ξ] = ZB
√σ uaTab
BY ξbd2x
gene a ing quasilocal ime ansla ion along
3B
ime di ec ion
ξa
; when
3B
ex ends o
inni y, his Hamil onian con e ges o ADM o Bondi mass.
Thus
Hg a
∂
iewable as hi d ype o bounda y ime gene a o , a same le el as Null
Modula and sca e ing end gene a o s.
Deni ion 3.5
(Unied Bounda y Time Gene a o )
.
Gi en bounda y iple
(∂M,A∂, ω∂)
,
call sel -adjoin ope a o
H∂
unied bounda y ime gene a o i :
1. In sca e ing ep esen a ion,
H∂
's spec al decomposi ion measu e o ene gy a i-
able
ω
equi alen o
dµsca
∂(ω)
;
2. In algeb aic ep esen a ion, modula Hamil onian sa ises
K∂= 2πβ−1H∂
o some
posi i e
β
, p oducing Tomi aTakesaki modula ow;
3. In geome ic ep esen a ion, B ownYo k Hamil onian w i able as
Hg a
∂[ξ] =
⟨H∂, J[ξ]⟩
whe e
J[ξ]
is bounda y cha ge unc ional associa ed wi h Killing ec o
ξ
.
Theo em 3.6
(D: Bounda y T ini y P inciple)
.
Assume ollowing ma ching s uc u e
exis s:
1. Sca e ing sys em's incoming/ou going channels embeddable in o sepa able subal-
geb a o some QFT's bounda y algeb a
A∂
, making sca e ing phase
φ(ω)
consis en wi h
modula Hamil onian's spec al phase in asymp o ic egions;
2. QFT's s essene gy enso expec a ion alue connec ed o B ownYo k bounda y
s ess enso ia holog aphic dic iona y o semiclassical Eins ein equa ions;
3. Bounda y Killing ime and sca e ing ene gy no maliza ion cons an s sa is y he -
mal ime hypo hesis no maliza ion condi ion: modula ow pa ame e die s om phys-
ical ime by cons an scale.
7
Unde hese condi ions, exis s unique (up o global ane ans o ma ion) unied
bounda y ime gene a o
H∂
making
Sca e ing ime delay
⇐⇒
Modula ow pa ame e
⇐⇒
B ownYo k bounda y ime
equi alen in common domain.
Mo e specically, posi i e cons an s
c1, c2
exis making
φ′(ω)
π=ξ′(ω) = 1
2π Q(ω),
KD= 2πZTσσgσ,
Hg a
∂[ξ] = Z√σ uaTab
BYξb
sa is y
H∂=Zω dµsca
∂(ω) = c1KD+c−1
2Hg a
∂
gi ing h ee ealiza ions o same one-pa ame e g oup
e−i H∂
in die en ep esen a ions
o same bounda y Hilbe space.
Theo em D doesn' claim unied gene a o au oma ically cons uc ible in a bi a y
heo ies, bu poin s ou unde abo e ma ching condi ions, h ee h eads na u ally com-
p ess on o same bounda y ime objec .

4 P oo s
P o ide p oo skele ons o main heo ems; ne echnical de ails and special cases in
appendices.
4.1 P oo Skele on o Theo em A
Bi manK ein iden i y shows unde Assump ion S.1, spec al shi unc ion
ξ(ω)
exis s
such ha
de S(ω) = exp(−2πiξ(ω)),
hus
Φ(ω) = a g de S(ω) = −2πξ(ω).
Hal -phase
φ(ω) = −πξ(ω)
. Die en ia ing a smoo h poin s:
φ′(ω) = −πξ′(ω).
On o he hand, EisenbudWigne Smi h ime delay heo y shows: in po en ial sca -
e ing sys ems sa is ying mode a e decay and egula i y condi ions, Wigne Smi h ime
delay ope a o denable as
Q(ω) = −iS(ω)†∂ωS(ω),
p o ing i s ace sa ises
Q(ω) = 2π ξ′(ω).
8
Combining wo equa ions gi es
φ′(ω)
π=ξ′(ω) = 1
2π Q(ω),
comple ing Theo em A p oo . Rigo ous ea men equi es conside ing excep ional ze o-
measu e se s on
ω
; see Appendix A.
[P oo s o Theo ems B, C, D condensed o space...]

5 Model Applica ions
Show unied bounda y amewo k applicabili y in h ee ypical physical scena ios: black
hole he modynamics, AdS/CFT, ni e-scale sca e ing ne wo ks.
5.1 Black Hole The modynamics
Fo s a ic black hole, e en ho izon is null bounda y; su ace g a i y
κ
and Hawking
empe a u e
TH=κ/(2π)
join ly cha ac e ize he mal p ope ies. QFT acuum s a e
ou side ho izon in wedge egion's modula ow equi alen o Euclidean ime ansla ion
a ound ho izon, pe iod being
βH= 1/TH
.
In unied bounda y amewo k:
1.
Sca e ing end
: Conside xed-ene gy sca e ing ou side black hole; g oup delay
Q(ω)
ene gy dependence encodes eec i e po en ial ba ie and quasino mal mode
s uc u e nea ho izon;
2.
NullModula end
: Ho izon i sel iewable as pa o NullModula double
co e ; modula Hamil onian localized on null gene a o , p opo ional o
T
in eg al;
3.
G a i y end
: Fo bounda y wo-sphe e su ounding black hole, B ownYo k
quasilocal ene gy app oaches ADM mass a inni y, o ms Legend e s uc u e wi h Bekens ein
Hawking en opy and empe a u e p oduc nea ho izon.
Unied gene a o
H∂
gi es compa ible ime ansla ion and hea geome y ela ions a
h ee endpoin s, es a ing black hole he modynamics as pu ely bounda y phenomenon.
5.2 AdS/CFT and Holog aphic Time Recons uc ion
In AdS/CFT duali y, bounda y CFT ime e olu ion gene a ed by Hamil onian
HCFT
;
modula Hamil onian and gene alized en opy ex emal su aces join ly de e mine bulk
geome ic esponse.
Unied bounda y amewo k p o ides na u al language:
•
Sca e ing end
: Bulk
high-ene gy p ocess sca e ing delay co esponds o bounda y CFT co ela ion unc ion
phase s uc u e;
•
NullModula end
: Bounda y CFT sphe ical egion modula ow
co esponds ia JLMS o bulk Killing ow and minimal su ace;
•
G a i y end
: These
Killing ows gene a ed by B ownYo k bounda y Hamil onian, ac ion eadable om GHY
e m Hamil onJacobi a ia ion.
Thus holog aphic ime econs uc ion unde s andable as unied gene a o
H∂
's wo
ep esen a ions on CFT and AdS sides.
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