Inertial Geometry from Information Rate Conservation: Microscopic Unification of Mass, Time Dilation, and Energy--Frequency Relations

Abs ac
In adi ional o mula ions, ime dila ion and  ela i is ic mass inc ease in special ela i i y
a e usually iewed as wo independen kinema ic eec s, while quan um mechanics di ec ly e-
la es ene gy o equency h ough he Planck ela ion
E=ℏω
. Fo a massi e pa icle, when i s
eloci y app oaches he speed o ligh , o al ene gy g ows apidly, co esponding o g ow h o
plane wa e phase equency
ω
; ye p ope ime ow o he same pa icle exhibi s Lo en z dila-
ion, wi h in e nal clocks appea ing o slow down. This image o slowe clock co esponding o
g ea e ene gy c ea es ension a he in ui i e le el. This pape in oduces a wo-dimensional
in o ma ion a e ci cle wi hin he amewo k o quan um cellula au oma a (QCA) and in-
o ma ion a e conse a ion (op ical pa h conse a ion): iewing o al in o ma ion upda e a e
as cons an
c
and making an o hogonal decomposi ion be ween ex e nal spa ial displacemen
eloci y
ex
and in e nal s a e e olu ion eloci y
in
, sa is ying
2
ex + 2
in =c2
. Unde his
s uc u e, es mass
m0
is cha ac e ized as he Comp on equency
ω0=m0c2/ℏ
o in e nal
quan um s a e in he es ame, while he mo ing s a e co esponds o esou ce ealloca ion be-
ween he wo equency componen s o in e nal e olu ion and ex e nal ansla ion in Hilbe
space. We p o e ha : o al equency o plane ma e wa e sa ises
ω o =γω0
; in e nal clock
equency unde goes edshi wi h eloci y as
ωclock =ω0/γ
; and hese wo equency ypes
a e bo h uni o mly embedded in he equency iden i y
ω2
o =ω2
0+ω2
space
induced by ene gy
momen um ela ion
E2=m2
0c4+p2c2
. On he in o ma ion a e ci cle, his co esponds o a
simple ule o ine ial geome y: when ex e nal eloci y app oaches
c
, in e nal e olu ion eloci y
in →0
, and co esponding eec i e ine ial impedance
me ∝ −3
in
di e ges. Ine ia is hus
in e p e ed as ene gy cos equi ed o main ain opological s uc u e om collapsing in he
in e nal- ime eezing limi . This amewo k gi es geome ic and in o ma ion- heo e ic uni-
ca ion be ween
E=mc2
and
E=ℏω
wi hou modi ying any e ied ela i is ic and quan um
mechanical p edic ions.
Keywo ds:
In o ma ion a e conse a ion; Quan um cellula au oma on; Ine ial geome y;
Comp on clock; Rela i is ic ime dila ion; Ene gy equency co espondence; FubiniS udy me ic
1 In oduc ion and His o ical Con ex
1.1 Appa en Con adic ion o Mass, Time, and F equency
Special ela i i y cen e s on he ene gymomen um ela ion
E2=m2
0c4+p2c2,
dening
m0
as a Lo en z in a ian , while  ela i is ic mass
γm0
mainly se ed as a pedagogical
con enience in ea ly eaching; i s physical con en has g adually been weakened o abandoned
in mode n li e a u e, shi ing o emphasizing in a ian mass and co a ian desc ip ion o ou -
momen um.
On he o he hand, quan um heo y di ec ly ela es ene gy o equency h ough he Planck
ela ion
E=ℏω.
Fo a plane wa e solu ion o a ee pa icle
ψ∼exp[i(kx −ω )]
, o al ene gy is de e mined by
empo al phase equency
ω
. De B oglie u he p oposed ha e e y pa icle wi h es mass
m0
ca ies an in insic oscilla ion equency in i s es ame:
ω0=m0c2
ℏ,
now commonly called he Comp on equency o in e nal clock.
This gi es ise o an appa en ly con adic o y pic u e:
1

Rela i i y p edic s: when a pa icle mo es wi h eloci y
, i s p ope ime sa ises
dτ= d /γ
,
i.e., he in e nal clock uns slowe  ela i e o labo a o y ime.

Unde he quan um wa e pic u e, o al ene gy
E=γm0c2
co esponds o plane wa e phase
equency
ω o =E/ℏ=γω0
which inc eases wi h
γ
, i.e.,  equency becomes as e .
I  equency is nai ely unde s ood as  hy hm o in e nal oscilla ions o he pa icle, hen
 ime slowing and  o al equency inc easing seem mu ually con adic o y. Indeed, he e has
been ex ensi e discussion and expe imen al p oposals ega ding he ela ionship be ween de B oglie
in e nal clock, Comp on equency, and ma e wa e equency, including  ocks a e clocks and
de ec ion o Comp on clocks in a omic in e e ome e s.
1.2 Quan um Cellula Au oma a and Disc e e Rela i is ic Dynamics
Quan um cellula au oma a (QCA) model he uni e se as disc e e, local, and s ic ly uni a y upda e
ules ac ing on la ice si es
Λ⊂Zd
, whose con inuum limi s can gi e ise o eld equa ions such as
Di ac, Weyl, and Maxwell. In pa icula , one-dimensional Di ac- ype QCA has been cons uc ed
and p o en o p ecisely ep oduce he e olu ion ope a o and dispe sion ela ion o he ee Di ac
equa ion in he long-wa eleng h limi .
In Feynman's checke boa d model, p opaga ion o spin-
1/2
e mions is ep esen ed as pa h
sums on space ime la ice poin s ad ancing a ligh speed and weigh ed by mass pa ame e s a
u ning poin s; his model likewise yields he Di ac equa ion in he con inuum limi . These wo ks
indica e:
(1)
F ee ela i is ic pa icle dynamics can eme ge in disc e e, ni e-in o ma ion amewo ks.
(2)
Mass pa ame e s a e na u ally connec ed o local s uc u es such as  u ning a e and
in e nal ip a e in disc e e models.
This p o ides a na u al en y poin o unde s anding ine ia and mass om in o ma ion and
compu a ion pe spec i es.
1.3 Quan um E olu ion Geome y and In e nal E olu ion Speed
Ano he impo an clue comes om quan um s a e space geome y. Anandan and Aha ono poin ed
ou ha on p ojec i e Hilbe space
P(H)
, quan um s a e e olu ion can be desc ibed by Fubini
S udy me ic geodesic leng h, wi h e olu ion  eloci y sa is ying
ds
d =2
ℏ∆H,
whe e
∆H
is ene gy unce ain y. These esul s, oge he wi h quan um speed limi s (Mandels am
Tamm and Ma golusLe i in bounds), indica e ha unde xed ene gy esou ces, he e olu ion
 a e o quan um s a es has an uppe bound.
I we unde s and in e nal ime ow as  a e a which s a es e ol e in Hilbe space, hen o a
gi en ee pa icle, i s a ailable e olu ion bandwid h can be iewed as ni e esou ce. When
his esou ce is used mo e o change spa ial posi ion (ex e nal mo ion), he sha e o in e nal
phase/spin/ opological s uc u e e olu ion mus necessa ily dec ease.
1.4 Goals and Con ibu ions o This Pape
Agains he abo e backg ound, his pape in oduces he concep o in o ma ion a e conse a ion:
iewing global e olu ion o a massi e pa icle as in o ma ion a e alloca ion be ween ex e nal
2
posi ion deg ees o eedom and in e nal s a e deg ees o eedom, and assuming he e exis s a
uni e sal uppe bound
c
such ha
2
ex + 2
in =c2,
whe e
ex
is ex e nal g oup eloci y and
in
cha ac e izes he  a e o in e nal quan um s a e
e olu ion unde FubiniS udy me ic, bo h join ly o ming a wo-dimensional in o ma ion a e
ci cle. On his basis, his pape achie es he ollowing:
(1)
P o ides an ine ial geome y model s a ing om QCA and quan um e olu ion geome y,
iewing special- ela i is ic ime dila ion as ealloca ion o in o ma ion a e be ween in e nal and
ex e nal componen s.
(2)
P o es ha o Di ac- ype ee pa icles, plane wa e o al equency, spa ial equency, and
es Comp on equency sa is y
ω2
o =ω2
0+ω2
space
, ob aining concise pa ame ic ela ions
ω o =γω0
and
ωclock =ω0/γ
on he in o ma ion a e ci cle.
(3)
S a ing om classical ela i is ic mechanics, ew i es longi udinal eec i e ine ial mass
me =γ3m0
as
me ∝ −3
in
, hus in e p e ing ine ia as  opological impedance in he in e nal ime
eezing limi .
(4)
Combined wi h exis ing Comp on clock expe imen s and ma e wa e in e e ence expe -
imen s, discusses how o es his ine ial geome yin o ma ion a e pic u e and i s possible
co ec ion e ms a he expe imen al le el.
This pape 's posi ion is: wi hou modi ying e ied ene gymomen um ela ions and quan um
measu emen ules, bu imposing a unied geome icin o ma ion- heo e ic in e p e a ion on hese
ela ions, making mass equency ime dila ion in eg a e in o one.
2 Model and Assump ions
2.1 Di acQCA Eec i e Model and F ee Pa icle Sec o
Conside a Di ac- ype quan um cellula au oma on on one-dimensional space, wi h la ice se
Λ=∆xZ
; each la ice si e ca ies in e nal Hilbe space
Hcell ∼
=C2
, co esponding o le - igh
p opaga ion modes o spin up/down wo in e nal deg ees o eedom. O e all Hilbe space is
H=O
n∈Λ
H(n)
cell.
E olu ion is gi en by local uni a y ope a o
U
, iewable as combina ion o in e nal o a ion
and condi ional ansla ion. Exis ing esea ch has shown ha unde condi ions o homogenei y,
locali y, and disc e e causali y, one can cons uc such a QCA class whose eec i e Hamil onian on
he single-pa icle sec o in he long-wa eleng h limi is
H=cαˆp+βm0c2,
whe e
ˆp
is one-dimensional momen um ope a o ,
α, β
a e Pauli ma ices sa is ying
α2=β2=I
and
an icommu a ion ela ion
{α, β}= 0
.
This Hamil onian's eigen alues on plane wa e basis
ψp(x)∼exp(ipx/ℏ)
a e
E(p) = ±qm2
0c4+p2c2,
he s anda d ene gymomen um dispe sion ela ion.
The e o e, in subsequen de i a ions, we need only use his ene gymomen um ela ion and
QCA's disc e e on ology as concep ual backg ound, wi hou depending on specic cell ule de ails.
3
2.2 In o ma ion Ra e Ci cle and In e nalEx e nal Decomposi ion
We in oduce he ollowing basic axiom.
Axiom 1
(In o ma ion Ra e Conse a ion)
.
Fo any ee pa icle, on one o i s wo ldlines pa ame ized
by some ex e nal e e ence ime
, he e exis wo non-nega i e unc ions
ex ( )
and
in ( )
, deno ed
espec i ely as ex e nal displacemen a e and in e nal s a e e olu ion a e, sa is ying
2
ex ( ) + 2
in ( ) = c2.
whe e:

ex ( )
in he con inuum limi equals he g oup eloci y o pa icle cen e posi ion
( ) =
∂E/∂p
.

in ( )
cha ac e izes he quan um s a e e olu ion a e in he in e nal deg ees o eedom
di ec ion o he same pa icle, wi h dimensions o eloci y, ob ained linea ly om Fubini
S udy e olu ion eloci y
ds/d
ia a xed scale ac o .
This ela ion geome ically cons ains
( ex , in )
o a wo-dimensional ci cle o adius
c
, hence
called he in o ma ion a e ci cle.
To emain consis en wi h special ela i i y, we iden i y ex e nal eloci y as
ex = ,
and acco dingly dene
in ( ) := pc2− 2=cp1−β2, β :=
c.
This deni ion gi es
in =c
in he es ame
( = 0)
, i.e., all in o ma ion a e is used o in e nal
e olu ion; while in he limi
→c
,
in →0
, co esponding o he ex eme case o in e nal ime
eezing.
2.3 Time Dila ion and In e nal Clock F equency
In special ela i i y, p ope ime
τ
and labo a o y ime
sa is y
dτ
d =p1−β2=1
γ, γ := 1
p1−β2.
Compa ing wi h he abo e, we can na u ally iew
in
c=dτ
d
as he no malized a e o in e nal ime ow pe uni labo a o y ime. Tha is, he magni ude o
in e nal eloci y
in
is equi alen o measu ing p ope ime ow a e.
I in he es ame he pa icle ca ies Comp on equency
ω0=m0c2
ℏ,
hen along he wo ldline pa ame ized by p ope ime, i s in e nal phase can be w i en as
φ(τ) = ω0τ.
4
F om he labo a o y ime
pe spec i e, he obse able in e nal clock equency is
ωclock := dφ
d =ω0
dτ
d =ω0
in
c=ω0
γ.
This is p ecisely he equency o m o ime dila ion: he in e nal clock o a mo ing pa icle unde goes
edshi ela i e o labo a o y ime.
2.4 Ene gyF equency Rela ion and Spa ial F equency Componen
On he o he hand, in momen um eigens a es, he plane wa e solu ion o Di ac pa icles has phase
ψ(x, )∼exp i
ℏ(px −E )= exp [i (kx −ω o )] ,
whe e
k:= p
ℏ, ω o := E
ℏ.
F om he ene gymomen um ela ion:
ω2
o =E2
ℏ2=m2
0c4+p2c2
ℏ2=m0c2
ℏ2
+pc
ℏ2=ω2
0+ω2
space,
whe e
ωspace := c|k|=pc
ℏ
can be in e p e ed as spa ial di ec ion equency componen , co esponding o spa ial oscilla ions
b ough by he wa e ec o .
This equency iden i y has close pa allelism wi h he in o ma ion a e ci cle:
ω0
plays he ole
o  es in e nal equency, co esponding o
in
a es ;
ωspace
co esponds o ex e nal momen um
and g oup eloci y.
In wha ollows, we will show how o uni y
ω o
,
ω0
,
ωspace
and
ex
,
in
unde QCA's in e nal
ex e nal deg ees o eedom decomposi ion in o a amewo k o ine ial geome y.
3 Main Resul s: Theo ems and Alignmen s
This sec ion p esen s main heo ems and s uc u al conclusions o his pape .
3.1 Ine ial Geome y Theo em: Geome ic Rew i e o Time Dila ion
Theo em 2
(Ine ial Geome y and Time Dila ion)
.
Unde Axiom
??
(in o ma ion a e conse a-
ion), dening ex e nal eloci y as
ex =
and in e nal eloci y as
in =pc2− 2,
he ollowing s a emen s a e equi alen :
(1)
Special- ela i is ic ime dila ion o mula
dτ= d 1− 2
c2,
i.e.,
dτ/d = in /c
.
5

(2)
The pa icle's eloci y ec o in he wo-dimensional in o ma ion a e plane
u= ( ex , in )
has xed modulus
|u|=c
.
In o he wo ds, ime dila ion can be unde s ood as: when an objec 's mo ion speed in ex e nal
space inc eases, in e nal e olu ion speed is o ced o dec ease on a ci cle o adius
c
.
3.2 F equency Geome y Theo em: Coo dina ion o In e nal Clock and To al
F equency
Theo em 3
(F equency Geome y and Ene gyMomen um Rela ion)
.
Fo a plane wa e s a e o a
ee Di ac pa icle, dene es Comp on equency
ω0=m0c2
ℏ,
o al equency
ω o =E
ℏ,
and spa ial equency
ωspace =pc
ℏ.
Then he e holds equency iden i y
ω2
o =ω2
0+ω2
space.
Fu he mo e, dening in e nal clock equency
ωclock := ω0
γ=ω0
in
c,
hen o al equency and in e nal clock equency sa is y
ω o =γω0=ω2
0
ωclock
,
i.e.,
ωclock ·ω o =ω2
0.
This shows:

Fo gi en in a ian
ω0
, when in e nal clock slows down in labo a o y ime (
ωclock
dec eases),
o al equency
ω o
necessa ily inc eases.

F equency slowing and speeding ac ually poin o wo die en p ojec ions: in e nal clock
and ex e nal plane wa e phase.
6
3.3 In o ma ion Geome ic Exp ession o Ine ial Mass Amplica ion
In ela i is ic mechanics, accele a ion
a∥
along eloci y di ec ion and applied o ce
F∥
sa is y
F∥=m0γ3a∥,
om which one can dene longi udinal eec i e ine ia
m∥
e := γ3m0.
Using
in =c/γ
, his can be ew i en as
m∥
e =m0c
in 3
.
Theo em 4
(Ine ia as In e se Cube o In e nal Time Ra e)
.
Unde he in o ma ion a e ci cle
amewo k, longi udinal eec i e ine ial mass a ies as he in e se cube o in e nal e olu ion eloci y
in
:
m∥
e ∝ −3
in .
The e o e, when
→c
,
in →0
, and
m∥
e → ∞
. F om an in o ma ion- heo e ic pe spec i e, his
co esponds o: in he limi o nea ly ozen in e nal ime, o a sys em o main ain s abili y o
i s own opology and quan um co ela ion s uc u e, i s esponse igidi y o ex e nal o ces ends o
inni y.
3.4 Ene gyIn e nal Ra e Iden i y and Unica ion o
E=ℏω
S a ing om ime dila ion ela ion
dτ/d = in /c
and in a ian
m0
, one can dene an ene gy
quan i y di ec ly ela ed o in e nal a e:
E o := m0c2d
dτ=m0c2c
in
=m0c3
in
.
On he o he hand, om ene gymomen um ela ion,
E o =γm0c2=ℏω o
.
Theo em 5
(Ene gyIn e nal Ra e Iden i y)
.
Unde he in o ma ion a e ci cle amewo k, o al
ene gy o a ee pa icle can be w i en bo h as
E o =γm0c2,
and as
E o =m0c3
in
,
compa ible wi h he Planck ela ion
E o =ℏω o .
This es ablishes
ω o =E o
ℏ=m0c3
ℏ in
=γω0.
This shows:

E=mc2
and
E=ℏω
a e essen ially wo ways o w i ing he same iden i y in die en
a iables.

The in o ma ion a e ci cle gi es a iple unica ion among ene gy equencyin e nal ime
a e.
P oo s o hese heo ems will be gi en in subsequen P oo s sec ion and appendices.
7
4 P oo s
This sec ion p o ides main de i a ions o abo e heo ems, wi h mo e echnical ope a o and geo-
me ic a gumen s mo ed o appendices.
4.1 P oo o Theo em ??: Minkowski Geome y and Ci cula Repa ame iza-
ion
Fou - eloci y is dened as
uµ=dxµ
dτ= (γc, γ ).
I s Minkowski no m sa ises
uµuµ=−c2γ2+ 2γ2=−c2.
Le
ex := , in := c 1− 2
c2.
Then
2
ex + 2
in = 2+c21− 2
c2=c2.
On he o he hand, om ime dila ion
dτ
d = 1− 2
c2= in
c,
we see he ela ion be ween
dτ/d
and
in
is p ecisely he adial p ojec ion o he in o ma ion
a e ci cle. Thus, Minkowski ou - eloci y iden i y and in o ma ion a e ci cle a e jus die en
pa ame iza ions o he same cons ain . This comple es Theo em
??
.
4.2 P oo o Theo em ??: Minkowski Geome y o F equency
Ene gymomen um ela ion
E2=m2
0c4+p2c2
di ided on bo h sides by
ℏ2
gi es
E
ℏ2
=m0c2
ℏ2
+pc
ℏ2.
Le
ω o := E
ℏ, ω0:= m0c2
ℏ, ωspace := pc
ℏ,
we ob ain
ω2
o =ω2
0+ω2
space.
On he o he hand, eloci y can be w i en as
=∂E
∂p =pc2
E=ωspacec2/c
ω o
=cωspace
ω o
.
Thus
β= /c =ωspace/ω o
. F om his:
γ=1
p1−β2=ω o
ω0
,
8
hence
ω o =γω0.
W i ing ime dila ion as
dτ/d = 1/γ
, in e nal clock equency
ωclock =ω0
dτ
d =ω0
γ
ob iously sa ises
ωclock ·ω o =ω0
γ·(γω0) = ω2
0.
Theo em
??
ollows.
4.3 P oo o Theo em ??: Rela i is ic Dynamics and In e nal Ra e Rew i e
Rela i is ic mechanics along eloci y di ec ion gi es
F∥=d
d (γm0 ) = m0γ3a∥,
a s anda d ex book esul . In oducing
m∥
e := γ3m0,
we ha e
F∥=m∥
e a∥.
Using
in =c 1− 2
c2=c
γ,
we ob ain
γ=c
in
, γ3=c
in 3
,
hus
m∥
e =γ3m0=m0c
in 3
.
When
→c
,
in →0
; om he abo e equa ion we di ec ly see
m∥
e
di e ges, i.e., Theo em
??
.
4.4 P oo o Theo em ??: T iple Iden i y o Ene gyIn e nal Ra eF equency
F om ime dila ion
d
dτ=γ
and
E o =γm0c2,
his can be iewed as
E o =m0c2d
dτ.
On he o he hand, om
in =c/γ
we ha e
d
dτ=c
in
,
9
[10] P. M. B own, On he concep o mass in ela i i y, a Xi :0709.0687 (2007).
[11] H. Ma, Uni e sal Conse a ion o In o ma ion Cele i y, p ep in (2025).
A Di acQCA Hamil onian and Ope a o P oo o F equency O -
hogonali y
This appendix p o ides ope a o - o m p oo o equency o hogonal ela ion in Theo em
??
and
demons a es i s connec ion wi h Di acQCA eec i e Hamil onian.
A.1 Di ac Hamil onian Squa e and Ene gyMomen um Rela ion
Conside one-dimensional Di ac- ype Hamil onian
H=cαˆp+βm0c2,
whe e
α, β
a e
2×2
Pauli ma ices sa is ying
α2=β2=I,{α, β}=αβ +βα = 0.
Compu ing
H2
:
H2=c2αˆp·αˆp+cαˆp·βm0c2+βm0c2·cαˆp+β2m2
0c4
=c2α2ˆp2+cm0c2ˆp(αβ +βα) + β2m2
0c4.
Using
α2=β2=I
and
{α, β}= 0
, middle c oss e m comple ely cancels, yielding
H2=c2ˆp2+m2
0c4.
On momen um eigens a e
ˆpψp=pψp
,
H2ψp=E2ψp
, om which we ob ain
E2=c2p2+m2
0c4,
he s anda d ene gymomen um ela ion. This de i a ion emains alid in Di acQCA's con inuum
limi , as QCA's one-s ep e olu ion ope a o
U
has eec i e ep esen a ion
U= exp(−iH∆ /ℏ)
.
A.2 F equency O hogonal Rela ion
Di iding bo h sides o abo e by
ℏ2
gi es
E
ℏ2
=m0c2
ℏ2
+pc
ℏ2.
Le
ω o := E
ℏ, ω0:= m0c2
ℏ, ωspace := pc
ℏ,
we ob ain
ω2
o =ω2
0+ω2
space,
p ecisely he Py hago ean ela ion in equency space. I embodies o hogonali y o mass e m
and momen um e m in he Hamil onian a ope a o le el: an icommu a ion ela ion ensu es no
c oss e m appea s in
H2
, hus p ese ing igh iangle algeb aic s uc u e.
16

A.3 Co espondence wi h In o ma ion Ra e Geome y
In in o ma ion a e ci cle,
ex
and
in
sa is y
2
ex + 2
in =c2.
Via ene gymomen um ela ion, eloci y and equency can be connec ed:
=cωspace
ω o
, γ =ω o
ω0
, in =c 1− 2
c2=c
γ=cω0
ω o
.
The e o e
 ex
c2=ωspace
ω o 2
, in
c2=ω0
ω o 2
.
This shows ha geome ic s uc u e o in o ma ion a e ci cle in eloci y space comple ely pa allels
o hogonal ela ion o
ω0
and
ωspace
in equency space, u he suppo ing a ionali y o ine ial
geome y as unied desc ip ion.
B De ailed De i a ion o Longi udinal Eec i e Ine ia
This appendix p o ides de ailed expansion o de i a ion o
F∥=m0γ3a∥
and gi es i s ew i e in
in o ma ion a e a iables.
B.1 Rela i is ic Momen um and Accele a ion Decomposi ion
Rela i is ic momen um is dened as
p=γm0 .
Accele a ion along eloci y di ec ion
a∥:= d
d ,
applied o ce
F∥:= dp
d .
Compu ing
F∥=d
d (γm0 )
=m0γd
d + dγ
d .
F om
γ=1
p1−β2, β =
c,
we ge
dγ
d =dγ
dβ
dβ
d =βγ3
c
d
d =γ3
c2a∥.
Subs i u ing
F∥=m0γa∥+ γ3
c2a∥=m0γ+γ3 2
c2a∥.
17
Using
γ2=1
1−β2⇒γ2−1 = β2
1−β2=γ2 2
c2,
i.e.,
γ2 2
c2=γ2−1,
hen
γ+γ3 2
c2=γ+γ(γ2−1) = γ3.
Thus
F∥=m0γ3a∥,
he o igin o longi udinal eec i e ine ial mass
m∥
e =γ3m0
.
B.2 Rew i e Using In e nal Ra e
in
In o ma ion a e ci cle gi es
in =c 1− 2
c2=c
γ,
hus
γ=c
in
, γ3=c
in 3
.
Hence
m∥
e =γ3m0=m0c
in 3
.
This shows sensi i e dependence o
m∥
e
on in e nal a e
in
: when
in
dec eases due o high
ex e nal eloci y, eec i e ine ia g ows apidly cubically.
In in o ma ion geome ic pic u e, his means:

La ge eec i e ine ia does no indica e mo e ma e , bu indica es in e nal ime ow com-
p essed o ex emely low a e, making any a emp o change i s ex e nal mo ion s a e ha e
o le e  a nea ly ozen in e nal s uc u e, hus appea ing ex emely dicul .
C In e nal E olu ion Speed and FubiniS udy Me ic
This appendix explains how o ela e in e nal eloci y
in
o e olu ion speed in quan um s a e space
and discusses i s ela ionship wi h quan um speed limi s.
C.1 FubiniS udy E olu ion Veloci y
In p ojec i e Hilbe space
P(H)
, e olu ion o s a e ec o
|ψ( )⟩
independen o o e all phase can
be cha ac e ized by FubiniS udy line elemen
ds2= 4 1− |⟨ψ( )|ψ( + d )⟩|2.
AnandanAha ono p o ed ha o e olu ion d i en by Hamil onian
H
, e olu ion  eloci y sa ises
ds
d =2
ℏ∆H,
18
whe e
∆H=p⟨H2⟩−⟨H⟩2
is ene gy unce ain y. This ela ion shows: unde gi en ene gy dis-
pe sion esou ces, geome ic e olu ion a e o s a e in
P(H)
is limi ed. When sys em app oaches
quan um speed limi , i s e olu ion speed app oaches
2∆H/ℏ
.
C.2 In e nal Veloci y Scaling and Sa u a ion
Fo single-pa icle sec o o ee Di ac pa icles, o e all Hilbe space can be spli in o ex e nal
(posi ion) and in e nal (spin/pa iclean ipa icle) wo pa s o deg ees o eedom. Unde high
symme y, ene gy unce ain y can be expec ed o be mainly de e mined by in e nal s uc u e, while
g oup eloci y is ela ed o ex e nal momen um.
One can hen dene in e nal e olu ion eloci y
in := ℓin
ds
d =ℓin
2
ℏ∆Hin ,
whe e
ℓin
is a xed leng h scale used o con e dimensionless geome ic eloci y o quan i y wi h
eloci y dimensions. Choose
ℓin
such ha a es
in ( = 0) = c,
es ablishing one- o-one co espondence be ween in e nal e olu ion eloci y and
in
in in o ma ion
a e ci cle. In con inuum limi , Di acQCA na u ally p o ides such leng h and ime uni s (la ice
spacing and s ep size), so in e nal e olu ion eloci y can be iewed as a io o FubiniS udy dis ance
a e sed by in e nal s a e change pe s ep o s ep size.
When sys em ene gy is en i ely con ibu ed by es mass, in e nal e olu ion app oaches quan um
speed limi ,
in ≈c
; when mos sys em ene gy con e s o ex e nal momen um, in e nal e olu ion
eloci y dec eases,
in < c
, co esponding o deec ion on in o ma ion a e ci cle.
C.3 Quan um Speed Limi and In o ma ion Ra e Uppe Bound
Quan um speed limi gi es minimum ime equi ed o e ol e om one s a e o o hogonal s a e:
⊥≥max πℏ
2∆E,πℏ
2¯
E,
whe e
∆E
and
¯
E
a e ene gy unce ain y and a e age ene gy espec i ely. I in e nal deg ees o
eedom a e iewed as subsys em mainly esponsible o s a e o hogonal change, uppe bound o
in e nal e olu ion eloci y
in
can be iewed as geome ic embodimen o quan um speed limi .
In o ma ion a e ci cle assumes
in ≤c
as absolu e uppe bound, sa u a ing his bound a es .
This assump ion is o mally compa ible wi h quan um speed limi concep :

S a iona y pa icle in e nal e olu ion app oaches ex eme speed, able o comple e s a e dis-
inc ion in minimum ime.

Mo ing pa icle alloca es pa o  eloci y budge  o ex e nal displacemen , hus in e nal
e olu ion slows, co esponding o longe s a e o hogonal ime.
This pic u e p o ides na u al quan um in o ma ion backg ound o in o ma ion a e ci cle, also
poin ing ou ha u u e mo e ened quan um speed limi expe imen s can u he es his ame-
wo k.
19