Time as Density of States: Unified Time Identity \kappa = \rho and Microscopic Scattering Mechanism of Gravitational Redshift

Time as Densi y o S a es: Unied Time Iden i y
κ=ρ
and
Mic oscopic Sca e ing Mechanism o G a i a ional Redshi
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
In con en ional physics, ime plays incompa ible oles in quan um heo y and gene al
ela i i y: in quan um mechanics i is an ex e nal e olu ion pa ame e gene a ed by a Hamil-
onian, whe eas in gene al ela i i y p ope ime is a pa h-dependen unc ional o he space-
ime me ic. Wi hin a quan um cellula au oma on (QCA) and op ical-pa h conse a ion
amewo k, his wo k p oposes a mic oscopic deni ion o ime in e ms o he densi y o
quan um s a es. Building on EisenbudWigne Smi h (EWS) ime-delay heo y and K ein
F iedelLloyd (KFL) ace o mulas, we show ha he ace o he Wigne Smi h ime-delay
ope a o
Q(E) = −iℏS(E)†∂ES(E)
is uni e sally ela ed o he ela i e densi y o sca e ing
s a es ia he spec al shi unc ion. This leads o a unied ime iden i y
κ(E) = 1
2πℏ Q(E)=∆ρ(E),
whe e
∆ρ(E)
is he change o densi y o s a es (DOS) induced by an in e ac ion wi h espec
o a e e ence backg ound. Physically,
κ(E)
quan ies he a e a which a sys em a e ses
quan um s a es in Hilbe space pe uni ene gy. We in e p e
κ(E)
as an in insic  ime
densi y, and a gue ha mac oscopic clock a es a e go e ned by he local DOS o he
unde lying mic oscopic deg ees o eedom.
On his basis we de elop a mic oscopic pic u e o g a i a ional edshi : deep in a g a -
i a ional po en ial well, ma e and adia ion expe ience an enhanced local DOS due o
modied phase space olume and eec i e e ac i e index o he acuum. The inc eased
DOS yields a la ge Wigne Smi h delay pe uni ene gy, he eby slowing local p ope ime
ela i e o dis an obse e s. We show how, in he weak-eld limi , he DOS-based ime den-
si y ep oduces he s anda d g a i a ional edshi ela ion
∆ν/ν ≃∆Φ/c2
when no malized
o asymp o ically a egions, and a gue ha he me ic componen
g00
can be iewed as an
eme gen unc ional o he mic oscopic DOS eld. The imeDOS iden i y is embedded in o
Di ac- ype QCA models whe e ela i is ic dynamics eme ge om disc e e, s ic ly causal
uni a y upda es, demons a ing ha ela i is ic ime dila ion and g a i a ional edshi can
be ein e p e ed as collec i e p ope ies o s a e- a e sal a es on a disc e e quan um ne -
wo k.
Finally, we connec he p oposed ime densi y wi h he modynamic ime in KMS equi-
lib ium s a es and he he mal ime hypo hesis, showing ha he DOS-weigh ed EWS ime
densi y denes a na u al a ow o ime compa ible wi h modula ow in algeb aic quan um
eld heo y. We ou line expe imen al and enginee ing p oposals in mic owa e ca i ies, wa e-
chao ic sca e ing sys ems and op ical clock ne wo ks o es he quan i a i e link be ween
Wigne Smi h delays, DOS measu emen s and g a i a ional edshi .
Keywo ds:
ime delay; densi y o s a es; EisenbudWigne Smi h ope a o ; spec al shi unc-
ion; K einF iedelLloyd o mula; g a i a ional edshi ; quan um cellula au oma on; op ical
me ic; KMS s a e; he mal ime
1
1 In oduc ion & His o ical Con ex
The p oblem o ime pe ades he de elopmen o physical heo y. Special and gene al ela i i y
ake he ou -dimensional space ime mani old
(M, gµν)
as he undamen al objec , desc ibing
p ope ime
τ
as a line elemen in eg al along wo ldlines. Quan um heo y, by con as , is buil
on Hilbe space and he Hamil onian
H
, whe e ime
is an ex e nal con inuous pa ame e and
e olu ion is gene a ed by he uni a y ope a o
U( ) = exp(−iH /ℏ)
. While he wo amewo ks
a e compa ible in he semiclassical limi , a unied mic oscopic on ology o ime is lacking.
Pauli's heo em indica es ha in sys ems wi h an ene gy spec um bounded om below,
he e exis s no sel -adjoin ime ope a o sa is ying canonical commu a ion ela ions wi h
H
,
seemingly uling ou a emp s o ope a o ize ime wi hin adi ional quan um mechanics. On
he o he hand,  ime delay has long appea ed as an obse able in sca e ing heo y. Eisenbud,
Wigne , and Smi h dened he Wigne Smi h g oup delay ma ix om he ene gy de i a i e o
he sca e ing ma ix
S(E)
as
Q(E) = −iℏS(E)†∂ES(E),
whose diagonal elemen s
Qαα(E)
gi e he ime delay in each channel, while he ace
Q(E)
cha ac e izes he o al empo al esponse o he sys em o a amily o sca e ing s a es.
In pa allel, K ein, F iedel, Lloyd and o he s es ablished he celeb a ed K einF iedelLloyd
o mula in sca e ing spec al heo y, ela ing he ene gy de i a i e o sca e ing phase shi s o
he change in DOS
∆ρ(E)
. In ni e-box no maliza ion, his o mula can be w i en simply as
∆ρ(E) = 1
2πi S(E)†∂ES(E)=1
π∂Eδ o (E),
whe e
δ o (E)
is he o al sca e ing phase shi .
Mode n spec al shi unc ion heo y u he shows ha he de i a i e o he spec al shi
unc ion
ξ(E)
is p ecisely he DOS die ence induced by he in e ac ion, and
ξ(E)
can i sel be
cha ac e ized by he de e minan phase o
S(E)
.
These esul s hin a a p o ound ac : he ime delay and he densi y o s a es in sca e ing
sys ems a e no wo independen quan i ies, bu a he die en p ojec ions o he same spec al
s uc u e. Building on his ounda ion, he p esen wo k p oposes a unied ime iden i y ha
s ic ly equa es he ime ow a e
κ(E)
wi h he DOS
ρ(E)
.
On he o he hand, gene al ela i i y in e p e s g a i a ional edshi as he spa ial a ia ion
o he
g00
componen in s a ic space ime me ics. In he weak-eld limi , he s a ic g a i a ional
po en ial
Φ(x)
and he me ic sa is y
g00(x)≃ −1 + 2Φ(x)/c2,
and he p ope ime o s a iona y obse e s sa ises
dτ=√−g00d
, so ha o wo posi ions
x1,x2
a die en po en ial alues we ha e
∆ν
ν≃Φ(x2)−Φ(x1)
c2,
a ela ion ha has been p ecisely e ied in he PoundRebka
γ
- ay expe imen , he Ha ele
Kea ing a ound- he-wo ld a omic clock igh , and ecen op ical la ice clock heigh compa ison
expe imen s.
Howe e , he geome ic desc ip ion o g a i a ional edshi s ill lea es he o igin o  ime
ow in he con inuous eld o he me ic enso , lacking a di ec connec ion o he s uc u e o
quan um s a es. Quan um cellula au oma on and disc e e quan um walk s udies ha e shown
ha unde app op ia e symme y condi ions, he Di ac and Weyl equa ions can eme ge as
he con inuum limi o local uni a y disc e e dynamics, hus p o iding igo ous models o he
disc e e pic u e o  he uni e se as quan um compu a ion.
2
In such a disc e e on ology, he  ime s ep is a disc e e pa ame e o he local upda e ule,
a he han an ex e nal con inuous a iable. The na u al ques ion is: how can we econs uc
con inuous p ope ime wi hin his disc e e amewo k, and ela e i o g a i a ional edshi ?
The basic iewpoin o his pape is:
ime is an alias o densi y o s a es
. Mo e p ecisely,
h ough he EWS ime-delay ope a o and he KFL ace o mula, we p o e ha unde qui e
gene al sca e ing se ings, he ollowing unied ime iden i y holds:
κ(E) = 1
2πℏ Q(E)=∆ρ(E)
whe e
κ(E)
is in e p e ed as he  ime ow densi y pe uni ene gy in e al, i.e., he eec i e
ime delay expe ienced by he sys em pe uni ene gy in he neighbo hood o ene gy
E
. We
u he show ha , unde app op ia e coa se-g aining and no maliza ion choices, he spa ial
a ia ion o his quan i y can es a e g a i a ional edshi in gene al ela i i y.
2 Model & Assump ions
2.1 Sca e ing Sys em and Densi y o S a es
Le
H0
be he ee Hamil onian and
H=H0+V
he in e ac ing Hamil onian con aining a local
po en ial
V
. We make he ollowing assump ions:
1.
H0, H
a e sel -adjoin ope a o s on he same Hilbe space
H
, and
V
is a sucien ly apidly
decaying bounded o ela i ely
H0
-bounded pe u ba ion, so ha he wa e ope a o s
Ω±= s-lim
→±∞ eiH /ℏe−iH0 /ℏ
exis and a e comple e.
2. The sca e ing ope a o
S= (Ω+)†Ω−
decomposes in he ene gy ep esen a ion in o a
amily o uni a y ma ices
S(E)
, ac ing on he channel space
HE
on each ene gy shell.
3. The spec al measu es o bo h ee and in e ac ing sys ems a e absolu ely con inuous plus
a ni e numbe o bound s a es, so ha he well-dened s a e-coun ing unc ions exis :
N0(E)=#{λn(H0)≤E}, N(E)=#{λn(H)≤E},
whose de i a i es gi e he densi ies o s a es
ρ0(E), ρ(E)
. The change in DOS induced by
he in e ac ion is
∆ρ(E) = ρ(E)−ρ0(E).
The spec al shi unc ion
ξ(E)
is dened by he K ein ace o mula, ia
 (H)− (H0)=Z+∞
−∞
′(E)ξ(E) dE
o sucien ly smoo h unc ions
. An app oxima ion o he indica o unc ion yields
ξ(E) =
N(E)−N0(E)
, whose de i a i e sa ises
ξ′(E) = ∆ρ(E).
On he o he hand, he de e minan o he sca e ing ma ix and he spec al shi unc ion
a e ela ed by
de S(E) = exp−2πiξ(E),
so ha
1
2πi∂Eln de S(E) = ξ′(E) = ∆ρ(E).
3
2.2 Wigne Smi h Time-Delay Ope a o
In he ene gy ep esen a ion, he Wigne Smi h g oup delay ma ix is dened as
Q(E) = −iℏS(E)†∂ES(E),
whose diagonal elemen
Qαα(E)
gi es he ime delay o channel
α
, and he ace
Q(E) = −iℏ S(E)†∂ES(E)
cha ac e izes he o al g oup delay o all channels. The sel -adjoin ness and obse abili y o
Q(E)
can be e ied by die en ia ing he uni a i y o he sca e ing ma ix
S(E)†S(E) = I
.
Taking he ace and compa ing wi h he spec al shi ela ion om he p e ious sec ion,
we immedia ely ob ain
∆ρ(E) = 1
2πi∂Eln de S(E) = 1
2πi
1
de S(E)∂Ede S(E) = 1
2πi S†∂ES=1
2πℏ Q(E).
This is he spec al- heo e ic ounda ion o he unied ime iden i y in he p esen wo k.
2.3 QCA Con inuum Limi and Sca e ing
Quan um cellula au oma a a e a class o s ic ly local, disc e e- ime, uni a y e olu ions
U
de-
ned on a la ice
Λ
, which in he single-pa icle subspace a e equi alen o disc e e- ime quan um
walks. A conside able body o wo k has shown ha , in he limi o sucien ly small wa e ec-
o and mass, app op ia ely cons uc ed QCA e olu ions can app oxima e Di ac equa ions in
a ious dimensions and hei gene aliza ions in cu ed space ime.
In he QCA amewo k, ime is essen ially he disc e e s ep numbe
n∈Z
, and ene gy
is gi en by he quasi-ene gy spec um
e−iε(k)
o he single-s ep e olu ion ope a o
U
. Local
de ec s o ex e nal po en ials can be implemen ed by modi ying he local upda e ope a o in a
ni e egion, co esponding o Floque sca e ing scena ios whe e he sca e ing ma ix
S(ε)
is
closely ela ed o con inuous- ime sca e ing. Th ough he con inuum-limi mapping
ε7→ E
,
he Floque EWS ime delay in QCA can be unied wi h
Q(E)
in con inuous sca e ing heo y.
We make he ollowing model assump ions:
1. The uni e se can be desc ibed mic oscopically by a class o Di ac- ype QCA models, whose
single-pa icle subspace app oxima es he s anda d Di ac equa ion in he long-wa eleng h
limi .
2. Mac oscopic g a i a ional elds co espond o slow spa ial a ia ions o he QCA back-
g ound cell upda e ules, i.e., g adual changes in local p opaga ion eloci y and phase
esponse. This can be likened o he cons uc ion o quan um walks in cu ed space ime.
3. In his amewo k, all mac oscopic ime obse a ions can ul ima ely be educed o phase
die ences and g oup delay measu emen s in some kind o sca e ing o in e e ence ex-
pe imen s, and he e o e can be uni o mly desc ibed by he EWSKFL s uc u e.
3 Main Resul s (Theo ems and Alignmen s)
This sec ion p esen s he unied ime iden i y and i s main consequences o g a i a ional ed-
shi , he he modynamic a ow o ime, and he QCA con inuum limi .
Deni ion 3.1
(Time Densi y)
.
Fo a sca e ing sys em sa is ying he abo e assump ions, dene
he ime densi y a ene gy
E
as
κ(E)≡1
2πℏ Q(E),
whe e
Q(E) = −iℏS(E)†∂ES(E)
is he Wigne Smi h g oup delay ope a o .
4
Theo em 3.2
(Unied Time Iden i y)
.
Unde he assump ions ha he K einF iedelLloyd
o mula and spec al shi unc ion exis and a e die en iable, he ime densi y and DOS change
sa is y
κ(E)=∆ρ(E) = ρ(E)−ρ0(E).
In o he wo ds,
he ime densi y equals he ela i e DOS in oduced by he in e ac ion
.
This heo em unies he ollowing h ee objec s as die en exp essions o he same unc ion:
1. The ene gy de i a i e o he o al sca e ing phase
φ(E) = a g de S(E)
:
κ(E) = 1
2π∂Eφ(E).
2. The DOS change
∆ρ(E)
induced by he in e ac ion.
3. The no malized ace o he Wigne Smi h delay ma ix
(2πℏ)−1 Q(E)
.
See Appendix A o he de ailed p oo .
P oposi ion 3.3
(Time Flow Ra e as Func ion o DOS)
.
Suppose a mac oscopic clock is d i en
by a amily o sca e ing s a es concen a ed nea ene gy
E0
, and i s ou pu pe iod
T
is de e mined
by he ene gy dependence o he o al phase
φ(E)
. Then in he na ow-band app oxima ion, he
p ope ime inc emen o he clock sa ises
dτ
d ≃κ e (E0)
κ(E0)=∆ρ e (E0)
∆ρ(E0),
whe e  e  deno es a xed e e ence posi ion (e.g., he asymp o ically a egion a inni y).
This shows ha wi hin a gi en ene gy window,
he ime ow a e is in e sely p opo ional
o he local DOS
.
Theo em 3.4
(G a i a ional Redshi in he Weak-Field Limi )
.
Conside a weak g a i a ional
eld wi h s a ic po en ial
Φ(x)
sa is ying
|Φ|/c2≪1
. Suppose ha he Hamil onian o some
quan um eld in a local ine ial ame is app oxima ely
H(x,p)≃pm2c4+c2p2+mΦ(x),
and he local DOS
ρ(E, x)
is gi en by he semiclassical Weyl o mula. Then o ligh ly bound
s a es in he non ela i is ic limi
E≃mc2
, we ha e
ρ(E, x)≃ρ∞(E)1−αΦ(x)
c2,
whe e
ρ∞(E)
is he DOS a inni y, and he cons an
α
depends on he dimension and he specic
model (in he h ee-dimensional non ela i is ic gas model
α= 3/2
). Adop ing he ime-densi y
no maliza ion
dτ(x)
d =κ∞(E)
κ(E, x)=ρ∞(E)
ρ(E, x),
a  s -o de expansion yields
dτ(x)
d ≃1 + αΦ(x)
c2.
By app op ia ely choosing an in o ma ion weigh  o dening he DOS (e.g., conside ing
ene gy-weigh ed DOS o op ical mode DOS), one can se
α= 1
, he eby ob aining a ime dila ion
ac o consis en wi h he weak-eld limi o gene al ela i i y:
dτ(x)
d ≃1 + Φ(x)
c2,
5

and eco e ing he g a i a ional edshi o mula
∆ν
ν≃Φ(x2)−Φ(x1)
c2.
See Appendix B o he de ailed de i a ion.
P oposi ion 3.5
(Time Densi y and KMS The mal Time)
.
Le
(A, α )
be a
C∗
dynamical
sys em, and
ωβ
a KMS s a e a empe a u e
T= 1/(kBβ)
. The ene gy-weigh ed DOS is
ρβ(E) = Z−1ρ(E)e−βE, Z =Zρ(E)e−βEdE.
Dene he a e age ime densi y in he he mal s a e as
¯κβ=Zκ(E)ρβ(E) dE=Z∆ρ(E)ρβ(E) dE.
Then
¯κβ
inc eases mono onically wi h ene gy densi y, is one- o-one co ela ed wi h he imagi-
na y ime pe iod
β
o he KMS ow, he eby p o iding a DOS- ime ealiza ion o he he mody-
namic a ow o ime compa ible wi h he he mal ime hypo hesis and he Un uhKMS s uc u e.
P oposi ion 3.6
(Time Densi y in QCA Con inuum Limi )
.
In a Di ac- ype QCA model, he
single-s ep e olu ion ope a o
U
has quasi-ene gy spec um
ε(k)
, and sca e ing de ec s in oduce
he Floque sca e ing ma ix
S(ε)
. Dene he Floque EWS ope a o
QF(ε) = −iUe (ε)†∂εUe (ε).
In he con inuum limi
ε→E
, i s ace con e ges o he con inuous sca e ing ime delay,
sa is ying
κ(E) = 1
2π∂Eφ(E) = 1
2πℏ Q(E) = lim
ε→E
1
2π QF(ε),
so ha he  ime densi y o disc e e s eps in QCA is consis en wi h he DOS deni ion in
con inuous heo y.
4 P oo s
This sec ion p o ides p oo ou lines o he unied ime iden i y and i s main co olla ies; com-
ple e echnical de ails a e placed in he appendices.
4.1 P oo o he Unied Time Iden i y (Theo em 3.2)
The p oo p oceeds in wo s eps.
S ep one: K einF iedelLloyd o mula and DOS.
Conside he one-dimensional case. Place he sys em in a ni e box o leng h
L
wi h ap-
p op ia e bounda y condi ions. Fo he ee sys em, he momen um quan iza ion condi ion is
knL=nπ
, and he numbe o s a es is
N0(k)≃L
πk, ρ0(k) = dN0
dk=L
π.
A e adding he po en ial
V(x)
, sca e ing bounda y condi ions gi e he phase-shi -co ec ed
quan iza ion condi ion
knL+δ(kn) = nπ,
so ha
N(k) = L
πk+1
πδ(k),∆ρ(k) = ρ(k)−ρ0(k) = 1
π∂kδ(k).
6
Using
E=ℏ2k2/(2m)
and he chain ule, we con e o he ene gy ep esen a ion o ob ain
∆ρ(E) = 1
π∂Eδ(E).
Fo mul i-channel and highe -dimensional cases, his can be gene alized ia pa ial-wa e
decomposi ion and spec al shi unc ion heo y o
∆ρ(E) = 1
2πi S†∂ES=1
2π∂Eφ(E),
whe e
φ(E) = a g de S(E)
, i.e., he K einF iedelLloyd o mula.
S ep wo: EWS ime delay and he ace o
Q(E)
.
The EWS ope a o is dened as
Q(E) = −iℏS†(E)∂ES(E),
whose ace is
Q(E) = −iℏ S†∂ES.
Compa ing his wi h he DOS exp ession we ob ain
∆ρ(E) = 1
2πi (S†∂ES) = 1
2πℏ Q(E).
This is p ecisely he unied ime iden i y in he ime densi y deni ion. Sel -adjoin ness
ollows di ec ly om
S†S=I⇒(∂ES†)S+S†(∂ES)=0,
which yields
Q†(E) = +iℏ(∂ES†)S=−iℏS†(∂ES) = Q(E),
so
Q(E)
is an obse able.
4.2 Time Flow Ra e and DOS (P oposi ion 3.3)
Conside a na ow-band wa epacke whose ene gy dis ibu ion
|a(E)|2
is concen a ed nea
E0
,
sa is ying
Z|a(E)|2dE= 1,⟨E⟩=E0,∆E≪E0.
The sca e ed s a e in he a egion can be w i en as
ψou ( ) = Za(E)e−iE /ℏS(E)|E⟩dE.
Expand
S(E)≃eiφ(E)˜
S(E)
, whe e
˜
S(E)
has a slowly a ying ma ix s uc u e, and he o al
phase is
φ(E) = a g de S(E)
. By compa ing he ansla ion o he ee p opaga ion s a e, we
can dene he g oup delay o a ce ain a poin as
τ(E0) = ∂Eφ(E)E0.
The e e ence s a e (e.g., dis an a egion) has a co esponding delay
τ e (E0)
de e mined
by he sca e ing ma ix
S e (E)
, wi h ime densi y
κ(E) = 1
2π∂Eφ(E), κ e (E) = 1
2π∂Eφ e (E).
Wi hin he same ex e nal coo dina e ime
, he in e nal subjec i e ime inc emen
dτ
is
p opo ional o he o al phase inc emen scanned by he wa epacke , yielding
dτ
d ∝∂Eφ e
∂Eφ=κ e (E)
κ(E)=∆ρ e (E)
∆ρ(E).
Unde app op ia e no maliza ion, he p opo ionali y cons an can be aken as 1, and he
p oposi ion ollows.
7
4.3 DOS and G a i a ional Redshi (Theo em 3.4 Ou line)
G a i a ional edshi essen ially a ises om Killing ene gy conse a ion in s a ic space ime and
he spa ial a ia ion o p ope ime o s a iona y obse e s. We wish o show ha in he
weak-eld limi , his eec can be es a ed as a spa ial a ia ion o he local DOS.
In he New onian limi , he non ela i is ic pa icle Hamil onian is
H(x,p) = p2
2m+mΦ(x),
and he DOS is gi en by he phase space olume:
ρ(E, x)≃1
(2πℏ)3ZδE−H(x,p)d3p.
The in eg al can be compu ed explici ly o gi e
ρ(E, x)∝E−mΦ(x)1/2.
Fo
E≃mc2
and
|Φ|/c2≪1
, expanding yields
ρ(E, x)≃ρ∞(E)1−1
2
Φ(x)
c2.
Fo h ee-dimensional ela i is ic models o sys ems wi h mul iple deg ees o eedom, he
exponen changes, gi ing he gene al o m
ρ(E, x)≃ρ∞(E)1−αΦ(x)
c2,
whe e
α
depends on he specic sys em. Theo em 3.4 indica es ha by in oducing an app o-
p ia e in o ma ion weigh , dening he local ime ow a e as
dτ(x)
d =ρin o,∞(E)
ρin o(E, x),
and aking
ρin o(E, x) = ρ(E, x)β
o some unc ion class, one can adjus he exponen
β
so ha he  s -o de expansion coecien
equals 1. Physically, his is he eedom o choose a  ime scale among die en mic oscopic
models, analogous o adop ing die en coo dina e scales in gene ally co a ian heo ies. Ma ch-
ing his deni ion o classical g a i a ional edshi expe imen s xes
β
and yields
dτ(x)
d ≃1 + Φ(x)
c2.
See Appendix B o he comple e calcula ion and no maliza ion p ocedu e.
4.4 Time Densi y and KMS The mal Time (P oposi ion 3.5)
Fo a many-body sys em wi h a con inuous spec um, he KMS condi ion is
ωβAα (B)=ωβα −iβ(B)A,
whe e
α
is he Heisenbe g e olu ion. Fo ope a o s
AEE′
in he ene gy ep esen a ion, he
ealiza ion o he KMS condi ion elies on he Bol zmann weigh
e−βE
weigh ing he DOS.
8
Dene he a e age ime densi y unde DOS weigh ing:
¯κβ=Zκ(E)ρβ(E) dE=Z∆ρ(E)ρβ(E) dE.
Then
¯κβ
is he a e age ime delay densi y o he sys em a empe a u e
T
. As he ene gy
densi y inc eases, he spec al weigh shi s o highe ene gies,
¯κβ
inc eases mono onically, co e-
sponding o s onge in o ma ion conges ion and slowe mac oscopic ime. This is compa ible
wi h he he mal ime hypo hesis iew ha  ime is he modula ow o he s a e on he algeb a:
when local ene gy densi ies die , he a io be ween he modula ow pa ame e and geome ic
ime changes, co esponding o die en g a i a ional ime dila ions and empe a u es.
4.5 Unied Time Iden i y in QCA Con inuum Limi (P oposi ion 3.6)
In a Di ac- ype QCA, he single-s ep e olu ion
U
can be diagonalized in he momen um ep e-
sen a ion as
U|k, σ⟩= e−iεσ(k)|k, σ⟩,
whe e
σ
labels in e nal deg ees o eedom. In he p esence o de ec s, a Floque sca e ing
ma ix
S(ε)
can be cons uc ed in he long- ime limi , wi h EWS ope a o
QF(ε) = −iS(ε)†∂εS(ε),
comple ely pa allel o he con inuous- ime case. Combining he QCA con inuum limi
ε→E
wi h he p e ious sca e ingDOS heo y, we ob ain
lim
ε→E
1
2π QF(ε)=∆ρ(E),
i.e., he ime densi y deni ion is consis en be ween disc e e and con inuous desc ip ions. Re-
la ed cons uc ions can be ound in s udies o Di ac QCA and quan um walks in cu ed space-
ime.
5 Model Apply
This sec ion discusses applica ions o he unied ime iden i y in se e al physical scena ios.
5.1 Wa e-Chao ic Ca i ies and Mic owa e Sca e ing
In wa e-chao ic ca i ies and mul i-po elec omagne ic sca e ing s uc u es, he Wigne Smi h
ma ix
Q(E)
has long been used o cha ac e ize mode-a e aged g oup delay and dwell ime dis-
ibu ions, and i s ela ion o DOS in he andom ma ix heo y amewo k has been ex ensi ely
e ied.
In hese sys ems, expe imen al es s o he unied ime iden i y can be ealized in he ol-
lowing ways:
1. Measu e he mul i-po sca e ing ma ix
S(ω)
, nume ically die en ia e o ob ain
Q(ω) =
−iS†∂ωS
and
Q(ω)
.
2. Independen ly ob ain DOS
ρ(ω)
and he ela i e
∆ρ(ω)
wi h espec o he emp y ca i y
h ough eigen equency s a is ics o G een's unc ion measu emen s.
The unied ime iden i y p edic s
1
2π Q(ω) = ∆ρ(ω).
De ia ions om his ela ion can be a ibu ed o loss o non-conse a i e eec s, equi ing
co ec ion using a gene alized (non-uni a y) Wigne Smi h ma ix.
9
so
dτ(x)
d ≃1 + βαΦ(x)
c2.
Choosing
β= 1/α
eco e s he GR linea coecien 1. Physically, his co esponds o
choosing a DOS deni ion p opo ional o  he numbe o dis inguishable mic oscopic s a es pe
uni ene gy in a many-body sys em, a he han simple single-pa icle phase space olume. I s
specic o m equi es u he de i a ion om ela i is ic eld heo y and many-body co ela ion
unc ions, which is beyond he scope o his pape .
C Time Densi y, KMS S a es, and The mal Time
This appendix b iey explains he ela ionship be ween ime densi y and KMS modula ow.
Le
(A, α )
be a one-pa ame e g oup on a
C∗
algeb a, whe e
α
is gene a ed by a Hamil onian
H
. The KMS condi ion is: o any
A, B ∈ A
, he e exis s an analy ic unc ion
FA,B(z)
sa is ying
FA,B( ) = ωβ(Aα (B)), FA,B( −iβ) = ωβ(α (B)A).
In he GNS ep esen a ion, he KMS s a e co esponds o a special ec o s a e on Hilbe
space, whose modula ow is gi en by Tomi aTakesaki heo y. The a io o he modula ow
pa ame e
s
o physical ime
can be unde s ood as  empe a u e o  ime scale.
The unied ime iden i y indica es ha ime densi y
κ(E)
is join ly de e mined by DOS and
EWS delay. Fo a gi en empe a u e
T
, he sys em is in ene gy p obabili y dis ibu ion
Pβ(E) = Z−1ρ(E)e−βE,
and he a e age ime scale o he KMS ow can be ela ed o
¯κβ=Zκ(E)Pβ(E) dE.
The he mal ime hypo hesis holds ha physical ime is p ecisely he na u al pa ame e o
he modula ow in a gi en s a e, and empe a u e is he p opo ionali y coecien be ween
he modula ow and geome ic ime. In his amewo k,
¯κβ
p o ides a way o di ec ly dene
his p opo ionali y coecien using spec al quan i ies.
D Time Densi y and Sca e ing in Di ac QCA
Di ac QCA models p o ide disc e e, local, uni a y mic oscopic upda e ules, whose single-
pa icle subspace app oxima es he Di ac equa ion in he con inuum limi .
In he one-dimensional case, he single-s ep e olu ion can be w i en as
U=X
x|x+ 1⟩⟨x|⊗C++|x−1⟩⟨x|⊗C−,
whe e
C±
a e coin ope a o s ac ing on in e nal deg ees o eedom. De ec s can be implemen ed
by modi ying
C±
o adding local phases in a ni e egion. In he quasi-ene gy ep esen a ion, he
sca e ing ma ix
S(ε)
is simila o
S(E)
in con inuous- ime sca e ing, and he EWS ope a o
is dened as
QF(ε) = −iS(ε)†∂εS(ε).
The unied ime iden i y in he Floque scena io becomes
κ(ε) = 1
2π QF(ε)=∆ρF(ε),
whe e
∆ρF(ε)
is he quasi-ene gy DOS change induced by he in e ac ion. The exis ence o he
Di ac con inuum limi
ε→E
ensu es ha he abo e ela ion can be seamlessly mapped o he
con inuous ene gy ep esen a ion, so ha he b idge be ween disc e e ime s eps in QCA and
con inuous ime densi y is join ly buil by DOS and EWS delay.
16

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18