Gauge Fields as Geometry of Information Transport: Deriving Maxwell and Yang--Mills Equations in Quantum Cellular Automata

Abs ac
In s anda d quan um eld heo y, gauge elds a e in oduced by demanding local in-
a iance o a Lag angian unde a p esc ibed Lie g oup, and a e geome ically in e p e ed as
connec ions on p incipal b e bundles. In a uni e se desc ibed by a quan um cellula au-
oma on (QCA), howe e , he mic oscopic on ology is a disc e e ne wo k o ni e-dimensional
quan um sys ems upda ed by causal local uni a ies. In such a disc e e on ology he e is no
p e e ed global e e ence ame o in e nal deg ees o eedom a die en la ice si es; each
cell only has access o a local Hilbe - ame.
This wo k de elops a amewo k in which gauge elds a ise as
ansla ion p o ocols
be ween local in o ma ion ames
on a QCA. S a ing om h ee assump ions(i) a
s ic ly causal, ansla ion-in a ian QCA on a egula la ice; (ii) local ame independence
o physical p edic ions; and (iii) a minimal-dis o ion p inciple o in o ma ion anspo we
show:
1. Local edundancy in he choice o in e nal basis en o ces he in oduc ion o link a i-
ables as pa allel- anspo ope a o s be ween neighbou ing cells, wi h he s anda d
la ice-gauge ans o ma ion law. Abelian phase edundancy yields a
U(1)
connec ion;
in e nal mul i-componen edundancy yields non-Abelian
G
-connec ions.
2. In he con inuum limi o a Di ac- ype QCA, he equi emen ha he disc e e dynamics
be co a ian unde local ame changes is equi alen o minimal coupling o a Di ac
eld o a gauge po en ial
Aµ
, and he gauge cu a u e is ob ained as he loga i hm o
elemen a y plaque e holonomies.
3. Imposing ha he dynamics o hese link a iables minimize a local, gauge-in a ian ,
quad a ic in o ma ion-dis o ion unc ional uniquely selec s, unde mild egula i y and
iso opy assump ions, an eec i e ac ion which educes in he con inuum o he Maxwell
o YangMills ac ion.
4. The mic oscopic gauge coupling can be exp essed in e ms o a ios o in o ma ion-
anspo a es be ween in e nal and in e -si e channels wi hin he QCA, connec ing
unning couplings o changes in eec i e connec i i y and ime-s ep a die en p obing
scales.
In his pic u e, undamen al in e ac ions a e no ex a en i ies added o ma e elds,
bu cons ain s on how in o ma ion can be anspo ed consis en ly ac oss a disc e e ne wo k
o local ames. Gauge cu a u e measu es he ailu e o in o ma ion-pa allel anspo o
be pa h-independen , and he gauge eld equa ions a ise as Eule Lag ange equa ions o an
in o ma ion-dis o ion unc ional dened on QCA link a iables.
Keywo ds:
Quan um Cellula Au oma a; Gauge In a iance; Wilson Loops; In o ma ion
T anspo ; Maxwell Equa ions; YangMills Theo y; La ice Gauge Theo y
1 In oduc ion & His o ical Con ex
F om he pe spec i e o con inuous eld heo y, he in oduc ion o gauge elds ypically ol-
lows he "gauge p inciple": gi en a ee Lag angian wi h global symme y g oup
G
, p omo e
he symme y o a space ime poin -dependen local symme y
G(x)
, and es o e in a iance by
in oducing gauge po en ial
Aµ(x)
and co a ian de i a i e
Dµ=∂µ−igAµ
. Fo
G=U(1)
,
his yields elec omagne ic elds; o
G=SU(2) ×U(1)
,
SU(3)
, e c., yields weak and s ong
in e ac ions in he S anda d Model. Geome ically, gauge po en ial is a connec ion one- o m on
a p incipal bundle, and cu a u e
Fµν
co esponds o he cu a u e wo- o m o he connec ion.
In he non-pe u ba i e egime, disc e iza ion o gauge heo y was pionee ed by Wilson's
la ice gauge heo y: space ime is disc e ized in o la ice si es and links, gauge elds a e g oup
elemen s
Ux,µ ∈G
alued on links, cu a u e is gi en by g oup elemen p oduc s on closed
Wilson loops, and in he con inuum limi he Wilson ac ion con e ges o YangMills ac ion.
La ice gauge heo y no only p o ided he  s con olled desc ip ion o low-ene gy beha io in
s ong in e ac ions bu also supplies a na u al disc e e s uc u e o quan um simula ion.
1
On he o he hand, Quan um Cellula Au oma a (QCA) model he physical uni e se as
a collec ion o ni e-dimensional quan um sys ems on disc e e la ice si es, globally upda ed
by local uni a y ope a o s a disc e e ime s eps. Re iews by A ighi, Fa elly, e al. show
ha QCA can uni y nume ous disc e e space ime quan um models, including quan um walks,
la ice eld heo ies, and pa ially disc e ized quan um eld heo ies, and can eme ge Di ac,
Weyl, and Maxwell equa ions in app op ia e con inuum limi s. Fu he mo e, A naul e al.
demons a ed ha disc e e- ime quan um walks (DTQW) can ealize
U(1)
and non-Abelian
disc e e gauge heo ies wi h s ic la ice gauge in a iance, simula ing Di ac ma e coupling o
elec omagne ic/YangMills elds in he con inuum limi .
These esul s indica e ha in oducing gauge elds in disc e e amewo ks is no dicul ; he
dicul y lies in hei on ological s a us. In adi ional cons uc ions, e en on la ices, gauge elds
a e o en iewed as "nume ical ools o simula ing a gi en con inuous gauge eld heo y." In he
disc e e on ology pe spec i e o QCA, howe e , la ice si es and links a e he mos undamen al
"uni e se p imi i es," wi h no deepe con inuous eld as e e ence. F om his pe spec i e, na u al
ques ions a ise:

Can gauge elds be unde s ood as a geome ic cons ain on in o ma ion anspo in QCA
ne wo ks, a he han addi ional "elds"?

Can gauge symme y eme ge om " edundancy o local in o ma ion e e ence ames" and
"consis ency o in o ma ion anspo ," a he han as an a p io i symme y assump ion?
This pape aims o p o ide such an in e p e a ion om he QCA pe spec i e. Specically:
1. Viewing each cell's in e nal Hilbe space
Hx
as a local in o ma ion e e ence ame, assume
physical p edic ions a e insensi i e o he choice o in e nal basis a each poin ;
2. P o e ha o main ain uni a y and causally consis en in o ma ion anspo wi hou global
basis, one mus necessa ily in oduce "pa allel- anspo ope a o s"
Ux,µ
as connec ion
elds be ween la ice si es, wi h ans o ma ion law iden ical o la ice gauge heo y;
3. In he smoo h con inuum limi , combining locali y, iso opy, and uni a y e olu ion con-
s ain s, cha ac e ize he minimal-dis o ion ac ion o such connec ion elds, p o ing i s
con inuum limi uniquely selec s Maxwell/YangMills ac ion;
4. Using he p e iously p oposed p inciple o conse a ion o in o ma ion a e, ela e gauge
coupling cons an o a ios o QCA in e nal/ex e nal in o ma ion channel a es, he eby
gi ing mic oscopic in o ma ion- heo e ic-geome ic in e p e a ion o cha ge and coupling
cons an s.
In his amewo k, gauge elds a e no longe "auxilia y objec s a icially in oduced o
main ain Lag angian in a iance unde a local symme y g oup," bu "geome ic cos s ha mus
be paid o allow in o ma ion o be s ably ansmi ed and die en obse e s o each consis en
desc ip ions in a disc e e uni e se wi hou global basis." Gauge cu a u e cha ac e izes he ex en
o which in o ma ion-pa allel anspo is no longe pa h-independen .
2 Model & Assump ions
This sec ion p esen s he QCA model, local e e ence ame edundancy and in o ma ion ans-
po p inciples, and denes gauge s uc u e as he minimal geome ic objec sa is ying hese
p inciples.
2
2.1 Basic S uc u e o Quan um Cellula Au oma a
Conside a
d
-dimensional egula la ice
Λ = aZd
wi h la ice spacing
a
and ime s ep
∆
. A
each la ice si e
x∈Λ
, place a ni e-dimensional Hilbe space
Hx∼
=CNin
, called in e nal
deg ees o eedom. The global Hilbe space is
H=O
x∈Λ
Hx.
One e olu ion s ep o he QCA is gi en by a uni a y ope a o
G:H → H
sa is ying he
ollowing condi ions:
1. Causali y: he e exis s ni e adius
R
such ha
G
's ac ion on local algeb a
Ax
a some
la ice si e
x
depends only on deg ees o eedom wi hin adius
R
neighbo hood o
x
;
2. T ansla ion in a iance: he e exis s a amily o ansla ion ope a o s
Ta
such ha
G
com-
mu es wi h
Ta
;
3. Local decomposi ion:
G
can be w i en as ni e-dep h local quan um ci cui
G=QℓGℓ
,
whe e each
Gℓ
ac s on ni ely many adjacen la ice si es.
Gi en in e nal dimensionali y and locali y cons ain s, one can cons uc Di ac- ype QCA
whose con inuum limi yields Di ac equa ion. This pape does no depend on a specic con-
s uc ion, bu assumes exis ence o such a class o " ee ma e QCA" whose single-pa icle
eec i e Hamil onian is
H0=X
x,µ ψ†(x)Kµψ(x+aˆµ)+h.c.+X
x
ψ†(x)Mψ(x),
whe e
ψ(x)∈CNin
,
Kµ, M
a e xed ma ices, and in he con inuum limi
H0
co esponds o
some Lo en z-in a ian ee eld (such as Di ac eld).
2.2 Local In o ma ion Re e ence F ame and Gauge Redundancy
In disc e e uni e se on ology, he e is no ex e nal "God's-eye- iew" global e e ence ame. Each
cell can only access i s own in e nal Hilbe space
Hx
's local basis choice. Fo each la ice si e
x
, choose an o hono mal basis
{|ei(x)⟩}Nin
i=1
, hen s a e ec o 's coo dina e ep esen a ion is
ψx=X
i
ψi
x|ei(x)⟩.
Local basis choice is a bi a y: o any
Vx∈G⊂U(Nin )
, pe o ming local ans o ma ion
|ei(x)⟩ 7→ X
j
(Vx)ji|ej(x)⟩, ψx7→ Vxψx,
physical p edic ions should emain unchanged. The se
{Vx}x∈Λ
cons i u es local gauge g oup
G=Y
x∈Λ
Gx, Gx∼
=G.
This embodies " edundancy o local in o ma ion e e ence ame."
Depending on in e nal s uc u e and physical con ex , conside wo ypical cases:
1.
G=U(1)
: only global phase edundancy, co esponding o cha ge
U(1)
gauge;
2.
G=SU(Nc)
o mo e gene al compac Lie g oup: in e nal deg ees o eedom ha e "colo "
o "a o " mul i-componen s uc u e, co esponding o non-Abelian gauge g oup.
Subsequen discussion unies using gene al compac Lie g oup
G
, wi h Abelian case as special
case.
3
2.3 In o ma ion T anspo and Deni ion o Connec ion Field
In QCA, spa ial p opaga ion o in o ma ion is ealized by coupling be ween adjacen cells. In
idealized desc ip ion xing some global basis, ee Hamil onian
H0
con ains e ms
ψ†(x)Kµψ(x+aˆµ),
whose physical meaning is "hopping ampli ude om
x
o
x+aˆµ
." Howe e , om on ology
pe spec i e wi hou global basis,
ψ(x)
and
ψ(x+aˆµ)
a e ep esen ed in espec i e local bases
and canno be di ec ly added. To desc ibe such c oss-cell in o ma ion anspo , a "pa allel-
anspo ope a o "
Ux,µ
mus be in oduced be ween la ice si es, mapping
Ux,µ :Hx−→ Hx+aˆµ.
Deni ion 1 (Connec ion eld/link a iable).
Gi en gauge g oup
G
, a amily o ope -
a o s
{Ux,µ}
is called connec ion eld o he QCA i o each link
(x, x +aˆµ)
,
Ux,µ ∈G⊂U(Hx,Hx+aˆµ),
and unde local gauge ans o ma ion
{Vx}
sa ises
Ux,µ 7→ U′
x,µ =Vx+aˆµUx,µV†
x.
Unde his deni ion, QCA ma e Hamil onian con aining connec ion eld na u ally w i es
as
Hma e [U] = X
x,µ ψ†(x)KµUx,µψ(x+aˆµ)+h.c.+X
x
ψ†(x)Mψ(x),
o mally iden ical o "inse ing g oup elemen on link" in la ice gauge heo y, bu he e
Ux,µ
is
geome ic quan i y necessa ily in oduced by local e e ence ame edundancy and in o ma ion
anspo p inciple, no a ick " o disc e izing some con inuous gauge eld heo y."
2.4 In o ma ion Dis o ion and P inciple o Gauge Field Ac ion
Connec ion elds a e no a bi a y deg ees o eedom. Physical e olu ion o QCA should ans-
po in o ma ion as "gen ly" as possible, i.e., minimizing dis o ion in pa allel anspo while
main aining uni a i y and causali y. Fo his, in oduce he ollowing p inciple:
P inciple A (Local in o ma ion dis o ion minimiza ion).
Among all connec ion
eld dynamics compa ible wi h gi en ma e Hamil onian
Hma e [U]
and sa is ying local gauge
in a iance, ac ual physical e olu ion co esponds o ajec o ies ex emizing some local gauge-
in a ian unc ional
Sgauge[U]
.
To embody locali y and iso opy,
Sgauge[U]
should depend only on pa allel- anspo ope a o
p oduc s on minimal closed loops (plaque es), i.e., Wilson loops
W□= U□, U□=Ux,µUx+aˆµ,νU†
x+aˆν,µU†
x,ν,
and con e ge in con inuum limi o some in eg able local ac ion densi y
L(Fµν)
.
S anda d la ice gauge heo y shows ha unde second-de i a i e and locali y assump ions,
he unique gauge-in a ian quad a ic o m in con inuum limi is p ecisely YangMills ac ion
(FµνFµν)
. This pape ein e p e s his conclusion as "unde local e e ence ame edundancy
and in o ma ion dis o ion minimiza ion p inciples, eec i e ac ion o connec ion eld necessa -
ily educes o Maxwell/YangMills ac ion."
3 Main Resul s (Theo ems and Alignmen s)
This sec ion p esen s co e heo ems and explains hei co espondence wi h exis ing heo ies.
4
3.1 Theo em 1 (Local Basis Redundancy
⇒
Gauge Connec ion T ans o ma-
ion Law)
Le
G
be a compac Lie g oup,
G⊂U(Nin )
some uni a y ep esen a ion. Conside a gi en
ee QCA, in oducing local basis ans o ma ion
Vx∈G
a each la ice si e
x
, and assume
ma e Hamil onian
Hma e [U]
is physically equi alen unde
G=QxGx
. I equi ing ha
unde a bi a y local ans o ma ion
Vx
, c oss-la ice ansi ion ampli ude
ψ†(x)KµUx,µψ(x+aˆµ)
is physically in a ian , hen he e exis s and is unique a amily o link ope a o s
Ux,µ ∈G
whose
ans o ma ion law unde
G
is
Ux,µ 7→ Vx+aˆµUx,µV†
x.
This ans o ma ion law is comple ely consis en wi h link a iable ans o ma ion in la ice
gauge heo y.
3.2 Theo em 2 (Minimal Coupling in F ee QCA Con inuum Limi )
Le a class o QCA wi hou connec ion eld ha e single-pa icle eec i e Hamil onian in long-
wa eleng h limi yielding ee Di ac Hamil onian
H0=Zddx ψ†(x)−iγ0γi∂i+mγ0ψ(x),
sa is ying ansla ion, o a ion, and disc e e
CPT
symme y. Applying Theo em 1's amewo k o
his QCA, in oducing
G
- alued connec ion eld
Ux,µ
on links, and equi ing e olu ion co a ian
unde local gauge ans o ma ion. Then in smoo h limi whe e
a, ∆ →0
and
Ux,µ →I
, single-
pa icle eec i e Hamil onian o
Hma e [U]
is equi alen o
H=Zddx ψ†(x)−iγ0γiDi+mγ0ψ(x), Dµ=∂µ−igAµ(x),
whe e
Aµ(x)∈g
is Lie algeb a- alued gauge po en ial, wi h ela ion o
Ux,µ
:
Ux,µ = exp (−igaAµ(x)) + O(a2).
In pa icula , o
G=U(1)
yields cha ged Di acMaxwell coupling, o non-Abelian
G
yields
Di acYangMills minimal coupling. This esul is consis en wi h DTQW model conclusion
ha "Di acgauge eld coupling eme ges in con inuum limi ."
3.3 Theo em 3 (Local In o ma ion Dis o ion Func ional
⇒
Maxwell / Yang
Mills Ac ion)
Le
Sgauge[U]
be ac ion dened on link a iables, sa is ying:
1. Locali y:
Sgauge
can be w i en as sum o local unc ions o each ni e Wilson loop;
2. Gauge in a iance: in a ian unde
Ux,µ 7→ Vx+aˆµUx,µV†
x
;
3. Iso opy: in a ian unde ac ion o disc e e o a ion g oup on la ice;
4. Smoo h limi : when
Ux,µ
alues nea iden i y, ac ion densi y is posi i e deni e quad a ic
o m o
Fµν
wi h no de i a i es highe han second o de .
5

Then in con inuum limi
a→0
,
Sgauge[U]
is equi alen o
Sgauge → −1
2Zdd+1x (FµνFµν) + O(a2),
whe e gauge cu a u e
Fµν =∂µAν−∂νAµ−ig[Aµ, Aν],
educes o Maxwell ac ion o
G=U(1)
, yields YangMills ac ion o non-Abelian
G
. This
conclusion is consis en wi h con inuum limi o Wilson ac ion.
3.4 Theo em 4 (In o ma ion-Theo e ic Cha ac e iza ion o Gauge Coupling
Cons an )
In a QCA uni e se sa is ying conse a ion o in o ma ion a e
2
ex + 2
in =c2,
le each cell ha e
Nch
channels a ailable o ex e nal in o ma ion anspo (links), wi h coo di-
na ion numbe
z
. Deno e
in =
eec i e a e o in e nal phase upda e pe ime s ep
o al in o ma ion a e
, hop =
eec i e a e o in e -la ice hopping pe ime s ep
o al in o ma ion a e
,
hen unde na u al no maliza ion, gauge coupling
g
and dimensionless ne s uc u e cons an
α=g2/(4π)
can be w i en as
g2∝ in
z Nch hop
, α ∝ in
z Nch hop
.
When p obing scale changes lead o changes in eec i e coo dina ion numbe and channel num-
be ,
α
will " un" acco dingly, he eby gi ing disc e e geome ic mechanism o eno maliza ion
g oup ow in QCA. This esul connec s he p e iously p oposed "uni e sal conse a ion o
in o ma ion a e" wi h gauge coupling cons an .
4 P oo s
This sec ion p o ides p oo ou lines o he abo e heo ems, wi h de ailed de i a ions in appen-
dices.
4.1 P oo o Theo em 1: Gauge Connec ion T ans o ma ion Law
Wi hou connec ion eld, ma e Hamil onian w i es as
H0=X
x,µ ψ†(x)Kµψ(x+aˆµ)+h.c.+X
x
ψ†(x)Mψ(x).
Unde local basis ans o ma ion
ψ(x)7→ Vxψ(x)
, c oss-la ice e m becomes
ψ†(x)Kµψ(x+aˆµ)7→ ψ†(x)V†
xKµVx+aˆµψ(x+aˆµ).
I
Vx
a ies wi h
x
, his e m's ma ix s uc u e gene ally changes, b eaking QCA's ansla ion
in a iance and o m o o iginal coupling ma ix
Kµ
. To main ain physical equi alence o "ma e 
connec ion sys em" a e local ans o ma ion, link a iable
Ux,µ
mus be inse ed in c oss-la ice
e m, equi ing a e ans o ma ion
ψ†(x)KµUx,µψ(x+aˆµ)7→ ψ†(x)KµU′
x,µψ(x+aˆµ),
6
i.e., he e exis s some
U′
x,µ
making new and old Hamil onian densi y o ms consis en .
W i ing local ans o ma ion explici ly:
ψ†(x)KµUx,µψ(x+aˆµ)7→ ψ†(x)V†
xKµUx,µVx+aˆµψ(x+aˆµ).
To ew i e as
ψ†(x)KµU′
x,µψ(x+aˆµ)
, need
KµU′
x,µ =V†
xKµUx,µVx+aˆµ.
Assuming
Kµ
in e ible in ep esen a ion space, hen
U′
x,µ =K−1
µV†
xKµUx,µVx+aˆµ.
To make
U′
x,µ
s ill ake alues in
G
and main ain ansla ion-in a ian o m o
Kµ
, na u al
equi emen is ha
Kµ
commu es wi h
G
's ep esen a ion, making
K−1
µV†
xKµ=V†
x
. This can
be sa ised by choosing
Kµ
scala on in e nal space o compa ible wi h
G
's ep esen a ion. Thus
ob aining
U′
x,µ =Vx+aˆµUx,µV†
x,
he s anda d ans o ma ion law o la ice gauge heo y. Since
Vx
a bi a y and
G
compac g oup,
can p o e his law uniquely main ains Hamil onian s uc u e and local ansla ion symme y
gi en
H0
, ob aining Theo em 1.
4.2 P oo o Theo em 2: Minimal Coupling in Con inuum Limi
Conside unied ep esen a ion o Abelian case
G=U(1)
and non-Abelian case. Le
Ux,µ = exp (−igaAµ(x)) , Aµ(x)∈g.
In long-wa eleng h limi , assume
ψ(x)
a ies slowly on la ice, can w i e
ψ(x+aˆµ) = ψ(x) +
a∂µψ(x) + O(a2)
. C oss-la ice e m is
ψ†(x)KµUx,µψ(x+aˆµ) = ψ†(x)Kµexp (−igaAµ(x)) ψ(x) + a∂µψ(x) + O(a2).
Expanding o
O(a)
:
ψ†(x)Kµψ(x) + aψ†(x)Kµ∂µψ(x)−igaψ†(x)KµAµ(x)ψ(x) + O(a2).
Fo con inuum limi o ee QCA, exis ing esul s show he e exis s choice such ha
X
µ
ψ†(x)Kµ∂µψ(x)→ψ†(x)γ0γi∂iψ(x),
and mass e m gi en by
Pxψ†(x)Mψ(x)
. On his basis, abso bing
−igaψ†KµAµψ
e m in o
de i a i e, can eplace spa ial de i a i e wi h co a ian de i a i e
∂µ7→ Dµ=∂µ−igAµ(x),
ob aining Di acgauge eld minimal coupling o m. Time di ec ion coupling can be ob ained
h ough simila cons uc ion o QCA's ime upda e ope a o . In ac ual DTQW/QCA li e a u e,
s ic ly la ice gauge-in a ian quan um walks ha e been demons a ed, whose con inuum limi
yields Di ac eld coupled Hamil onian wi h ex e nal elec omagne ic eld; in e p e ing "manually
inse ed" connec ion eld he ein as abo e cons uc ion yields p ecise s a emen o Theo em 2.
7
4.3 P oo o Theo em 3: Con inuum Limi o In o ma ion Dis o ion Func-
ional
By P inciple A,
Sgauge[U]
should be cons uc ed om Wilson loops on each minimal plaque e.
Fo a plaque e
□
dene
U□=Ux,µUx+aˆµ,νU†
x+aˆν,µU†
x,ν.
Requi ing
Sgauge
in a ian unde
Ux,µ →Vx+aˆµUx,µV†
x
means
Sgauge
mus be cons uc ed
om
(U□)
and conjuga e. Simples local iso opic unc ional is Wilson ac ion
SW[U] = X
□
β
Nin
Re I−U□,
whe e
β
ela ed o coupling cons an
g
. Fo
Ux,µ ≈exp(−igaAµ)
expanding in small
a
limi ,
ob ain
U□= exp −iga2Fµν(x) + O(a3),
see Appendix A o BCH expansion calcula ion. Thus
I−U□= iga2Fµν (x) + 1
2g2a4F2
µν(x) + O(a6),
eal pa o ace o
O(a4)
is
Re I−U□=1
2g2a4 F2
µν(x)+O(a6).
App oxima ing
P□
as
Rdd+1x/ad+1
, ob ain
SW[U]→βg2
2Nin Zdd+1x FµνFµν,
which is YangMills ac ion. Reduces o Maxwell ac ion o
G=U(1)
. Con e sely, can p o e
ha gi en abo e ou condi ions, any o he local gauge-in a ian quad a ic o m in con inuum
limi only adds highe -de i a i e co ec ions o his ac ion, hus Theo em 3 holds.
4.4 P oo o Theo em 4: Coupling Cons an and In o ma ion Ra e
In p e ious wo k, conse a ion o in o ma ion a e p inciple s a es: o any local exci a ion, i s
ex e nal g oup eloci y
ex
and in e nal s a e e olu ion eloci y
in
sa is y
2
ex + 2
in =c2,
whe e
c
is ligh -cone eloci y o QCA. In e nal s a e e olu ion eloci y can be cha ac e ized by
spec al wid h and phase o a ion a e o local Hamil onian, ex e nal g oup eloci y gi en by
in e -la ice ansi ion ampli ude and coo dina ion numbe .
In QCA wi h gauge eld, pa o in e nal phase e olu ion is ca ied by " o a ion" o gauge
connec ion, co esponding om pa icle pe spec i e o coupling wi h gauge eld. Deno ing
in = 2
in
c2, ex = 2
ex
c2, in + ex = 1,
can decompose
in
in o gauge- ela ed pa
gauge
and "ba e in e nal deg ees o eedom" pa
ba e
. In a QCA wi h
z
nea es neighbo s,
Nch
anspo channels pe link, eec i e in e -link
in o ma ion anspo capaci y inc eases wi h
zNch
; o main ain o al a e conse a ion,
gauge
ela i e sha e should dec ease acco dingly. Na u al scaling ela ion is
g2∝ gauge
z Nch ex
.
In weak coupling limi ,
gauge
is dominan pa o
in
, can app oxima e
gauge ≈ in
, hus
ob aining Theo em 4's ela ion. When p obing ene gy scale ises, QCA's eec i e la ice spac-
ing
a
and isible coo dina ion numbe
z
change, leading o scaling changes in
g2
and
α
, hus
co esponding o eno maliza ion g oup ow in con inuous eld heo y.
8
5 Model Apply
This sec ion gi es se e al conc e e models, demons a ing how he abo e gene al amewo k is
implemen ed in specic QCA and co esponds o amilia physical heo ies in con inuum limi .
5.1
U(1)
Gauge Field in One-Dimensional Di ac QCA
Conside one-dimensional Di ac- ype QCA, whose ee e olu ion can be w i en as spli -s ep
quan um walk o m: a each la ice si e has wo-componen spin
ψ(x) = (ψ↑, ψ↓)T
, ime upda e
gi en by al e na ing o a ion and condi ional shi . App op ia ely choosing o a ion angle and
condi ional shi , can p o e con inuum limi yields one-dimensional Di ac equa ion.
On his basis, in oduce
U(1)
phase
Ux,±= exp[−igaA±(x)]
on each link, co esponding
o le / igh anspo channels espec i ely. Requi ing ull-s ep e olu ion co a ian unde local
U(1)
phase ans o ma ion
ψ(x)7→ eiα(x)ψ(x)
, ob ain
Ux,+7→ eiα(x+a)Ux,+e−iα(x), Ux,−7→ eiα(x−a)Ux,−e−iα(x).
This is p ecisely Abelian special case o Theo em 1. Expanding con inuum limi ,
Ux,±≈
exp[−igaAx(x)]
, yields Di acelec omagne ic minimal coupling. Full-ensemble Wilson loops
gi e disc e e cu a u e o one-dimensional elec ic eld, ac ion adop ing Wilson- ype unc ion
yields Maxwell ac ion.
5.2 Two-Dimensional QCA and Non-Abelian
SU (2)
Gauge Field
On wo-dimensional la ice, conside QCA wi h in e nal space
Nin = 4
, spin and "colo " each
occupying wo componen s. Le gauge g oup
G=SU(2)
ac on colo space, in e nal Hamil onian
Kµ
and
M
scala o colo space. In oduce link a iables
Ux,µ ∈SU(2), Ux,µ = exp[−igaAa
µ(x)Ta],
whe e
Ta
a e
SU(2)
gene a o s. Local gauge ans o ma ion
ψ(x)7→ Vxψ(x)
,
Vx∈SU(2)
ac ing
only on colo space, causes link a iables o ans o m acco ding o Theo em 1's ule. Choosing
Wilson ac ion o es ablish gauge eld dynamics, con inuum limi yields s anda d equa ions o
SU(2)
YangMills eld.
This cons uc ion echoes A naul e al.'s wo k on "quan um walks and non-Abelian disc e e
gauge heo y," which explici ly demons a ed disc e e- ime quan um walks wi h exac disc e e
U(N)
gauge in a iance, yielding YangMillsDi ac coupling in con inuum limi .
5.3 Elec omagne ic Field om In o ma ion Geome y Pe spec i e
In abo e QCAgauge s uc u e, elec ic and magne ic elds can co espond o he ollowing
in o ma ion geome ic quan i ies:
1. Elec ic eld
Ei
: phase g ow h a e die ence on same link be ween adjacen ime s eps,
cha ac e izing desynch oniza ion deg ee o local clocks. Disc e ely can w i e
Ei(x)∼1
ga∆ a g hUx,i( + ∆ )U†
x,i( )i.
2. Magne ic eld
Bi
: phase o Wilson loop on minimal plaque e in spa ial plane, cha ac e -
izing incompa ibili y deg ee o pa allel anspo on spa ial ci cula pa h. Disc e ely can
w i e
Bi(x)∼1
ga2a g U□(x),
whe e
□
is plana plaque e pe pendicula o
i
di ec ion.
9