Quantum--Classical Bridge on Time Scale:\\ Phase, Time Delay, Redshift, and Gravity--Entropy Geometry

Quan umClassical B idge on Time Scale:
Phase, Time Delay, Redshi , and G a i yEn opy
Geome y
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 20, 2025
Abs ac
Unde he unied  ime scale pe spec i e, his pape sys ema ically cons uc s a
se o equi alence ela ions among quan um phase, p ope ime, sca e ing g oup de-
lay, cosmological edshi , and bounda y en opy e olu ion, o ganizing hem in o an
axioma izable quan umclassical b idge amewo k. On he geome ic end, p ope
ime
dτ=p−gµνdxµdxν
and gene alized en opy
Sgen
on causal bounda ies a e
aken as undamen al objec s; on he quan um end, pa h in eg al phase
ϕ=−S/ℏ
,
o al phase
Φ(ω)
o sca e ing ma ix
S(ω)
, and Wigne Smi h g oup delay ope -
a o
Q(ω) = −iS(ω)†∂ωS(ω)
a e aken as undamen al objec s. We p opose and
adop he unied scale iden i y
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
whe e
φ(ω)
is no malized o al phase,
ρ el
is ela i e s a e densi y. We p o e:
unde semiclassical limi and app op ia e egula i y assump ions
1. When single-pa icle wa e packe p opaga es along classical geodesic, i s phase
ϕ
is equi alen o a linea unc ion o p ope ime accumula ed along ha
wo ldline
ϕ=−mc2Rdτ/ℏ
, hus p ope ime scale can be iewed as geome ic
pa ame e o phase;
2. In s a ic o asymp o ically a g a i a ional elds, g a i a ional ime delay o
pho ons o ma e wa es equals de i a i e o sca e ing phase wi h espec o
equency, i.e., equals ace o Wigne Smi h g oup delay, hus g a i a ional
ime dila ion can be in e p e ed as cu a u e o phase equency geome y;
3. In FRW cosmological backg ound, edshi
1 + z=a( 0)/a( e)
can be equi -
alen ly w i en as a io o phase equency
(dϕ/d )
on same pho on wo ldline
a emission and de ec ion e en s, hus cosmological edshi becomes phase
exp ession o cosmic ime scale shea ;
4. In local causal diamonds, aking ex emali y and mono onici y o gene alized
en opy
Sgen =A/(4Gℏ) + Sou
(along null gene a o pa ame e
λ
) as axioms,
Eins ein equa ions wi h cosmological cons an can be de i ed in semiclassical
holog aphic window, iewing g a i a ional geome y as eec i e equa ions o
how en opy o ganizes along ime scale on causal bounda ies.
1
This yields a unied  imephaseen opygeome y co espondence diag am.
Appendices p o ide: de i a ion o scale iden i y be ween Wigne Smi h g oup delay
and spec al shi s a e densi y; semiclassical p oo o phasep ope ime equi a-
lence unde wo ldline pa h in eg al; enemen o edshi phase exp ession in FRW
cosmology; and de ailed p oo ou line o de i ing Eins ein equa ions om gene al-
ized en opy ex emali y and quan um ene gy condi ions on local causal diamonds.
Keywo ds:
Time Scale; Wigne Smi h G oup Delay; P ope Time; Cosmological Red-
shi ; Gene alized En opy; En opic In e p e a ion o Eins ein Equa ions
MSC (2020):
83C45, 81T20, 81U40, 83C57

1 In oduc ion
Time plays die en oles in classical physics and quan um heo y: in gene al ela i i y,
ime and space oge he o m ou -dimensional space ime mani old, whose p ope ime
scale
dτ
is de e mined by me ic
gµν
; in quan um mechanics and quan um eld heo y,
ime is mo e mani es ed h ough phase ac o
exp(−iE /ℏ)
and uni a y e olu ion ope a o
U( )
. Wigne Smi h g oup delay ope a o
Q(ω)
in oduced in sca e ing heo y p o ides
an ope a ional scale o  ime as phase de i a i e wi h espec o equency; cosmological
edshi eec s mac oscopic shea o  ime hy hm h ough changes in equency o
wa eleng h due o cosmic scale ac o e olu ion.
On he o he hand, black hole he modynamics, Jacobson- ype en opic in e p e-
a ion, and holog aphicen anglemen geome y esea ch show: g a i a ional geome y
can be iewed as mac oscopic equa ions o ce ain en opyin o ma ion o ganiza ion,
especially on causal bounda ies and local ho izons, ex emali y and mono onici y o gen-
e alized en opy
Sgen
impose manda o y cons ain s on space ime cu a u e.
Behind hese seemingly dispa a e phenomena is a common s uc u e: hey all use
 ime scale as b idge, connec ing quan um phase, classical clocks, sca e ing delay, ed-
shi , and en opy ow. The goal o his pape is o sys ema ically, clea ly, and ax-
ioma ically cha ac e ize his s uc u e, p o iding igo ous equi alence o co espondence
ela ions.
This pape de elops a ound he ollowing main ques ions:
1. How o iew quan um phase, g oup delay, p ope ime, and cosmological edshi
as die en c oss-sec ions o he same  ime geome y unde unied scale?
2. Unde wha assump ions can we equa e how gene alized en opy o ganizes along
ime on causal bounda ies wi h mac oscopic g a i a ional equa ions (Eins ein equa-
ions)?
3. How do hese equi alence ela ions na u ally connec quan um wi h classical, mi-
c oscopic wi h mac oscopic in semiclassical limi ?
To his end, Sec ion 2 gi es no a ions and axioma ic scale iden i y; Sec ions 35 succes-
si ely cons uc p ecise co espondences o phasep ope ime,  ime delayg a i a ional
ime dila ion,  edshi  ime scale shea ; Sec ion 6 gi es en opic geome y heo em o
gene alized en opy e olu iong a i a ional equa ions in causal diamond amewo k.
Sec ion 7 summa izes hese b idges as a geome ic pic u e. Appendices p o ide de ailed
p oo s o main echnical esul s.

2
2 No a ions and Basic S uc u es
2.1 Geome ic End: Space ime, P ope Time, and Causal Dia-
monds
Le
(M, gµν)
be geodesically comple e Lo en zian space ime mani old, me ic signa u e
(−+ ++)
. P ope ime o imelike cu e
γ(λ)
is
dτ= −gµν
dxµ
dλ
dxν
dλdλ.
In local discussions, choose poin
p∈M
and small pa ame e
≪Lcu
, dene small
causal diamond a ha poin
Dp, := J+(p−)∩J−(p+),
whe e
p±
a e poin s ose by p ope ime
±
along some chosen imelike di ec ion.
Bounda y o
Dp,
is gene a ed by wo amilies o null geodesics, o ming local causal
bounda y.
2.2 Quan um and Sca e ing End: S-Ma ix and Wigne Smi h
G oup Delay
Le
H0
be ee Hamil onian,
H
be Hamil onian wi h in e ac ion e m. Unde s anda d
sca e ing assump ions, wa e ope a o s
Ω±= s- lim
→±∞ eiH e−iH0
exis and a e comple e, sca e ing ope a o dened as
S:= (Ω+)†Ω−.
In ene gy o equency ep esen a ion, absolu ely con inuous spec um
ω∈I⊂R
gi es o each
ω
ni e-dimensional channel space
Hω≃CN(ω)
, wi h uni a y ma ix
S(ω)
.
Dene o al sca e ing phase
Φ(ω) := a g de S(ω).
Deni ion 2.1
(Wigne Smi h G oup Delay Ope a o (Deni ion 2.1))
.
Unde equency
die en iabili y assump ion, dene
Q(ω) := −i S(ω)†∂ωS(ω),
called Wigne Smi h g oup delay ope a o .
Q(ω)
is sel -adjoin ma ix, whose eigen-
alues deno ed
τj(ω)
can be in e p e ed as g oup delays o espec i e channels. T ace
gi es o al g oup delay
T Q(ω) = ∂ωΦ(ω).
3
2.3 Spec al Shi Func ion and Rela i e S a e Densi y
Unde sui able aceable pe u ba ion assump ion, le
ξ(ω)
be K en spec al shi unc-
ion,
ρ el(ω)
be ela i e s a e densi y. Classical esul gi es
Φ(ω) = −2π ξ(ω), ρ el(ω) = −∂ωξ(ω).
Thus
∂ωΦ(ω) = 2π ρ el(ω).
2.4 Unied Scale Iden i y
Combining Sec ions 2.22.3, we ob ain he ollowing scale iden i y.
Axiom 2.2
(Time Scale Unica ion (Axiom 2.2))
.
Fo all conside ed sca e ing congu-
a ions and ene gy windows, he e exis well-dened gene alized phase
φ(ω)
and ela i e
s a e densi y
ρ el(ω)
such ha
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
whe e
is ace on channel space,
Q(ω) := Q(ω)
.
This iden i y unies h ee ypes o objec s on same ime scale:
1. Phase de i a i e
φ′(ω)
: cu a u e o phase equency geome y;
2. Rela i e s a e densi y
ρ el(ω)
: co ec ion o spec als a e densi y;
3. G oup delay ace
Q(ω)
: a i al ime ose ela i e o ee p opaga ion.
All ime scale- ela ed quan i ies in wha ollows will be w i en back o his iden i y
as much as possible.
2.5 Cosmological Time and Redshi
Conside spa ially homogeneous iso opic FRW me ic ( aking
k= 0
empo a ily)
ds2=−d 2+a( )2dx2.
Fo pho on p opaga ing along como ing coo dina e
x( )
, null geodesic condi ion
ds2=
0
gi es
dx
d =±1
a( )ˆ
n.
Cosmological edshi dened as
1 + z:= λ0
λe
=νe
ν0
=a( 0)
a( e),
whe e subsc ip s
e, 0
deno e emission and obse a ion e en s espec i ely.
4
2.6 Bounda y Gene alized En opy and Time Pa ame e
In quan um eld heo y wi h g a i y, conside bounda y
Σ
o some causal egion and i s
ex e nal (o in e nal) quan um eld deg ees o eedom, dene gene alized en opy
Sgen(Σ) := A ea(Σ)
4Gℏ+Sou (Σ),
whe e
Sou
is on Neumann en opy ela i e o
Σ
. When de o ming
Σ
along some null
gene a o amily
γ(λ)
, aking ane pa ame e
λ
as bounda y ime, s udy ex emali y
and mono onici y p ope ies o
Sgen(λ)
, gi ing how en opy o ganizes wi h ime.
Axiom 2.3
(Bounda y En opy Time E olu ion Axiom (Axiom 2.3))
.
Unde app op ia e
ene gy condi ions and semiclassical assump ions, o small causal diamond
Dp,
a any
poin
p
, he bounda y admi s a amily o local cu s
{Σλ}
such ha
1. Unde app op ia e cons ain s (xed local ene gy o eec i e olume),
Sgen(λ)
akes
ex emum a
λ= 0
;
2. Fo ex apola ion along any null di ec ion,
Sgen(λ)
sa ises app op ia e mono onic-
i y o con exi y condi ions (such as QNEC/QFC ype inequali ies) unde physical
e olu ion.
We will use his axiom and scale iden i y as basis o gi e en opic geome ic in e p e-
a ion o g a i a ional equa ions.

3 Equi alence o Phase and P ope Time
This sec ion discusses semiclassical p opaga ion o single pa icle o na ow wa e packe in
cu ed space ime, explaining ha essence o quan um phase is p ope ime accumula ed
along wo ldline, he eby connec ing quan um and classical on ime scale.
3.1 Wo ldline Ac ion and Pa h In eg al
Fo poin pa icle o mass
m
, classical wo ldline ac ion can be aken as
S[γ] = −mc2Zγ
dτ=−mc2Z −gµν
dxµ
dλ
dxν
dλdλ.
Quan um ampli ude in wo ldline pa h in eg al amewo k is o mally w i en as
A(x , xi)≃Zγ:xi→x
Dγexpi
ℏS[γ].
In semiclassical limi
ℏ→0
, main con ibu ion comes om s a iona y phase ajec-
o ies, i.e., wo ldlines
γcl
sa is ying geodesic equa ion.
5

3.2 PhaseP ope Time Theo em
Theo em 3.1
(PhaseP ope Time Equi alence (Theo em 3.1))
.
Le na ow wa e packe
p opaga e in cu ed space ime, whose cen e ajec o y
γcl
is imelike geodesic o pa icle
o mass
m
. Then in semiclassical app oxima ion, phase e olu ion o wa e packe cen e
is
ϕ=−1
ℏS[γcl] = mc2
ℏZγcl
dτ,
hus ins an aneous phase equency sa ises
dϕ
dτ=mc2
ℏ.
P oo .
Cons uc na ow wa e packe ini ial s a e in some local Fe mi no mal coo -
dina e sys em, concen a ing i in phase space a classical ini ial condi ion
(xi, pi)
. Pa h
in eg al can be expanded using s a iona y phase app oxima ion in semiclassical limi . Le
γcl
be unique geodesic sa is ying gi en bounda y condi ions, hen
1. Fo each pa h
γ
, ampli ude phase is
ϕ[γ] = S[γ]/ℏ
;
2. In limi
ℏ→0
, ajec o y wi h maximum weigh is ha wi h
δS = 0
, i.e., geodesic
γcl
;
3. On ha ajec o y
S[γcl] = −mc2Rdτ
.
The e o e, o al phase o dominan s a e is
ϕ=−1
ℏS[γcl] = mc2
ℏZdτ.
Taking de i a i e wi h espec o p ope ime gi es
dϕ
dτ=mc2
ℏ,
comple ing he p oo .
□
Co olla y 3.2
(Quan um Time Scale (Co olla y 3.2))
.
Fo pa icle o mass
m
, i s phase
o a ion equency on p ope ime scale is cons an
mc2/ℏ
. The e o e, p ope ime
dτ
is
equi alen o quan um phase die ence
dϕ
, die ing only by cons an ac o :
dϕ=mc2
ℏdτ.
This shows: on geome ic end, p ope ime is in insic scale along wo ldline; on quan-
um end, phase is angula coo dina e unde ha scale. Linea equi alence be ween
hem p o ides unied backg ound o subsequen  ime delayg oup delay,  edshi 
phase hy hm.

6
4 Sca e ing Time Delay and G a i a ional Time Dila-
ion
This sec ion examines p opaga ion o ligh o ma e wa es in s a ic o asymp o ically a
g a i a ional elds, explaining ha g a i a ional ime delay equals de i a i e o sca e ing
phase wi h espec o equency, hus can be scaled by Wigne Smi h g oup delay.
4.1 S a ic Me ic and Re ac i e Index Pe spec i e
Conside s a ic me ic
ds2=−V(x)c2d 2+gij(x) dxidxj,
whe e
V(x)>0
. In oduce  ime e ac i e index
n (x) := V(x)−1/2= (−g (x))−1/2.
Fo wa e eld o xed equency
ω
, eikonal equa ion in geome ical op ics limi can
be w i en as
gµν∂µϕ ∂νϕ= 0,
whose solu ion
ϕ
gi es wa e on phase. I aking
ϕ=−ω +S(x)
, spa ial pa sa ises
op ical Fe ma p inciple simila o e ac i e index
n (x)
.
4.2 Time Delay and Phase De i a i e
Le he e be wo pa hs: one in g a i a ional eld
γg
, one in a backg ound
γ0
, co e-
sponding o p opaga ion imes
Tg(ω)
and
T0(ω)
espec i ely. Dene ime delay
∆T(ω) := Tg(ω)−T0(ω).
On he o he hand, g a i a ional eld co ec ion o o al phase
∆Φ(ω)
is ac ion die -
ence along classical pa h di ided by
ℏ
. Fo xed- equency wa e, ac ion die ence mainly
comes om ime e ac i e index co ec ion, can be w i en as
∆Φ(ω)≃ −ω∆T(ω),
whe e e ms weakly dependen on equency a e igno ed. Thus
∆T(ω) = −∂ω∆Φ(ω)
In desc ip ion wi h
ω
as spec al pa ame e , his is p ecisely deni ion o g oup delay.
Combining scale iden i y
∂ωΦ(ω) = T Q(ω),
we ob ain:
7
Theo em 4.1
(G a i a ional Time DelayG oup Delay Equi alence (Theo em 4.1))
.
Unde app op ia e geome ical op ics and semiclassical assump ions, o xed- equency
wa e p opaga ing in s a ic g a i a ional eld backg ound, mac oscopically obse able g a -
i a ional ime delay
∆T(ω)
is equi alen o de i a i e o o al sca e ing phase
Φ(ω)
wi h
espec o equency, i.e., equi alen o ace o Wigne Smi h g oup delay ope a o
Q(ω)
:
∆T(ω) = ∂ωΦ(ω) = T Q(ω).
P oo ou line.
In eikonal app oxima ion, phase unc ion
ϕ
sa ises Hamil onJacobi
equa ion, phase accumula ion on pa h p opo ional o classical ac ion. Fo xed- equency
s a e, ime di ec ion ac ion con ibu ion is
−ωT
. Compa ing congu a ions wi h/wi hou
g a i a ional eld, ac ion die ence is
−ω∆T
, hus o al phase die ence sa ises
∆Φ =
−ω∆T
. Die en ia ing wi h espec o equency gi es
∆T=−∂ω∆Φ
. Meanwhile, om
sca e ing heo y
∂ωΦ = T Q
. Iden i ying bo h gi es his equi alence ela ion.
□
This heo em shows: mac oscopic  ime dila ion o  ime delay usually unde s ood
as caused by space ime cu a u e, is comple ely equi alen o sca e ing g oup delay in
equency domain. The e o e, h ough scale iden i y, g a i a ional ime eec s can be
desc ibed by unied ime scale objec
ρ el(ω)
.

5 Cosmological Redshi as Time Scale Shea
This sec ion conside s ligh p opaga ion and edshi in FRW uni e se, explaining ha
edshi can be iewed as shea o  ime scale on same pho on wo ldline, exp essible as
a io o phase equency a die en e en s.
5.1 FRW Geome y and Redshi Fo mula
As in Sec ion 2.5, o a FRW uni e se, me ic is
ds2=−d 2+a( )2dx2.
Fo pho on p opaga ing along como ing coo dina e, using con o mal ime
η
dened
by
dη= d /a( )
, me ic w i en as
ds2=a(η)2(−dη2+ dx2).
Null geodesic sa ises
dx
dη=±ˆ
n,
i.e., pho on p opaga es in con o mal imespace as i in a Minkowski space.
Conside measu emen o equency
ν
: o obse e a es in como ing coo dina es,
ou - eloci y is
uµ= (1,0,0,0)
, pho on ou -momen um
kµ
sa ises
ν∝ −kµuµ
. Ob ain
ν∝1
a( ),
hus edshi
1 + z=νe
ν0
=a( 0)
a( e).
8
5.2 Geome ic In e p e a ion o Phase Rhy hm
Le pho on phase unc ion be
ϕ(x)
. Along null geodesic
γ
wi h pa ame e
λ
ha e
dϕ
dλ=kµ
dxµ
dλ,
whe e
kµ
is ou -wa e ec o . Fo some obse e , p ope ime is p ope ime
τ
. F e-
quency dened as
ν:= 1
2π
dϕ
dτ.
Fo como ing obse e ,
τ=
. Thus
ν( ) = 1
2π
dϕ
d .
F om p e ious esul ,
ν∝1/a( )
, he e o e can w i e
dϕ
d  = e
dϕ
d  = 0
=νe
ν0
= 1 + z.
This gi es:
P oposi ion 5.1
(Redshi Phase Rhy hm Equi alence (P oposi ion 5.1))
.
In FRW uni-
e se, a io o phase ime de i a i es a emission e en
e
and obse a ion e en
0
on same
pho on wo ldline equals cosmological edshi :
1 + z=νe
ν0
=
dϕ
d e
dϕ
d 0
.
Since
ϕ
also sa ises Hamil onJacobi equa ion o geome ical op ics, i can also be
unde s ood as angula coo dina e o ce ain cosmic ime scale. Redshi is he e o e
a io measu emen o his scale a wo epochs, i.e., mac oscopic mani es a ion o  ime
scale shea .
5.3 Rela ion o Unied Scale Iden i y
I iewing cosmological p opaga ion as ce ain eec i e sca e ing p ocess, can o mally
in oduce equi alen sca e ing ma ix
S(ω)
, whose o al phase
Φ(ω)
can be gi en by
eikonal in eg al along con o mal ime pa h. Then edshi can be iewed as die ence in
phase g adien  o
Φ(ω)
a die en cosmic ime c oss-sec ions. Th ough scale iden i y
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
can cha ac e ize edshi 's eec on s a e densi y and g oup delay in equency space,
hus b inging cosmological obse a ions in o same ime scale amewo k.

9
D De i a ion Ou line o Gene alized En opy Ex emal-
i y and Eins ein Equa ions
This appendix gi es de ailed de i a ion amewo k o Theo em 6.2, ocusing on showing
how gene alized en opy o ganiza ion along bounda y ime cons ains geome y.
D.1 Small Causal Diamond and Null Gene a o Pa ame e
In neighbo hood o poin
p
, choose imelike ec o eld
ξµ
, cons uc small causal diamond
Dp,
. I s bounda y can be gene a ed by wo amilies o null geodesics, co esponding o
u u e and pas di ec ions espec i ely. Choose one amily o u u e null gene a o s,
pa ame ized by ane pa ame e
λ
, such ha
λ= 0
co esponds o c oss-sec ion passing
h ough
p
.
Fo each
λ
, dene ans e se cu
Σλ
, whose a ea is
A(λ)
, gene alized en opy is
Sgen(λ) = A(λ)
4Gℏ+Sou (λ).
D.2 Fi s Va ia ion: Ex emali y Condi ion
Fi s a ia ion o gene alized en opy is
dSgen
dλ=1
4Gℏ
dA
dλ+dSou
dλ.
A ea a ia ion sa ises
dA
dλ=ZΣλ
θdA,
whe e
θ
is expansion. Taking
λ= 0
, Axiom 6.1 equi es unde app op ia e cons ain
dSgen
dλλ=0
= 0.
On o he hand, linea esponse o
Sou
o small de o ma ion can be ela ed o expec-
a ion alue o modula Hamil onian, i.e.,
dSou
dλ= 2πZΣλ
λ⟨Tkk⟩dA+· · · ,
aking app op ia e limi nea
λ= 0
yields e m p opo ional o
⟨Tkk⟩
. This s ep elies
on local  s law and ela i e en opy linea esponse.
Combining,  s -o de ex emali y condi ion gi es p elimina y o m o p opo ionali y
ela ion be ween
Rkk
and
⟨Tkk⟩
.
D.3 Second Va ia ion and Raychaudhu i Equa ion
Conside second a ia ion
d2Sgen
dλ2=1
4Gℏ
d2A
dλ2+d2Sou
dλ2.
Second a ia ion o a ea uses Raychaudhu i equa ion:
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dθ
dλ=−1
2θ2−σµνσµν −Rµνkµkν.
A
λ= 0
can choose ini ial condi ion such ha
θ= 0
, shea con ibu ion abso bed by
highe -o de e ms, ob aining
dθ
dλλ=0
≈ −Rkk,
hus
d2A
dλ2λ=0
=ZΣ0
dθ
dλλ=0
dA≈ − ZΣ0
Rkk dA.
On o he hand,
d2Sou /dλ2
ela ed o ene gy ow uc ua ions and quan um ene gy
condi ions (such as QNEC), la e gi es
d2Sou
dλ2≤2πZΣ0
⟨Tkk⟩dA,
o equali y unde sa u a ion condi ion.
D.4 F om Scala Rela ion o Tenso Equa ion
Subs i u ing abo e exp essions in o gene alized en opy second a ia ion o mula, com-
bined wi h mono onici y o con exi y equi emen , ob ain
−1
4GℏZΣ0
Rkk dA+ 2πZΣ0
⟨Tkk⟩dA≥0,
equali y unde sa u a ion o ex emali y condi ion. Since cu and di ec ion
kµ
can
be a bi a ily chosen locally, abo e o mula holds o all null di ec ions and small cu s,
meaning a poin
p
Rµνkµkν= 8πG ⟨Tµν⟩kµkν
holds o all null ec o s
kµ
.
Thus enso
Eµν := Rµν −8πG ⟨Tµν⟩
sa ises
Eµνkµkν= 0
o all null ec o s, yielding
Eµν = Λgµν
, whe e
Λ
is some con-
s an . Using Bianchi iden i y
∇µ(Rµν −1
2Rgµν) = 0
and ene gymomen um conse a ion
∇µTµν = 0
, can iden i y
Λ
as cosmological cons an , ob aining
Rµν −1
2Rgµν + Λgµν = 8πG ⟨Tµν⟩.
P oo comple e.

This pape , unde unied ime scale pe spec i e, o ganizes quan um phase, p ope
ime, sca e ing g oup delay, cosmological edshi , and gene alized en opy e olu ion
in o sel -consis en geome icen opicspec al amewo k, on his basis gi ing en opic
geome ic in e p e a ion o mac oscopic g a i a ional equa ions.
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