Boundary as Clock: Time as Unified Translation Operator of Phase--Spectral Shift--Modular Flow

Bounda y as Clock:
Time as Unied T ansla ion Ope a o
o PhaseSpec al Shi Modula Flow
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
Agains backg ound o gene al
C∗
-algeb as and ope a o sca e ing heo y, con-
s uc amewo k o  ime = bounda y ansla ion. Time no iewed as p e-gi en
ow pa ame e in bulk domain bu dened as unique ansla ion scale gene a ed by
bounda y spec al da a, main aining sel -consis ency among phasespec al shi 
modula ow iple eadings.
Specically: Fi s , in sca e ing sys ems sa is ying Bi manK ein condi ions,
ake o al sca e ing phase
Φ(ω) = a g de S(ω)
, spec al shi unc ion
ξ(ω)
, ela-
i e s a e densi y
∆ρ(ω)
, Wigne Smi h ope a o
Q(ω) = −iS(ω)†∂ωS(ω)
as co e;
es ablish scale iden i y
φ′(ω)
π=ρ el(ω) = 1
2πT Q(ω), φ(ω) := 1
2Φ(ω),
in e p e ing as spec al ule o scaling ime by bounda y phase g adien .
Second, o gi en bounda y obse able algeb a
A∂
and ai h ul s a e
ω
's GNS
ep esen a ion, in oduce Tomi aTakesaki modula ope a o
∆
and modula ow
σω
; unde sca e ingKMS consis ency assump ion p o e: ime pa ame e
in-
e ed om
S(ω)
and
Q(ω)
iden ical wi h modula ime pa ame e unde app op i-
a e no maliza ion.
Finally p opose  ime equi alence p inciple and scale iden i y axiom: any
physical e olu ion o bulkex e io da a pai s equi alen ly ew i able as ansla ion
U( ) = e−i H∂
gene a ed by bounda y gene a o
H∂
, whe e
uniquely de e mined
by bounda y spec al measu e eadou unc ion
T
.
Unde na u al mono onici y and egula i y assump ions, p o e ime scale sa -
is ying hese axioms unique in sense o addi i e and p opo ional ans o ma ions.
Thus a pu ely ope a o geome ic and sca e ing in o ma ion le el, ime cha ac-
e ized as bounda y ansla ion pa ame e wi h phasespec al shi modula ow
iple eading sel -consis en , p o iding es able heo e ical basis o econs uc ing
space ime and dynamics om bounda y in o ma ion.
Keywo ds:
Time Essence; Sca e ing Phase; Spec al Shi Func ion; Wigne Smi h
Ope a o ; Modula Flow; Bounda y Algeb a; KMS S a e; Time Equi alence P inciple
MSC 2020:
81U40, 81Q10, 46L55, 58J40

1
1 In oduc ion and His o ical Con ex
Classical mechanics iews ime as absolu ely uni o mly owing ex e nal pa ame e ; in gen-
e al ela i i y, ime embedded as coo dina e unc ion wi h causal cone s uc u e; s anda d
quan um heo y mos ly uses in e nalex e nal pa ame e spli , aking ime as con inuous
pa ame e in Sch ödinge equa ion.
In con as , de elopmen o ope a o algeb as and quan um s a is ical mechanics
shows: gi en obse able algeb a
A
and s a e
ω
, can cons uc in insic au omo phism
g oup amily
σω
ia Tomi aTakesaki modula heo y, na u ally in e p e ed as modula
ime. This s uc u e plays cen al ole in KMS condi ions and equilib ium s a e heo y.
On o he hand, in sca e ing heo y, Wigne Smi h ime delay concep in e p e s o al
sca e ing phase equency de i a i e as a e age  esidence ime pa icle expe iences in
po en ial eld; his concep ex ensi ely gene alized and e ied in andom media, chao ic
sca e ing, elec omagne ic, acous ic sys ems.
In igo ous ope a o sca e ing amewo k, Bi manK ein o mula connec s sca e ing
de e minan wi h spec al measu e using spec al shi unc ion
ξ(λ)
, gi ing
de S(λ) = exp(−2πi ξ(λ)),
while K ein ace o mula connec s spec al shi unc ion be ween wo ope a o s wi h es
unc ion die ence ace. These chains uni y phasespec al shi  ela i e s a e densi y
as die en aspec s o same objec .
ConnesRo elli  he mal ime hypo hesis u he p oposes: in gene ally co a ian
quan um heo ies, physical ime ow shouldn' be gi en by p ese ex e nal pa ame e
bu join ly de e mined by sys em's s a is ical s a e and obse able algeb a; ime ow
ealized by s a e's modula au omo phism g oup. This makes  ime = modula ow
powe ul candida e answe .
Abo e h ee h eads espec i ely answe how o ead ou ime om sca e ing phase,
how o scale s a e densi y by spec al shi unc ion, how o cons uc ime ow om
s a e and algeb a.
This pape 's goal: wi hin single, geome ically minimal-s uc u e amewo k, uni y
hese h ee; gi e igo ous exis ence and uniqueness conclusions. Co e idea: in oduce
bounda y algeb a
A∂
as insideou side in o ma ion in e ace, equi ing:
1. All obse able ou pu s ul ima ely land on
A∂
; 2. Gi en ai h ul s a e
ω
and
sca e ing da a
S(ω)
, exis s unique (up o ane) ime ansla ion g oup
α
such ha :
•
α
gene a ed by sel -adjoin ope a o
H∂
in GNS ep esen a ion;
•α
consis en wi h
σω
;
•
Time ule unde
α
no malized by scale iden i y.
F om his pe spec i e, ime no longe ow a iable in bulk domain bu cha ac e ized
as unique ansla ion pa ame e ealizing alignmen be ween bounda y spec al da a and
modula ow.
This cha ac e iza ion main ains spi i ual con inui y wi h he mal ime hypo hesis bu
equi es addi ional obse able sca e ing da a as scale ancho , making ime ha e di ec
expe imen al eadou .

2 Model and Assump ions
Gi e model s uc u e and basic assump ions in abs ac amewo k. Goal: ob ain minimal
condi ion amily sucien o applying Bi manK ein o mula, spec al shi unc ion,
2
modula heo y wi hou elying on specic space ime geome y.
2.1 Bounda y Algeb a and S a e
Le
A∂
be sepa able
C∗
-algeb a ep esen ing bounda y obse ables. Selec ai h ul s a e
ω:A∂→C
; GNS ep esen a ion deno ed
(πω,Hω,Ωω)
sa is ying
ω(A) = ⟨Ωω, πω(A)Ωω⟩, A ∈ A∂,
Ωω
is cyclic and sepa a ing ec o .
Assume s ongly con inuous
C∗
-au omo phism amily exis s:
α :A∂→ A∂, ∈R,
ealized on GNS space by uni a y g oup
U( )
:
πω(α (A)) = U( )πω(A)U( )−1, U( )Ωω= Ωω.
Gene a o
H∂
is sel -adjoin ope a o sa is ying
U( ) = e−i H∂
.
Call
(A∂, ω, α )
**bounda y dynamical sys em**.
2.2 Sca e ing Sys em and Bi manK ein Se ing
Le
H0, H
be sel -adjoin ope a o s on sepa able Hilbe space
H
sa is ying ypical sca -
e ing assump ions: 1.
V:= H−H0
is ace-class pe u ba ion; 2.
H0
's absolu ely
con inuous spec um domina es on ene gy axis
I⊂R
; 3. Wa e ope a o s
W±
exis
making
W±= s-lim
→±∞ ei He−i H0Pac(H0);
4. Sca e ing ope a o
S:= W∗
+W−
is uni a y on
Pac(H0)H
.
In ene gy ep esen a ion,
S
decomposable as xed-ene gy sca e ing ma ix amily
S(λ) : K(λ)→ K(λ), λ ∈I,
whe e
K(λ)
is channel space a each ene gy. Assume
λ7→ S(λ)
sucien ly smoo h on
I
.
Unde hese assump ions, spec al shi unc ion
ξ(λ)∈L1
loc(I)
exis s sa is ying K ein
ace o mula
T ( (H)− (H0)) = ZI
′(λ)ξ(λ)dλ
o sucien ly la ge unc ion class.
Simul aneously, Bi manK ein o mula gi es sca e ing de e minan spec al shi
unc ion ela ion:
de S(λ) = exp(−2πi ξ(λ))
almos e e ywhe e on
I.
2.3 Rela i e Densi y o S a es, Sca e ing Phase, Wigne Smi h
Ope a o
Dene o al sca e ing phase
Φ(λ) := a g de S(λ), φ(λ) := 1
2Φ(λ).
3
F om Bi manK ein o mula:
Φ(λ)≡ −2πξ(λ) (mod 2π),
hus on locally con inuous ep esen a i e
Φ′(λ) = −2πξ′(λ).
Dene ela i e s a e densi y (DOS die ence)
∆ρ(λ) := ρ(λ)−ρ0(λ),
whe e
ρ, ρ0
a e s a e densi y unc ions o
H, H0
. Unde s anda d se ing,
∆ρ
and spec al
shi unc ion de i a i e sa is y
∆ρ(λ) = −ξ′(λ)⇒1
2πΦ′(λ) = ∆ρ(λ),
yielding
φ′(λ)
π= ∆ρ(λ).
On o he hand, o each ene gy, dene Wigne Smi h delay ope a o
Q(λ) := −iS(λ)†∂λS(λ)
on
K(λ)
.
Q(λ)
is sel -adjoin ope a o ; ace
T Q(λ)
equals o al sca e ing phase de i a i e unde s anda d sca e ing amewo k:
Φ′(λ) = T Q(λ).
Me ging abo e ela ions gi es scale iden i y
φ′(λ)
π= ∆ρ(λ) = 1
2πT Q(λ).
To a oid no a ion con usion, uni o mly deno e ene gy a iable as
ω
; w i e scale iden-
i y as
φ′(ω)
π=ρ el(ω) = 1
2πT Q(ω),
whe e
ρ el(ω) := ∆ρ(ω)
.
2.4 Modula Flow, KMS Condi ion, The mal Time
On GNS ep esen a ion
(πω,Hω,Ωω)
, dene Tomi a ope a o
S0πω(A)Ωω=πω(A)∗Ωω, A ∈ A∂.
Closu e deno ed
S
; pola decomposi ion
S=J∆1/2
gi es an ilinea uni a y conju-
ga ion
J
and modula ope a o
∆
. Tomi aTakesaki heo em asse s modula au omo -
phism amily exis s:
σω
(A) := ∆i A∆−i , ∈R,
4
o ming one-pa ame e au omo phism g oup on
A∂
;
ω
sa ises KMS condi ion o
σω
.
Fo mally w i e modula gene a o
Kω:= −log ∆, σω
(A) = ei KωAe−i Kω.
ConnesRo elli he mal ime amewo k p oposes: in gene ally co a ian eld heo-
ies, physical ime ow de e minable by gi en s a e and obse able algeb a, specically
modula ow
σω
.
This pape es ic s his idea o bounda y algeb a
A∂
; equi es modula ime con-
sis en wi h ime pa ame e scaled om sca e ing phasespec al shi Wigne Smi h
ope a o .

3 Main Resul s (Theo ems and Alignmen s)
P opose  ime equi alence p inciple, scale iden i y axiom, modula consis ency axiom;
gi e exis ence and uniqueness heo em o unied ime scale.
3.1 Time Equi alence P inciple and Bounda y Gene a o
Axiom 1
(Time Equi alence P inciple)
.
Conside ealizable bulkex e io da a pai s
(Din, Dou )
as ec o s o s a es in
Hin,Hou
. Bounda y Hilbe space
H∂
, sel -adjoin
ope a o
H∂
, uni a y g oup
U( ) = e−i H∂, ∈R
exis such ha o any ealizable da a pai , eal numbe
exis s sa is ying
Kou
∂Dou =U( )Kin
∂Din.
Axiom 2
(Bounda y Gene a o Axiom)
.
C∗
-algeb a
A∂
and ai h ul s a e
ω
exis making
abo e
U( )
ealize bounda y dynamics on
Hω
:
α (A) = U( )AU( )−1, A ∈ A∂,
and
U( )Ωω= Ωω
.
Call
(A∂, ω, U( ))
**bounda y as clock** da a.
3.2 Gauge Fixing by PhaseSpec al-Shi WS T ace
Axiom 3
(Scale Iden i y Axiom)
.
Sca e ing sys em sa ises a o emen ioned Bi man
K ein and Wigne Smi h condi ions. In ene gy window
I
, phase
φ(ω)
, ela i e s a e
densi y
ρ el(ω)
, Wigne Smi h ope a o
Q(ω)
exis sa is ying
φ′(ω)
π=ρ el(ω) = 1
2πT Q(ω), ω ∈I.
Dene ime die en ial as
d := 1
2πT Q(ω)dω.
Gi en e e ence poin
ω0, 0
, ime scale de e mined by
(ω)− 0=Zω
ω0
1
2πT Q(˜ω)d˜ω.
5

Scale iden i y axiom ans o ms sca e ing spec al s uc u e on equency axis in o
bounda y ansla ion scale on ime axis.
3.3 Modula Consis ency and Unied Time Flow
Axiom 4
(Modula Consis ency Axiom)
.
Fo bounda y dynamical sys em
(A∂, ω, α )
,
assume cons an
c > 0
exis s such ha o all
A∈ A∂
α (A) = σω
c (A),
whe e
σω
is Tomi aTakesaki modula ow. Abso bing
c
in o ime uni , can losslessly
ew i e as
α (A) = σω
(A).
Thus bounda y gene a o
H∂
and modula gene a o
Kω
die only by cons an shi :
H∂=Kω+λ1, λ ∈R.
Deni ion 3.1
(Time S uc u e)
.
Call quad uple
T= (A∂, ω, α , S(ω))
ime s uc u e i sa is ying: 1.
ω
is ai h ul no mal s a e on
A∂
; 2.
α
ealized on GNS
ep esen a ion by
U( ) = e−i H∂
,
H∂
sel -adjoin ; 3. Sca e ing ma ix amily
S(ω)
and
co esponding
φ(ω), Q(ω), ρ el(ω)
exis sa is ying scale iden i y; 4.
α =σω
as au omo -
phism g oups consis en .
Theo em 3.2
(Time Scale Exis ence)
.
Le
T
be ime s uc u e; assume in ene gy window
I
,
ρ el(ω)
is in eg able con inuous unc ion nonze o on some in e al. Then local bijec ion
ω←→ (ω)
exis s gi en by scale iden i y axiom such ha :
1. Fo all
A∈ A∂
,
α (ω)(A) = σω
(ω)(A)=∆i (ω)A∆−i (ω);
2. Fo sca e ing side, can iew
S(ω)
as
S( )
sa is ying
d
d φ(ω( )) = π ρ el(ω( )) = 1
2T Q(ω( )),
ew i ing phase g adien , ela i e s a e densi y, Wigne Smi h ace as ime de i a i es.
In o he wo ds, ime pa ame e
simul aneously pa ame izes modula ow and sca -
e ing ime eadou s, making la e obse able scale o o me .
Theo em 3.3
(Addi i eP opo ional Uniqueness o Scale)
.
Unde Theo em assump-
ions, u he assume: 1.
ρ el(ω)
s ic ly posi i e o s ic ly nega i e in conside ed ene gy
window; 2. Modula ow
σω
non- i ial: no nonze o ime
makes
σω
iden i y.
I ano he ime pa ame e
˜
and map
ω7→ ˜
(ω)
exis such ha : 1.
α˜
also ealizes as
modula ow:
α˜
=σω
˜
; 2. Scale iden i y holds unde
˜
in same ene gy window.
Then cons an s
a > 0
and
b∈R
exis making
˜
=a +b.
Time scale sa is ying axiom sys em unique in ane ans o ma ion sense; ime e e -
sal
(a < 0)
excluded.

6
4 P oo s
P o ide p oo s uc u e o main heo ems; concen a e echnical ope a o sca e ing and
modula heo y de ails in appendices.
4.1 Bi manK ein Iden i y and PhaseSpec al-Shi Rela ion
Unde p e ious assump ions, spec al shi unc ion
ξ(λ)
sa ises K ein ace o mula.
Taking smoo hed app oxima ion o
(λ) = χ(−∞,E](λ)
yields
ξ(E) = T (PH((−∞, E]) −PH0((−∞, E])),
hus
ξ′(λ) = −(ρ(λ)−ρ0(λ)) = −∆ρ(λ)
holds in dis ibu ional sense.
On o he hand, Bi manK ein o mula gi es
de S(λ) = exp(−2πi ξ(λ))
. Taking con-
inuous b anch and die en ia ing wi h espec o
λ
:
Φ′(λ) = −2πξ′(λ) = 2π∆ρ(λ),
i.e.,
1
2πΦ′(λ)=∆ρ(λ).
Wi h
φ= Φ/2
ob ain
φ′(λ)
π= ∆ρ(λ).
4.2 Wigne Smi h T ace and Rela i e Densi y o S a es
Wigne Smi h delay ope a o dened as
Q(λ) = −iS(λ)†∂λS(λ).
In momen um o channel basis,
Q(λ)
is ni e o coun able-dimensional ma ix sa is-
ying
T Q(λ) = −iT (S(λ)†∂λS(λ)).
On o he hand, loga i hmic de i a i e o
de S(λ)
sa ises
∂λlog de S(λ) = T (S(λ)−1∂λS(λ)) = T (S(λ)†∂λS(λ)),
using
S(λ)
's uni a i y. Taking imagina y pa yields
∂λΦ(λ) = T Q(λ),
hus
∆ρ(λ) = 1
2πT Q(λ).
This di ec ly e iable h ough spec al ep esen a ion cons uc ion o
H, H0
and
S(λ)
in igo ous sca e ing heo y; widely used in mul iphysics applica ions.

7
5 Model Applica ions
Gi e se e al conc e e models illus a ing bounda y as clock ealiza ion in die en phys-
ical scena ios.
5.1 One-Dimensional Sch ödinge Sca e ing
Conside 1D Sch ödinge ope a o
H0=−d2
dx2, H =−d2
dx2+V(x)
on
H=L2(R)
; assume
V∈L1(R,(1 + |x|)dx)
eal- alued. Sca e ing heo y comple ely
sol able; eec ion, ansmission ampli udes
(k), (k)
exis ; ene gy
E=k2
.
Selec bounda y Hilbe space as momen um space channels
H∂≃L2(Rk)⊕L2(Rk);
bounda y algeb a
A∂
as closu e o bounded mul iplica ion ope a o s and ni e- ank pe -
u ba ions;
ω
as equilib ium s a e (e.g., Fe miDi ac o Bol zmann weigh ).
Unde app op ia e he mal equilib ium limi , modula ow o
A∂
and
ω
can co e-
spond o Sch ödinge e olu ion, ealizing consis ency be ween modula ime and sca e -
ing ime scale in ene gy window.
Time eadou
gi en by
(k)− 0=Zk
k0
1
2πT Q(˜
k)dE
d˜
kd˜
k=ZE
E0
∆ρ(˜
E)d˜
E,
ans o ming ene gy axis in o bounda y ime axis.
5.2 Local Algeb as and Rindle Wedge
In algeb aic quan um eld heo y, on Neumann algeb a
A(W)
associa ed wi h Minkowski
space wedge egion
W
's modula ow in acuum s a e gi en by BisognanoWichmann
heo em as Lo en z boos p ese ing wedge.
This means: o Rindle obse e , p ope ime ow p opo ional o modula ime on
A(W)
; ConnesRo elli he mal ime hypo hesis gene alizes his as  ime = modula ow
pa adigm in gene ally co a ian eld heo ies.
In his backg ound, can iew wedge bounda y (o mo e gene ally double cone bound-
a y) as his pape 's bounda y algeb a
A∂
; sca e ing ma ix cons uc ed om a - egion
eld inciden /ou going modes; Wigne Smi h delay ma ix cha ac e izes eld esidence
ime nea wedge.
Th ough scale iden i y, can align geome ically dened p ope ime wi h sca e ing
phase de i a i e, ealizing bounda y as clock in conc e e quan um eld heo y models.

6 Enginee ing P oposals
P opose se e al expe imen al and enginee ing schemes o es ing key equa ions and scale
iden i y cons uc ion o bounda y as clock on con ollable pla o ms.
8
6.1 Mic owa e Ne wo k wi h Vec o Ne wo k Analyze
In mic owa e enginee ing, complex ne wo ks (wa eguides, esonan ca i ies, couple s)
commonly desc ibed by mul i-po sca e ing ma ix
S(ω)
, di ec ly measu able by ec o
ne wo k analyze (VNA).
Cons uc mul i-po ne wo k app oxima ing dissipa ionless in wo king equency
band sa is ying sca e ing heo y egula i y equi emen s:
1. Measu e
S(ω)
wi h VNA; nume ically die en ia e o ge
∂ωS(ω)
; 2. Cons uc
Wigne Smi h ma ix
Q(ω) = −iS(ω)†∂ωS(ω)
; compu e
T Q(ω)
; 3. Th ough app op i-
a e ene gy equency no maliza ion, map
ω
axis o ime axis
(ω)− 0=Zω
ω0
1
2πT Q(˜ω)d˜ω;
4. View ne wo k as conc e e ealiza ion o bounda y algeb a: po modes span
H∂
;
ne wo k in e io bulkex e io domain dynamics p ojec on o po s gi ing
S(ω)
; 5.
Unde s a is ical s eady s a e, cons uc empi ical s a e
ωexp
o po exci a ion and
ou pu ; app oxima ely eco e eec i e modula ow h ough ene gy ow conse a ion
and equilib ium condi ions; es consis ency wi h ime ansla ion dened by
(ω)
in
co ela ion unc ions.
I measu ed
T Q(ω)
equency in eg al and ne wo k in e io a e age esidence ime
plus ene gy s o age a e sa is y scale iden i y, iewable as enginee ing-le el e ica ion o
bounda y phase g adien scales ime.

7 Discussion (Risks, Bounda ies, Pas Wo k)
Discuss applicabili y domain, po en ial isks, ela ion o exis ing wo k o bounda y as
clock amewo k.
1. **Dependence on sca e ing egula i y**: Scale iden i y depends on Bi manK ein
o mula and well-dened spec al shi unc ion, equi ing
H−H0
a leas ace-class pe -
u ba ion; sca e ing ma ix smoo h in ene gy window. Fo s ong coupling, many-body
pu e poin spec um dominan sys ems, amewo k equi es modica ion o gene aliza-
ion.
2. **Dynamical in e p e a ion o modula ow**: The mal ime hypo hesis c i i-
cism poin s ou modula ow may no always ob ain na u al dynamical in e p e a ion,
especially lacking geome ic backg ound o equilib ium s a e assump ions. This pape
by equi ing modula ow consis en wi h sca e ing ime scale ac ually selec s amily o
s a es and algeb as wi h good dynamical meaning; howe e , his selec ion i sel equi es
addi ional physical inpu and expe imen al calib a ion.
3. **Locali y and causal s uc u e**: This pape doesn' explici ly in oduce space-
ime causal s uc u e, only wo king a bounda y algeb a and sca e ing channel le el.
To ele a e bounda y as clock o comple e  ime geome y, equi es u he in oduc-
ing local subalgeb as, causal embedding, mac oscopic geome y econs uc ion p ocedu e.
Closely ela ed o algeb aic quan um eld heo y esea ch on econs uc ing space ime
s uc u e h ough local algeb as.
4. **Time a ow and i e e sibili y**: Scale iden i y only cha ac e izes ime pa am-
e e scale and di ec ion, no di ec ly explaining ime a ow o igin. Binding ime a ow
9