Time as Generalized Entropy Optimal Path:\\ Reconstruction of Time Arrow on Causally Consistent History Space

Time as Gene alized En opy Op imal Pa h:
Recons uc ion o Time A ow on Causally
Consis en His o y Space
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
This pape p oposes a unied amewo k ha econs uc s  ime as he op imal
pa h o a gene alized en opy unc ional. We no longe iew ime as a p ede e mined
one-dimensional pa ame e , bu dene he  ime a ow as an ex endable pa h on he
causally consis en his o y space ha makes a ce ain class o gene alized en opy
unc ional ake an ex emum (unde app op ia e cons ain s, a minimum), along
wi h i s pa ame iza ion equi alence class. Specically, on a gi en causal s uc u e
and obse able algeb a, we cons uc he ollowing h ee le els:
(1) S uc u al Le el
: View wo ld his o y as a cu e amily
γ:I→ C
on congu a ion space, whe e
C
is he s a e space sa is ying eld equa ions and
cons ain condi ions; in oduce he causally consis en subspace
Cons ⊂Pa hs(C)
,
composed o pa hs sa is ying local causali y, eco d ex endabili y, and conse a ion
laws.
(2) Func ional Le el
: On
Cons
, dene he gene alized en opy unc ional
Sgen[γ] = αS h[γ] + βSen [γ] + γD el[γ] + λB[γ],
whe e
S h
is coa se-g ained he modynamic en opy,
Sen
is en anglemen en opy
o gene alized en opy,
D el
is ela i e en opy- ype di e gence, and
B
is a bounda y
e m om bounda y geome y o ex insic cu a u e. The coecien s
α, β, γ, λ
a e
de e mined by physical scena ios and scale choices.
(3) Time Le el
: Dene he
ime a ow
as he pa h amily
γ⋆
ha makes
Sgen
sa is y he ex emum p inciple on
Cons
wi h non-nega i e local en opy p oduc ion
a e, and dene he ime scale equi alence class as all mono onic epa ame iza ions
7−→ ( ), ∈Di 1
+(I),
unde o bi s. Thus, ime is no longe an ex e nal pa ame e , bu he solu ion o a
causal consis ency + gene alized en opy op imiza ion p oblem.
A he sca e ing and spec al heo y end, we in oduce he unied scale mo he
ule
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
whe e
S(ω)
is he sca e ing ma ix,
Q(ω) = −iS(ω)†∂ωS(ω)
is he Wigne Smi h
delay ope a o ,
φ(ω) = 1
2a g de S(ω)
is he o al hal -phase, and
ρ el
is he ela-
i e s a e densi y. We p o e ha in a well-posed sca e inggeome yin o ma ion
1
se ing,
κ(ω)
can be used o conc e ize he  ime cos  o he gene alized en opy
unc ional as a spec al in eg al, he eby ob aining an obse able ime scale p oxy.
A he in o ma ion and causal end, aking ela i e en opy mono onici y and
QNEC/QFC- ype inequali ies as consis ency cons ain s, we p o e: i local ux
and en opy ow sa is y a se o na u al con exi y and posi i i y condi ions, hen
unde gi en causal s uc u e and bounda y da a, he causally consis en his o y
ha minimizes
Sgen
is unique unde mono onic epa ame iza ion, he eby econ-
s uc ing he ime a ow as he causally ex endable pa h wi h minimum gene alized
en opy cos . This amewo k p o ides a unied a ia ional in e p e a ion o he -
modynamic second law, en anglemen en opy g ow h, sca e ing g oup delay, and
cosmological edshi .
Keywo ds:
Time A ow; Gene alized En opy; Causal S uc u e; Rela i e En opy;
Sca e ing Phase; Wigne Smi h Time Delay; Scale Unica ion; Causally Consis en His-
o y

1 In oduc ion
1.1 Res a ing he P oblem o Time
In classical and quan um heo y,  ime is adi ionally iewed as a p ede e mined pa am-
e e : in gene al ela i i y i is he p ope pa ame e o imelike cu es o Killing/ADM
ime, in quan um heo y i is he e olu ion pa ame e o he Sch ödinge equa ion, and
in s a is ical physics i is he ime index o Ma ko p ocesses. Howe e , once we simul-
aneously conside he ollowing h ee classes o ac s:
1.
I e e sibili y
and a ow o ime in he modynamics and in o ma ion heo y;
2.
Gene alized en opy condi ions
in gene al ela i i y (such as gene alized en-
opy mono onici y, quan um null ene gy condi ion, quan um ocusing condi ion);
3.
Scale iden i y
among phasedelays a e densi y in sca e ing heo y and spec al
heo y,
we nd: ime is mo e like some selec ed s uc u e a he han a backg ound pa am-
e e w i en in he wo ld equa ions om he beginning.
The s a ing poin o his pape is: gi en causal s uc u e and obse able algeb as,
can we cha ac e ize ime as
he solu ion o an op imiza ion p oblem
among all
causally consis en his o y pa hs, selec he ex emal pa h o a ce ain class o
gene al-
ized en opy unc ionals
and i s mono onic pa ame iza ion equi alence class as he
essence o ime a ow and ime scale?
1.2 Co e Idea o This Pape
The co e idea o his pape can be b iey summa ized as a a ia ional p inciple:
On a gi en causally consis en his o y space, he eal wo ld co esponds o
he pa h ha makes a ce ain class o gene alized en opy unc ional
Sgen
ake
an ex emum (unde na u al assump ions, a minimum); he so-called  ime
2
a ow is p ecisely he mono onic pa ame iza ion equi alence class on hese
ex emal pa hs whe e he local en opy p oduc ion a e is non-nega i e.
Unlike he adi ional  ime a ow = en opy inc ease na a i e, he e:

We do no p esuppose en opy mus ine i ably inc ease wi h ime, bu dene
 ime i sel as  he pa h pa ame e ha ex emizes he gene alized en opy unc-
ional;

We a e no limi ed o he modynamic en opy, bu in oduce
gene alized en-
opy
: including he mal en opy, en anglemen en opy, ela i e en opy, and
bounda y geome ic e ms;

We equi e ha pa hs no only sa is y dynamical equa ions, bu also mee
causal
consis ency
,
eco d ex endabili y
, and
in o ma ion mono onici y
cons ain s.
Mo e specically, his pape will:
1. Cons uc he causally consis en his o y space and gene alized en opy unc ional;
2. P o e ha unde na u al con exi y and bound cons ain s, he minimal his o y is
unique unde mono onic epa ame iza ion;
3. Th ough he scale mo he ule in sca e ing heo y
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
uni y he abs ac ime scale wi h obse able phase de i a i es, g oup delays, and
s a e densi ies;
4. Discuss he ela ionship among local obse e s, measu emen eco ds, and ime
pe cep ion.
1.3 A icle S uc u e
The s uc u e o he ull ex is as ollows: Sec ion 2 gi es he o maliza ion o causal
s uc u e, his o y space, and gene alized en opy unc ional, and p oposes he axioma ic
econs uc ion o ime. Sec ion 3 p esen s he main heo em: unde well-posed condi ions,
he gene alized en opy unc ional's minimal causally consis en his o y is unique unde
mono onic epa ame iza ion, hus dening he ime a ow and ime scale equi alence
class. Sec ion 4 in oduces he scale mo he ule
κ(ω)
in he sca e ing and spec al
heo y con ex , embeds i in o he gene alized en opy amewo k, and gi es he obse -
able ealiza ion o ime scale. Sec ion 5 discusses he ela ionship among local obse e s,
eco ds, and subjec i e ime. Sec ion 6 p o ides simplied models o illus a e how he
amewo k wo ks. Appendices gi e s ic p oo s o main heo ems and echnical de ails.

3
2 Causally Consis en His o y Space and Gene alized
En opy Func ional
2.1 Space ime, Causali y, and Obse able Algeb a
Le
(M, g)
be a Lo en zian mani old wi h globally hype bolic s uc u e, ha ing he s an-
da d causal s uc u e
J±(·)
. Le
A
be he obse able algeb a associa ed wi h
M
(e.g.,
a ne o
C∗
-algeb as sa is ying HaagKas le axioms, o es ic ed algeb a on sca e ing
channels), and
ω
a s a e on i .
We conside he s uc u al iple sa is ying he ollowing p ope ies:
Da a = (M, g;A, ω;C),
whe e
C
is he eec i e s a e space, composed o s a es (o s a e equi alence classes)
sa is ying eld equa ions and cons ain condi ions, which can be iewed as an app op ia e
comple ion o some congu a ion space o phase space.
2.2 Wo ld His o y as Pa h: His o y Space and Causal Consis-
ency
Deni ion 2.1
(His o y)
.
A con inuous cu e
γ:I→ C, I ⊂R
is an in e al
,
is called a
wo ld his o y
o
his o y pa h
. Deno e all such cu es as
Pa hs(C)
.
Deni ion 2.2
(Causally Consis en His o y)
.
Gi en
Da a
, a his o y
γ∈Pa hs(C)
is
called
causally consis en
i he ollowing s uc u e exis s:
1. Fo each poin
γ( )
, he e exis s a co esponding spa ial slice o Cauchy sec ion
Σ ⊂M
such ha
1< 2⇒Σ 1⊂J−(Σ 2)
;
2. The s a e e olu ion
γ( )
induces es ic ed s a es
ω
on
A(Σ )
sa is ying local causal-
i y (e.g., sa is ying Eins ein causali y, mic olocali y condi ions);
3. The e exis s a amily o  eco d subalgeb as
R ⊂ A(Σ )
such ha o any
1< 2
,
he eco d dis ibu ion in
R 1
can be aceback- econs uc ed om eco ds in
R 2
( eco d ex endabili y).
Deno e he se o all causally consis en his o ies as
Cons(C)⊂Pa hs(C).
The abo e deni ion abs ac s he basic equi emen s o wo ld his o y, causali y, and
eco ding: no only mus e olu ion i sel con o m o causal s uc u e, bu i mus also
allow obus econs uc ion o pas eco ds, which is c ucial when dening he ime a ow.
4
2.3 Gene alized En opy Func ional and Time Cos
We in oduce a gene alized en opy unc ional dened on he causally consis en his o y
space.
Deni ion 2.3
(Gene alized En opy Func ional)
.
Fo
γ∈Cons(C)
, dene
Sgen[γ] = αS h[γ] + βSen [γ] + γD el[γ] + λB[γ],
whe e:
1.
S h[γ] = RIσ h(γ( ),˙γ( )) d
is he in eg al o coa se-g ained he modynamic en opy
densi y along he pa h;
2.
Sen [γ] = RIσen (γ( )) d
can be aken as he in eg al o en anglemen en opy o
gene alized en opy densi y;
3.
D el[γ] = RId(ω |ω(0)
) d
, whe e
d
is he ela i e en opy densi y and
ω(0)
is a e e -
ence s a e;
4.
B[γ]
is a unc ional o bounda y geome ic beha io , ypically w i en as
B[γ] = Z∂M[γ]
Lbdy(h, K) dΣ,
whe e
h
is he induced me ic and
K
is he ex insic cu a u e.
The coecien s
α, β, γ, λ
a e de e mined by physical scena ios and scale choices (e.g.,
unied ime scale).
Hypo hesis 2.4
(Posi i i y and Con exi y)
.
1.
σ h
and
σen
a e con ex and non-nega i e
in eloci y a iables;
2. The ela i e en opy densi y
d(·|·)
is s ic ly con ex in he  s a iable and sa ises
mono onici y;
3. The bounda y unc ional
B
is lowe semicon inuous on he allowed bounda y a i-
a ion space and has a good lowe bound.
Unde hese condi ions,
Sgen
is a well-dened lowe -bounded unc ional on
Cons(C)
.
2.4 Axioma ic Recons uc ion o Time A ow and Time Scale
We now p esen his pape 's axioma ic scheme o ime.
Axiom 2.1
(Causal P io i y)
.
Physically allowed wo ld his o ies mus belong o he
causally consis en his o y space
Cons(C)
.
Axiom 2.2
(Gene alized En opy Op imiza ion)
.
The eal wo ld co esponds o he
his o y amily ha makes he gene alized en opy unc ional
Sgen :Cons(C)→R
ake an ex emum unde gi en bounda y and ini ial s a e cons ain s, i.e., he e exis s
γ⋆∈Cons(C),Sgen[γ⋆] = in {Sgen[γ]|γ∈Admissible},
whe e
Admissible ⊂Cons(C)
is composed o ini ial s a es, cons ain condi ions, and
ene gy/ux bounds.
5

Axiom 2.3
(Time A ow Condi ion)
.
On he minimal his o y
γ⋆
, he e exis s a mono-
onic pa ame e
such ha he local en opy p oduc ion a e
˙sloc( ) := d
d (αs h( ) + βsen ( ) + γd )≥0
almos e e ywhe e
,
whe e
s h, sen , d
a e local densi ies along he his o y. This mono onic di ec ion denes
he ime a ow.
Deni ion 2.5
(Time Scale Equi alence Class)
.
Le
γ⋆:I→ C
be he minimal his o y.
I
:I→I′
is a s ic ly mono onic die en iable bijec ion, hen
˜γ=γ⋆◦ −1
is ano he
pa ame iza ion o he same his o y. Two pa ame iza ions
and
′
a e said o belong o
he same
ime scale equi alence class
i he e exis s
∈Di 1
+
such ha
′= ( )
.
Deno e his equi alence class as
[ ]
.
Thus, ime is no longe a p ede e mined backg ound axis, bu a s uc u e dened
by he minimal his o y and i s mono onic epa ame iza ion equi alence class.

3 Exis ence o Minimal His o y and Geome ic Unique-
ness o Time A ow
The goal o his sec ion is: unde easonable assump ions, p o e ha he minimal poin
o he gene alized en opy unc ional
Sgen
on he causally consis en his o y space exis s
and is unique unde mono onic epa ame iza ion, he eby ma hema ically suppo ing
he claim  ime = gene alized en opy op imal pa h.
3.1 Va ia ional Se ing and Topological S uc u e
Conside he unc ion space
Pa hs(C) = {γ:I→ C | γ
absolu ely con inuous
},
endowed wi h, e.g., a opology combining
W1,1
o
C0
wi h
L1
. Assume:
1.
C
is a comple e sepa able me ic space;
2.
Cons(C)⊂Pa hs(C)
is closed in he abo e opology;
3.
Admissible ⊂Cons(C)
, gi en by bounda y condi ions and ene gy/ux cons ain s, is
closed and has app op ia e compac ness (e.g., h ough A zelàAscoli o Dun o d
Pe is ype condi ions).
In his se ing, he gene alized en opy unc ional
Sgen
is lowe semicon inuous and
sa ises he ollowing p ope ies.
P oposi ion 3.1
(Lowe Bound and Compac ness)
.
Unde Hypo hesis 2.3.2 and he
abo e opological assump ions, he e exis s a cons an
C
such ha o all
γ∈Admissible
,
Sgen[γ]≥ −C.
Mo eo e , o any
s∈R
, he se
{γ∈Admissible | Sgen[γ]≤s}
is ela i ely compac in he chosen opology.
6
P oo ou line
: Using he lowe bound and con exi y o ela i e en opy and en opy
densi y, p o ide ene gy- ype es ima es o eloci y and s a e, hen apply s anda d com-
pac ness heo ems.
3.2 Exis ence o Minimal Poin
Theo em 3.2
(Exis ence o Gene alized En opy Minimal His o y)
.
Unde he abo e
assump ions, he gene alized en opy unc ional
Sgen
a ains a minimum on
Admissible
,
i.e., he e exis s
γ⋆∈Admissible
such ha
Sgen[γ⋆] = in {Sgen[γ]|γ∈Admissible}.
P oo ou line
: Take a minimizing sequence
(γn)⊂Admissible
such ha
Sgen[γn]→
in
. By P oposi ion 3.1.1's compac ness, he e exis s a subsequence (s ill deno ed
γn
)
con e ging o
γ⋆∈Admissible
in he chosen opology. Using lowe semicon inui y,
Sgen[γ⋆]≤lim in
n→∞ Sgen[γn] = in ,
hus
γ⋆
is he minimal poin .
Comple e igo ous p oo in Appendix A.
3.3 Local Eule Lag ange Equa ion and En opy P oduc ion Ra e
On he minimal his o y
γ⋆
, o local a ia ion
δγ
(p ese ing ini ial/nal condi ions and
causal consis ency), conside he  s -o de a ia ion
δSgen[γ⋆;δγ] = 0.
Fo mally, i w i ing he densi y as Lag angian ype
L(γ, ˙γ) = ασ h(γ, ˙γ) + βσen (γ) + γd(ω(γ)|ω(0)(γ)) + λ↕bdy(γ, ˙γ),
he minimal pa h sa ises he Eule Lag ange equa ion
d
d ∂L
∂˙γ−∂L
∂γ = 0,
plus eec i e mechanical equa ions om causal cons ain s and eco d cons ain s. The
en opy p oduc ion a e can be w i en as
˙sloc( ) = α˙s h( ) + β˙sen ( ) + γ˙
d .
In many physical scena ios (such as non-equilib ium he modynamics consis en wi h
local equilib ium, quan um channels consis en wi h comple e posi i i y and conse a ion,
and g a i a ional backg ounds sa is ying QNEC/QFC condi ions), i can be p o en ha
˙sloc( )≥0
almos e e ywhe e, he eby p o iding a a ia ional in e p e a ion o he ime
a ow.
7
3.4 Uniqueness and Time Scale Equi alence Class
To ex ac he  ime a ow om he minimal pa h, we need o p o e ha he minimal
pa h is unique unde mono onic epa ame iza ion.
Hypo hesis 3.3
(S ic Con exi y and Topological I educibili y)
.
1. Fo almos e -
e y
, he gene alized en opy densi y
(γ, ˙γ)7→ ασ h(γ, ˙γ) + βσen (γ) + γd(ω(γ)|ω(0)(γ))
is s ic ly con ex in
˙γ
;
2. The causally consis en his o y space
Cons(C)
is  opologically i educible unde
gi en ini ial/nal cons ain s: any wo easible pa hs ha induce almos e e ywhe e
iden ical eco d dis ibu ions a each ime slice a e equi alen unde mono onic
epa ame iza ion.
Unde his assump ion we ha e:
Theo em 3.4
(Mono onic Repa ame iza ion Uniqueness o Minimal Causally Consis-
en His o y)
.
Unde he abo e assump ions, i
γ1, γ2∈Admissible
a e bo h minimal poin s
o
Sgen
, hen he e exis s a s ic ly mono onic die en iable bijec ion
such ha
γ2( ) = γ1( −1( )),
i.e., he wo minimal his o ies die only by a mono onic epa ame iza ion, hus belong-
ing o he same ime scale equi alence class
[ ]
.
P oo key poin s
: S ic con exi y ensu es ha i wo die en minimal pa hs exis , he
in e pola ed pa h be ween hem will lowe he unc ional alue, con adic ion; opological
i educibili y ensu es he eedom o die en pa ame iza ions is exac ly he mono onic
epa ame iza ion g oup
Di 1
+
. See Appendix A o de ails.
This shows ha he  ime scale equi alence class is a s uc u e uniquely selec ed by
gene alized en opy op imiza ion, p o iding a geome ic a ia ional deni ion o ime.

4 Scale Mo he Rule in Sca e ing Theo y and Ob-
se able Realiza ion o Time Cos
To connec he abo e abs ac ime s uc u e wi h obse ables, we now u n o sca e ing
spec al heo y and in oduce he scale mo he ule
κ(ω)
.
4.1 Sca e ing Ma ix, G oup Delay, and Rela i e S a e Densi y
Conside a class o s a ic o s eady-s a e sca e ing sys ems whose sca e ing ma ix
S(ω)
is a uni a y ma ix on equency
ω
sa is ying app op ia e die en iabili y. Dene
he Wigne Smi h delay ope a o
Q(ω) = −iS(ω)†∂ωS(ω),
whose ace
τ(ω) := Q(ω)
8
gi es he o al g oup delay. On he o he hand, dene he o al sca e ing phase
Φ(ω) = a g de S(ω), φ(ω) = 1
2Φ(ω),
hen he hal -phase de i a i e
φ′(ω)
π
can be connec ed o he ela i e s a e densi y
ρ el(ω)
and he g oup delay ace. By
Bi manK en ype o mulas and F iedel ype ela ions,
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
holds igo ously wi hin he applicable ange.
Deni ion 4.1
(Scale Mo he Rule )
.
Dene he equency scale mo he ule
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
On one hand, his quan i y can be measu ed by sca e ing phase de i a i es; on he o he
hand, i can be measu ed by g oup delay ace o s a e densi y die ence, hus ha ing
clea scale meaning in expe imen s and heo y.
4.2 Time Cos and Coupling wi h Scale Mo he Rule
Suppose a class o physical p ocesses can be desc ibed in equency space by a measu e
µγ
induced by his o y
γ
: o each
ω
, le
µγ(dω)
desc ibe he weigh and ux o ha equency
mode in his o y
γ
. Then we can dene a class o spec al ime cos  unc ionals
T[γ] = Zκ(ω)µγ(dω).
In many sca e ing o open sys em scena ios, i can be p o en ha
T[γ]
is equi alen
o o bounded-con olled by some e ms in he gene alized en opy unc ional
Sgen[γ]
,
e.g.:

I
D el[γ]
is he ela i e en opy be ween inciden /ou going s a es, i s densi y can
be exp essed h ough ela i e s a e densi y
ρ el(ω)
, hus
D el[γ]≈Z (κ(ω)) µγ(dω),
holds o some con ex unc ion
;

I he bounda y e m
B[γ]
is ela ed o sca e ing c oss-sec ion o eec ion phase,
i can also be w i en as an in eg al o e eigen equencies.
This means ha , a he sca e ingspec al end, pa o he gene alized en opy unc-
ional can be w i en as a spec al in eg al weigh ed by he scale mo he ule , so ime
cos has di ec ly obse able scale p oxies.
9
C S anda d De i a ion Ou line o Sca e ing Scale Mo he
Rule
This appendix p o ides a s anda d de i a ion ou line o he scale mo he ule
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
C.1 Bi manK en Fo mula and Rela i e Spec al Shi
Le
H0
be he ee Hamil onian and
H=H0+V
he sca e ing Hamil onian, sa is ying
app op ia e ace class condi ions. Dene he ela i e spec al shi unc ion
ξ(λ)
whose
de i a i e
ξ′(λ)
gi es he ela i e s a e densi y
ρ el(λ) = ξ′(λ).
The Bi manK en o mula gi es
de S(λ) = e−2πiξ(λ),
hus
Φ(λ) = a g de S(λ) = −2πξ(λ)+2πk, k ∈Z.
Taking con inuous b anch and dening hal -phase
φ(λ) = 1
2Φ(λ)
,
φ′(λ)
π=−ξ′(λ) = ρ el(λ),
ob aining he  s equali y unde app op ia e sign and con en ion choices.
C.2 Wigne Smi h Delay Ope a o and T ace Fo mula
The Wigne Smi h delay ope a o is dened as
Q(λ) = −iS(λ)†∂λS(λ),
i
S(λ)
is sucien ly smoo h in
λ
, hen
Q(λ) = −i S(λ)†∂λS(λ)=−i∂λlog de S(λ) = −i∂λ(iΦ(λ)) = Φ′(λ).
By
φ=1
2Φ
,
φ′(λ) = 1
2Φ′(λ) = 1
2 Q(λ),
hus
φ′(λ)
π=1
2π Q(λ).
Combining C.1 and C.2 yields he scale mo he ule iden i y
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
p o iding a solid spec alsca e ing ounda ion o embedding he unied ime scale in o
he gene alized en opy op imal pa h amewo k in his pape .
16