Special Relativity as Emergence of Information Rate Conservation: Orthogonal Decomposition of Internal Time Flow and External Displacement

Abs ac
Classical special ela i i y akes he cons ancy o he speed o ligh and he p inciple o
ela i i y as axioms, adop ing Minkowski space ime as an a p io i geome ic s age. In his
adi ional pic u e, me ic s uc u e and Lo en z ac o s a e iewed as geome ic ac s, wi h
hei ela ionship o in o ma ion ow and compu a ional capaci y no made explici . On he
o he hand, quan um in o ma ion and condensed ma e heo y e eal ha physical sys ems a e
uni e sally cons ained by ni e in o ma ion p opaga ion speed and ni e quan um e olu ion
a e: local quan um la ice sys ems sa is y he LiebRobinson ni e g oup eloci y bound,
dening an eec i e "signal speed o ligh "; quan um speed limi heo ems p o ide he maximum
e olu ion a e o pu e s a es in p ojec i e Hilbe space, de e mined by ene gy unce ain y;
in o ma ion-physical limi analysis u he indica es ha e e y physical de ice has a maximum
compu a ional a e de e mined join ly by ene gy and numbe o deg ees o eedom.
This pape p oposes an eme gence scheme o special ela i i y cen e ed on in o ma ion a e
wi hin he amewo k o disc e e quan um cellula au oma on (QCA) on ology. Fo any local
exci a ion, we dene wo classes o in o ma ion upda e a es: one is he ex e nal g oup eloci y
ex
o he en elope cen e on he la ice, cha ac e izing changes in spa ial coo dina es; he o he
is he FubiniS udy eloci y in in e nal Hilbe space p ojec i e space, app op ia ely scaled and
deno ed as in e nal eloci y
in
, cha ac e izing he a e o in e nal quan um s a e e olu ion.
We p opose he
in o ma ion a e conse a ion axiom
: he e exis s a uni e sal cons an
c
such ha a any momen
2
ex + 2
in =c2,
and dene he p ope ime ow a e by in e nal eloci y as
dτ/d = in /c
. On his basis, we
p o e he ollowing main esul s:
(1)
F om a e conse a ion and he p ope ime deni ion, one di ec ly de i es
dτ= d /γ
,
whe e
γ= (1 − 2
ex /c2)−1/2
is he s anda d Lo en z ime dila ion ac o .
(2)
Rew i ing he in o ma ion a e ci cle in e ms o
d
,
dx
,
dτ
yields he in a ian line
elemen
ds2=−c2dτ2=−c2d 2+ dx2
, hus eco e ing Minkowski me ic s uc u e.
(3)
F om p ope ime and eloci y ela ions, cons uc ing ou - eloci y and ou -momen um
yields s anda d
E=γmc2
,
p=γm ex
, and he in a ian ela ion
E2=p2c2+m2c4
.
(4)
In he Di acQCA one-dimensional model, he in e nal Zi e bewegung- ype oscilla ion
equency
ω0
and eec i e mass
m
sa is y
mc2=ℏω0
, so mass can be in e p e ed as he minimum
in o ma ion a e equi ed o main ain he pa icle's in e nal equencya kind o in o ma ion-
heo e ic impedance.
In his pe spec i e, p ope ime is no longe a p esupposed ex e nal pa ame e bu he
dis ance a eled by in e nal quan um s a e in p ojec i e Hilbe space; he impossibili y o
accele a ing massi e pa icles o ligh speed s ems om he ac ha as
ex →c
, in e nal a e
in →0
, and he ene gy equi ed o main ain he same in e nal s uc u e ends o inni y. We
discuss he ela ionship o his amewo k wi h QCA ela i is ic eme gence li e a u e, quan um
speed limi heo y, and LiebRobinson ni e g oup eloci y, analyze limi a ions ega ding global
upda e ime pa ame e s and many-body en anglemen , and p opose se e al easible enginee ing
es s based on QCA quan um simula ion and ex eme-en i onmen in o ma ion sys ems.
Keywo ds:
In o ma ion a e conse a ion; Quan um cellula au oma on; P ope ime; Lo en z
ac o ; Quan um speed limi ; FubiniS udy me ic; Fou -momen um
1 In oduc ion and His o ical Con ex
Eins ein p oposed special ela i i y in 1905, s a ing om wo axioms:
(1)
P inciple o ela i i y: Physical laws ake he same o m in all ine ial ames.
(2)
Cons ancy o ligh speed: The p opaga ion speed
c
o ligh in acuum is he same in all
ine ial ames.
1
Minkowski subsequen ly geome ized special ela i i y, cons uc ing ou -dimensional pseudo-
Euclidean space ime wi h signa u e
(−+ ++)
and iewing ee pa icle mo ion as geodesics in
space ime. Lo en z ans o ma ions a e equi alen in his space ime o he g oup o linea ans o -
ma ions p ese ing he line elemen
ds2=−c2d 2+ dx2
. This geome y is o mal and success ul,
bu he on ological ques ion o why is i so has long emained un esol ed: Why does a limi ing
speed exis and equal he cons an
c
? Why mus he me ic necessa ily be Minkowski- ype a he
han some o he o m?
On he o he hand, de elopmen s in quan um in o ma ion and quan um s a is ical mechanics
ha e e ealed a se ies o uni e sal cons ain s ela ed o eloci y and ime. Lieb and Robinson
p o ed in s udying quan um spin sys ems ha in o ma ion p opaga ion has a ni e g oup eloci y
bound; he inuence o local ope a o s sp eads spa ially in a ni e ligh cone manne , dening
he so-called LiebRobinson eloci y. This esul p o ides a ligh -speed-like uppe bound in non-
ela i is ic la ice sys ems, embodying he p o ound connec ion be ween locali y and ni e signal
speed.
In quan um dynamics, Mandels amTamm and Ma golusLe i in heo ems p o ide quan um
speed limi s: he minimum ime equi ed o e ol ing om a pu e s a e o one o hogonal o
i is bounded by ene gy unce ain y and a e age ene gy lowe bounds. Aha ono and Anandan
in oduced he FubiniS udy me ic on p ojec i e Hilbe space, es a ing he imeene gy inequali y
as he ela ionship be ween quan um s a e cu e leng h and ene gy uc ua ion.
In esea ch on in o ma ion-physical limi s, Lloyd analyzed he limi ing compu a ional a e de-
e mined by
c
,
ℏ
, and
G
, poin ing ou ha he numbe o compu a ional s eps and in o ma ion
s o age capaci y achie able by any physical sys em a e bo h limi ed by ene gy and deg ees o ee-
dom. These wo ks collec i ely indica e ha ime and eloci y a e no me ely geome ic quan i ies,
bu eec ions o in o ma ion p opaga ion and p ocessing capaci y.
Quan um cellula au oma a p o ide a disc e e and s ic ly causal dynamical amewo k o he
concep ion o uni e se as quan um compu a ion. Local uni a y upda es ac ing on la ice si es
na u ally ha e ni e p opaga ion speed; unde app op ia e con inuum limi s, a se ies o wo ks
ha e p o en ha Di ac, Weyl, and e en Maxwell equa ions can eme ge om he long-wa eleng h
limi o QCA. These esul s sugges ha ela i is ic- ype eld heo y can be iewed as an eec i e
desc ip ion o unde lying disc e e in o ma ion p ocessing.
Agains his backg ound, his pape a emp s o answe a mo e undamen al ques ion: i we
in e p e
c
as  he o al in o ma ion upda e a e a ailable o a single local exci a ion pe uni coo -
dina e ime, can he en i e dynamics o special ela i i y eme ge om he cons ain o in o ma ion
a e conse a ion? Specically, we dis inguish wo ypes o in o ma ion upda es:
(1)
Ex e nal displacemen : p opaga ion o he exci a ion on he la ice, cha ac e ized by g oup
eloci y
ex
.
(2)
In e nal e olu ion: mo ion o he in e nal Hilbe space s a e in p ojec i e space, yielding
in
by scaling FubiniS udy eloci y.
The co e axiom p oposed in his pape is: o any exci a ion, he o al in o ma ion a e ec o
( ex , in )
has modulus cons ained a Planck scale by cons an
c
, iewed as he in o ma ion a e
budge a ailable o ha exci a ion. Ex e nal and in e nal e olu ion a e o hogonal alloca ions o
he same budge . We will hen p o e ha ime dila ion, Minkowski me ic, and s anda d ene gy-
momen um ela ions o special ela i i y can all be iewed as geome ic ew i es o his ci cula
budge cons ain .
2
2 Model and Assump ions
This sec ion p o ides he QCA backg ound, igo ous deni ions o ex e nal and in e nal eloci ies,
and o malizes he in o ma ion a e conse a ion axiom and p ope ime deni ion.
2.1 QCA Backg ound and Causal S uc u e
Conside a quan um cellula au oma on dened on a
d
-dimensional egula la ice
Λ⊂Zd
. Fo each
la ice si e
x∈Λ
, associa e a ni e-dimensional local Hilbe space
Hx
; he global Hilbe space is
H=O
x∈ΛHx.
Time is indexed by in ege s eps
n∈Z
, and each s ep is ealized by a global uni a y ope a o
U
decomposable in o a ni e-dep h a ay o local uni a y ga es. Locali y means: he e exis s a ni e
neighbo hood
N( )
such ha in each e olu ion s ep, he ope a o a any la ice si e couples only
wi h ope a o s on i s ni e neighbo hood.
Unde his se up, one can dene he suppo egion o a local ope a o
AX
a e
n
e olu ion
s eps, whose g ow h a e is con olled by he LiebRobinson bound: he e exis cons an s
LR >0
and decay unc ion
F
such ha he commu a o no m o ope a o s on egion
Y
sucien ly a om
X
wi h
AX(n)
decays exponen ially o supe -polynomially. The e o e, in o ma ion p opaga ion in
QCA has an uppe bound, which can be iden ied in he con inuum limi as he eec i e speed o
ligh
c
.
In his pape , we assume QCA e olu ion con e ging o Weyl/Di ac equa ions in he long-
wa eleng h limi has been ob ained h ough known cons uc ions. Thus, ex e nal g oup eloci y
ex
in he eec i e con inuum limi equals he g oup eloci y o ela i is ic wa epacke s, bu he
unde lying s uc u e o disc e e la ice and global upda e s eps is e ained.
2.2 Local Exci a ions and Ex e nal Veloci y
Conside he single-exci a ion subspace
H1p
o QCA, spanned by posi ion and in e nal deg ees
o eedom o a single pa icle on he la ice. Le
|ψp⟩
be a single-exci a ion eigenmode wi h
quasi-momen um
p
and eec i e dispe sion ela ion
ω(p)
. On his basis, cons uc a na owband
wa epacke
|Ψ( )⟩=Zdp (p−p0)e−iω(p) |ψp⟩,
whe e
is a weigh unc ion sha ply concen a ed a ound
p0
. The ime de i a i e o he en elope
cen e posi ion
x( )
gi es he g oup eloci y
ex (p0) = 
dx
d 
=
∂ω(p)
∂p p=p0
.
By he ni e signal speed p ope y o QCA,
| ex (p)| ≤ c
holds o all
p
.
In wha ollows, we abb e ia e
ex
as
, whose physical meaning is: he a e age displacemen
a e o he pa icle en elope cen e in space pe uni coo dina e ime.
2.3 In e nal Hilbe Space and FubiniS udy Veloci y
Each exci a ion ca ies a quan um s a e in a ni e-dimensional in e nal Hilbe space
Hin
ep esen -
ing spin, a o , in e nal oscilla ion modes, e c. Le
|ψ( )⟩∈Hin
be he in e nal s a e a coo dina e
3
ime
, e ol ing unde eec i e Hamil onian
H
:
iℏd
d |ψ( )⟩=H|ψ( )⟩.
Physical equi alence classes o in e nal s a es a e ep esen ed by p ojec i e Hilbe space
P(Hin )
,
i.e., pu e s a e o bi s wi h o e all phase emo ed. P ojec i e space is na u ally equipped wi h he
FubiniS udy me ic, whose inni esimal line elemen is
ds2
FS = 4 ⟨˙
ψ( )|˙
ψ( )⟩−|⟨ψ( )|˙
ψ( )⟩|2d 2,
and he co esponding ins an aneous geome ic eloci y is
FS( ) = dsFS
d =2∆H( )
ℏ,
whe e
∆H( ) = p⟨H2⟩ −⟨H⟩2
is he ene gy unce ain y.
This esul shows ha he in e nal s a e's e olu ion speed in p ojec i e space is con olled by
ene gy uc ua ion, and quan um speed limi heo ems gi e he minimum ime o e ol e om he
ini ial s a e o one o hogonal o i a his speed.
To place his geome ic eloci y in he same dimension as ex e nal g oup eloci y
ex
, in oduce
an in e nal leng h scale
ℓin >0
and dene in e nal eloci y
in ( ) = ℓin FS( ) = ℓin
2∆H( )
ℏ.
ℓin
can be iewed as a p opo ionali y ac o con e ing FubiniS udy dis ance o spa ial equi alen
leng h, i s alue chosen by calib a ion: in he es ame, ake some maximally ac i e in e nal
s a e such ha i s in e nal eloci y sa u a es a
c
. Quan i a i ely, on a wo-le el sys em wi h xed
spec al suppo , choose a supe posi ion s a e simul aneously sa u a ing Mandels amTamm and
Ma golusLe i in quan um speed limi s and equi e i o sa is y
in =c
in he es ame, hus
de e mining
ℓin
.
Unde his no maliza ion, o any in e nal s a e we ha e
0≤ in ≤c
.
2.4 In o ma ion Ra e Conse a ion Axiom
This pape p oposes he ollowing axiom as he s a ing poin o special ela i i y eme gence:
Axiom 1
(In o ma ion Ra e Conse a ion)
.
Fo any local exci a ion in QCA and any coo dina e
ime
, i s ex e nal g oup eloci y
ex ( )
and in e nal eloci y
in ( )
ob ained by scaling in e nal
FubiniS udy eloci y sa is y
2
ex ( ) + 2
in ( ) = c2,
whe e
0≤ ex ( )≤c
,
0≤ in ( )≤c
.
This axiom can be physically unde s ood as: he o al in o ma ion upda e a e a ailable o
each exci a ion pe uni coo dina e ime is bounded by uppe limi
c
; ex e nal displacemen and
in e nal e olu ion can only make o hogonal alloca ions wi hin his budge . The squa ed sum o m
is adop ed o ob ain geome ic s uc u e co esponding o a Euclidean ci cle, he eby in oducing
he in a ian o m o Minkowski me ic.
4
2.5 In e nal Deni ion o P ope Time
Based on he abo e in e nal eloci y deni ion, p ope ime
τ
is dened as he cumula i e amoun
o in e nal e olu ion in he in o ma ion leng h sense:
dτ
d = in ( )
c.
When he exci a ion is in he es ame, i he in e nal s a e is chosen as an ex emal s a e sa u a ing
he quan um speed limi , hen
in =c
, hus
τ= + cons
.
In a gene al mo ing s a e, inc easing ex e nal eloci y will comp ess in e nal eloci y
in
, hus
slowing p ope ime ow. This deni ion will be p o en in la e sec ions o exac ly ep oduce he
Lo en z ela ion be ween p ope ime and coo dina e ime in special ela i i y.
3 Main Resul s: Theo ems and Alignmen s
Based on he abo e model and axioms, we ob ain he ollowing main esul s.
Theo em 2
(Eme gence o Lo en z Time Dila ion)
.
Le a local exci a ion in some ine ial ame
ha e ex e nal eloci y
ex ( )
and in e nal eloci y
in ( )
sa is ying in o ma ion a e conse a ion
2
ex ( ) + 2
in ( ) = c2,
and p ope ime
τ
dened as
dτ/d = in ( )/c
. Then
dτ
d = 1− 2
ex ( )
c2,
i.e.,
dτ=d
γ( ), γ( ) = 1
p1− 2
ex ( )/c2.
This is he s anda d special- ela i is ic Lo en z ime dila ion o mula be ween p ope ime and coo -
dina e ime.
Theo em 3
(Minkowski Line Elemen as In o ma ion Ra e Iden i y)
.
In he same ine ial ame,
le he exci a ion's spa ial coo dina e be
x( )
sa is ying

dx
d 
= ex ( ).
Dene he space ime line elemen
ds2=−c2d 2+ dx2.
Then unde in o ma ion a e conse a ion and p ope ime deni ion,
ds2=−c2dτ2.
Thus
ds2
depends only on p ope ime inc emen , is an in a ian independen o specic ine ial
ame, and has a o m consis en wi h Minkowski me ic.
5

Theo em 4
(Fou -Veloci y and Fou -Momen um S uc u e)
.
Dene ou -coo dina es
xµ= (c , x)
,
p ope ime
τ
as be o e, ou - eloci y
Uµ=dxµ
dτ.
Le
m > 0
be he exci a ion's es mass; ou -momen um is dened as
Pµ=mUµ.
Then:
(1)
Fou - eloci y componen s a e
U0=cγ, U=γ ex , γ =1
p1− 2
ex /c2,
and sa is y he in a ian no maliza ion condi ion
UµUµ=−c2.
(2)
Fou -momen um componen s a e
P0=E
c=mcγ, P=p=mγ ex ,
and sa is y he in a ian mass shell condi ion
PµPµ=−m2c2,
hus
E2=p2c2+m2c4.
Theo em 5
(Mass as In e nal F equency and In o ma ion Impedance)
.
Suppose a s a iona y exci-
a ion in i s es ame has in e nal oscilla ion angula equency
ω0
, wi h in e nal s a e e ol ing in
p ope ime as
e−iω0τ|ψ0⟩
. Dene es mass as
mc2=ℏω0.
In a gene al ine ial ame, o al ene gy
E
is gi en by in e nal phase e ol ing wi h coo dina e ime
e−iE /ℏ
. I in o ma ion a e conse a ion and p ope ime deni ion hold, hen:
(1)
To al ene gy sa ises
E=γmc2,
whe e
γ
is he Lo en z ac o de e mined by
ex
.
(2)
In e nal eloci y sa ises
in =c
γ,
hus ene gy can also be exp essed as
E=mc2c
in
.
When
ex →c
,
in →0
,
E→ ∞
. The e o e, mass can be in e p e ed as: unde in o ma ion a e
conse a ion cons ain , he minimum in o ma ion impedance equi ed o main ain he pa icle's
in e nal oscilla ion equency
ω0
; he highe he ex e nal eloci y, he g ea e he o al ene gy needed
o p ese e ha in e nal s uc u e.
6
4 P oo s
This sec ion p o ides p oo s o he abo e heo ems, wi h echnical de ails supplemen ed in appen-
dices.
4.1 P oo o Theo em ??
F om in o ma ion a e conse a ion
2
ex ( ) + 2
in ( ) = c2
holding o any
, we ob ain
in ( ) = qc2− 2
ex ( ).
P ope ime is dened as
dτ
d = in ( )
c,
subs i u ing he abo e gi es
dτ
d =1
cqc2− 2
ex ( ) = 1− 2
ex ( )
c2.
Dene
γ( ) = 1
p1− 2
ex ( )/c2,
which can be w i en as
dτ=d
γ( ),
comple ely consis en wi h he ime dila ion ela ion in special ela i i y.
4.2 P oo o Theo em ??
In he same ine ial ame, he spa ial line elemen sa ises
dx2= 2
ex ( ) d 2.
By a e conse a ion, we can w i e
2
ex ( ) = c2− 2
in ( ),
hus he space ime line elemen
ds2=−c2d 2+ dx2=−c2d 2+c2− 2
in ( )d 2=− 2
in ( ) d 2.
On he o he hand, he p ope ime deni ion gi es
dτ2= in ( )
c2
d 2⇒ 2
in ( ) d 2=c2dτ2.
Subs i u ing back yields
ds2=−c2dτ2.
Thus he line elemen depends only on p ope ime, is an in a ian unde Lo en z ans o ma ions,
and has a o m consis en wi h Minkowski me ic.
7
4.3 P oo o Theo em ??
F om Theo em
??
,
dτ
d = 1− 2
ex
c2⇒d
dτ=γ, γ =1
p1− 2
ex /c2.
Fou -coo dina es a e dened as
xµ= (c , x)
, ou - eloci y as
Uµ=dxµ
dτ=d(c )
dτ,dx
dτ=cd
dτ,dx
d
d
dτ= (cγ, γ ex ).
I s no m
UµUµ=−(U0)2+U2=−c2γ2+γ2 2
ex =−c2γ21− 2
ex
c2=−c2.
This gi es he Lo en z-in a ian no maliza ion o ou - eloci y.
Fou -momen um is dened as
Pµ=mUµ
, hus
P0=mcγ, P=mγ ex .
Se ing
E=P0c
,
p=P
, we ha e
E=γmc2,p=γm ex .
Compu ing he in a ian
PµPµ=−(P0)2+P2=−m2c2γ2+m2 2
ex γ2=−m2c2,
yields
E2=p2c2+m2c4.
4.4 P oo o Theo em ??
In he es ame (
ex = 0
), suppose he in e nal s a e e ol es wi h p ope ime as
|ψ(τ)⟩= e−iω0τ|ψ0⟩,
whe e
ω0
is he in e nal oscilla ion equency. The quan um mechanical ene gy equency ela ion
gi es
E0=ℏω0.
Dene es mass
m
sa is ying
mc2=E0=ℏω0.
In a gene al ine ial ame, he coo dina e ime e olu ion o m is
|ψ( )⟩= e−iE /ℏ|ψ′
0⟩,
whe e
E
is he o al ene gy measu ed in ha ame. F om he p ope ime and coo dina e ime
ela ion
dτ=d
γ⇒d
dτ=γd
d ,
8
he in e nal s a e's de i a i e wi h espec o p ope ime is
d
dτ|ψ(τ)⟩=−iE
ℏγ|ψ(τ)⟩.
Compa ing wi h he es ame case
d
dτ|ψ(τ)⟩=−iω0|ψ(τ)⟩,
we ob ain
Eγ =ℏω0=mc2,
i.e.,
E=mc2
γ.
No e ha he e
γ
is dened as he Lo en z ac o o he es ame ela i e o ha ine ial ame, so i
we adop he con en ion o iewing
γ
as he mo ing ame ela i e o es ame ac o , we need o
handle he in e se. A mo e di ec app oach is o use he al eady-ob ained esul om Theo em
??
E=γmc2,
and combine wi h he in e nal eloci y exp ession.
F om in o ma ion a e conse a ion and Theo em
??
, we ha e
in =c 1− 2
ex
c2=c
γ.
Thus o al ene gy can be ew i en as
E=γmc2=mc2c
in
.
When
ex →c
,
γ→ ∞
,
in →0
, o al ene gy di e ges. This indica es: o main ain he o iginal
in e nal s uc u e unde nea ly comple ely ozen in e nal ime ow equi es inni e ene gy in es -
men . Mass
m
he e can be iewed as in o ma ion impedance o in e nal oscilla ion equency
ω0
:
he highe he equency, he g ea e he in o ma ion a e needed o p e en collapse o ha in e nal
s uc u e o xed ex e nal eloci y componen , mani es ing as la ge
m
and mo e d ama ic ene gy
g ow h.
5 Model Applica ions
This sec ion discusses applica ions and in e p e a ions o he abo e s uc u e in se e al ypical
physical si ua ions.
5.1 Two Limi s: Pho ons and S a iona y Pa icles
On he in o ma ion a e plane
( ex , in )
, in o ma ion a e conse a ion co esponds o a qua e -
ci cle o adius
c
. Two impo an limi s a e:
1. S a iona y massi e pa icle
:
ex = 0
,
in =c
. All in o ma ion budge is used o
in e nal e olu ion; p ope ime coincides wi h coo dina e ime; he pa icle's in e nal clock uns
a maximum a e.
9
A.4 In e nal Veloci y No maliza ion and Uppe Bound
This pape denes in e nal eloci y
in =ℓin FS =ℓin
2∆H
ℏ.
To x
ℓin
, we can adop he ollowing s a egy:
(1)
Selec some physically ealizable limi ing in e nal s a e amily, e.g., in wo-le el sys ems,
supe posi ion s a es sa u a ing Mandels amTamm and Ma golusLe i in uppe bounds, making
hem ep esen maximum in e nal ac i i y in he es ame.
(2)
Requi e hese s a es o ha e
in =c
in he es ame.
(3)
Sol e o ge
ℓin =cℏ
2∆Hmax
.
Since any s a e's ene gy unce ain y does no exceed
∆Hmax
unde ha suppo , o all s a es we
ha e
in ≤c.
Combined wi h QCA's ni e signal speed
| ex | ≤ c
, he in o ma ion a e conse a ion axiom equi es
physical s a es o be es ic ed wi hin a ci cle o adius
c
in he wo-dimensional a e plane, na u ally
in oducing subsequen Minkowski geome y s uc u e.
B Abs ac De i a ion om In o ma ion Ra e Ci cle o Minkowski
Me ic
This appendix p o ides de i a ion om in o ma ion a e ci cle o Minkowski me ic unde mo e
abs ac se up, highligh ing consis ency be ween geome ic s uc u e and algeb aic iden i ies.
B.1 Abs ac In o ma ion Ra e Plane
Conside wo-dimensional eal ec o space
R2
wi h coo dina es
(u1, u2)
ep esen ing wo ypes o
in o ma ion a es. Assume physically allowed a e pai s a e cons ained on a ci cle o adius cons an
c > 0
, i.e.,
u2
1+u2
2=c2.
In oduce pa ame e
θ∈[0, π/2]
, le ing
u1=csin θ, u2=ccos θ.
Iden i y
u1
as ex e nal eloci y
ex
and
u2
as in e nal eloci y
in
.
Assume coo dina e ime pa ame e
exis s; ex e nal displacemen sa ises

dx
d 
= ex ( ) = u1( ) = csin θ( ),
and dene p ope ime
dτ
d = in ( )
c=u2( )
c= cos θ( ).
16

B.2 Geome ic Rela ion o Lo en z Fac o
By deni ion
dτ
d = cos θ( )⇒d
dτ=1
cos θ( )≡γ( ).
On he o he hand,
ex =csin θ
, so
sin2θ= 2
ex
c2,cos2θ= 1 − 2
ex
c2,
hus
γ=1
cos θ=1
p1− 2
ex /c2,
p ecisely he Lo en z ac o . Thus, as long as an in o ma ion a e ci cle and p ope ime deni ion
exis as abo e, ime dila ion ela ion is au oma ically eco e ed.
B.3 Algeb aic Rew i e o Minkowski Line Elemen
Dene space ime line elemen
ds2=−c2d 2+ dx2.
F om
dx2= 2
ex d 2
, we ha e
ds2=−c2d 2+ 2
ex d 2=−(c2− 2
ex )d 2.
Using a e ci cle
c2− 2
ex = 2
in ,
we ob ain
ds2=− 2
in d 2.
On he o he hand, p ope ime deni ion gi es
dτ= in
cd ⇒ 2
in d 2=c2dτ2.
Thus
ds2=−c2dτ2.
This ela ion shows ha Minkowski line elemen s uc u e is equi alen o combined esul o in o -
ma ion a e ci cle and p ope ime deni ion. In o he wo ds, once we admi ex e nal eloci y and
in e nal eloci y o m ci cle o adius
c
on a e plane, Minkowski me ic is no longe an addi ional
assump ion bu an algeb aic ew i e o he ci cle.
C Mass and In e nal F equency in One-Dimensional Di acQCA
This appendix uses one-dimensional Di acQCA as example o illus a e ela ion be ween in e nal
oscilla ion equency and mass pa ame e , hus p o iding model-le el suppo o Theo em
??
.
17
C.1 One-Dimensional Di acQCA Cons uc ion
Conside one-dimensional la ice
Λ = aZ
wi h la ice spacing
a
. Each la ice si e ca ies wo-
dimensional in e nal Hilbe space
Hx∼
=C2
, iewable as spin-
1
2
deg ee o eedom. Dene shi
ope a o s
(S+ψ)x=ψx−1,(S−ψ)x=ψx+1.
Cons uc one-s ep e olu ion ope a o
U= exp −iθX
x
σy
x!exp −iπ
2X
x
σx
x!(S+⊗|↑⟩⟨↑|+S−⊗|↓⟩⟨↓|),
whe e
σx, σy
a e Pauli ma ices and
θ
is an adjus able pa ame e . Fou ie ans o ming his QCA
yields diagonalized o m in momen um space
U(p)=e−iHe (p)∆ /ℏ,
wi h eec i e Hamil onian con e ging in small
p
limi o one-dimensional Di ac Hamil onian
HD(p) = cp σz+mc2σx,
whe e
c
and
m
a e pa ame e s de e mined by
θ
,
a
,
∆
.
C.2 Dispe sion Rela ion and In e nal Oscilla ions
Diagonalizing he abo e Di ac Hamil onian yields ene gy spec um
E±(p) = ±p(cp)2+m2c4.
A es momen um
p= 0
,
E±(0) = ±mc2.
Conside in e nal s a e o med by equal-ampli ude supe posi ion o posi i e and nega i e ene gy
band eigens a es
|+⟩
,
|−⟩
:
|ψ(0)⟩=1
√2(|+⟩+|−⟩).
I s ime e olu ion is
|ψ( )⟩=1
√2e−imc2 /ℏ|+⟩+ eimc2 /ℏ|−⟩.
Fo sui able in e nal obse ables (e.g., some Pauli componen ), expec a ion alues will exhibi os-
cilla ions a equency
2mc2/ℏ
 his is he disc e e e sion o Zi e bewegung phenomenon.
In his pape 's amewo k, he undamen al equency
ω0
o in e nal oscilla ions can be aken as
mc2/ℏ
o a cons an mul iple o
2mc2/ℏ
; o b e i y, we dene
mc2=ℏω0,
iewing mass as linea unc ion o in e nal equency. The abo e cons uc ion shows ha his
ela ion has explici model ounda ion in Di acQCA.
18
C.3 Co espondence o In e nal Veloci y and Mass
A es momen um
p= 0
, in e nal s a e e ol es in p ojec i e space a equency
ω0
wi h Fubini
S udy eloci y
FS =2∆H
ℏ.
Fo equal-ampli ude supe posi ion s a e o wo-le el symme ic ene gy spec um
E±=±mc2
, en-
e gy unce ain y is
∆H=mc2,
so
FS =2mc2
ℏ= 2ω0.
Unde his pape 's chosen in e nal leng h scale
ℓin
, equi ing sa u a ed s a e in es ame o sa is y
in =c
, i.e.,
c=ℓin FS =ℓin 2ω0⇒ℓin =c
2ω0
.
Thus o gene al s a e we ha e
in =ℓin FS =c
2ω0
FS.
When conside ing cases including momen um, ene gy spec um expands o
E±(p)
, in e nal oscil-
la ion equency and ene gy unce ain y a y wi h
p
, and in e nal eloci y
in
will be below
c
.
Combined wi h in o ma ion a e conse a ion and p ope ime deni ion, we ob ain he eec o
ex e nal eloci y comp ession o in e nal eloci y as
p
inc eases, he eby de i ing ime dila ion and
ene gy g ow h ela ions.
The e o e, in Di acQCA models, he e exis s linea co espondence be ween mass
m
and in-
e nal oscilla ion equency
ω0
; in e nal eloci y no maliza ion and in o ma ion a e conse a ion
axiom ha e explici ealiza ion me hods in conc e e models, p o iding s ong suppo o he mass
equency ela ion adop ed by Theo em
??
.
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20