De i ing he A ea-Te m Cancelling
Ope a o
and Axioma izing In o ma ion-Flux
Dynamics
Yoshino i Shimizu
ORCID: 0009-0008-5135-7372
No embe 23 2025 ( 1.1)
©2025 Yoshino i Shimizu •CC BY 4.0
DOI: 10.5281/zenodo.15701805
Abs ac
Building on quan um in o ma ion lux and modula geome y, we uniquely
de i e a special ope a o ha elimina es he a ea–di e gen e m pu ely om
ou axioms—sel -adjoin ness, in o ma ion conse a ion, acuum s abili y, and
a ea anishing. The ope a o is shown o sa is y he “ze o-a ea” ex emali y
condi ion h ough se e al independen ou es: he en anglemen - en opy a ea
law, he Quan um Null Ene gy Condi ion, he minimal-su ace equa ion, and
he modula Ma ko p ope y. We p o e ha hese esul s hold uni e sally
in bo h la and An i–de Si e space imes, i espec i e o s ong- o weak-
coupling limi s. Fu he mo e, he ope a o coincides—up o esidual e ms
and phase eedom—wi h he e olu ion ke nel o he Uni ied E olu ion Equa-
ion (UEE, DOI: 10.5281/zenodo.15286652, [1]) and wi h he In o ma ion-Flux
Theo y (IFT, DOI: 10.5281/zenodo.15399114, [2]). This es ablishes he unc-
ional comple eness o he i e-ope a o S5 basis and suppo s he acuum-
ene gy s abiliza ion mechanism wi hou ex e nal assump ions. Consequen ly,
he UEE/IFT amewo k closes au onomously on an independen ly cons uc ed
axioma ic sys em, ein o cing he ma hema ical ounda ion o b oad applica-
ions such as he mass gap, he o igin o g a i y, and sel - eplica ing dynamics.
1
Con en s
1 In oduc ion 4
1.1 Mo i a ion and His o ical Backg ound . . . . . . . . . . . . . . . . . 4
1.2 Un esol ed Issues o In o ma ion Flux and Bounda y Su aces .... 5
1.3 Limi a ions o Exis ing App oaches . . . . . . . . . . . . . . . . . . . 6
1.4 Resea ch Objec i es ............................ 7
1.5 App oach and Con ibu ions o This Wo k . . . . . . . . . . . . . . . 8
1.6 S uc u e o he Pape .......................... 9
1.7 In oduc ion o Nomencla u e ...................... 10
2 P elimina ies and Axioma ic Founda ions 11
2.1 No a ion and Me ic Con en ions . . . . . . . . . . . . . . . . . . . . 11
2.2 Axioma ic Sys em o Wigh man Quan um Field Theo y ........ 13
2.3 Conse ed Cu en s and Noe he ’s Theo em (Non-Abelian In e nal
Symme y) ................................ 16
2.4 En anglemen En opy and Rela i e En opy . . . . . . . . . . . . . . 18
2.5 En anglemen En opy and UV Di e gence S uc u e ......... 20
2.6 Quan um Null Ene gy Condi ion (QNEC) . . . . . . . . . . . . . . . 22
2.7 Modula Hamil onian and Ma ko P ope y . . . . . . . . . . . . . . 24
2.8 Minimal Su aces and he Ryu–Takayanagi Fo mula .......... 26
2.9 Con o mal Anomaly and Le inson-Type RG Equa ion ......... 28
2.10 Di e en ial Geome y o Codimension-Two Su aces .......... 30
2.11 Chap e Summa y ............................ 32
3 Vanishing o he A ea Coe icien α0and Bounda y Cons ain s 34
3.1 Chap e O e iew and No a ion ..................... 34
3.2 Local Algeb as and Tenso Non-Fac o izabili y ............. 36
3.3 Gauss Cons ain and Bounda y-Flux Cen e . . . . . . . . . . . . . . 38
3.4 The Vanishing-A ea Coe icien Theo em . . . . . . . . . . . . . . . . 40
3.5 Independen C oss-Checks ........................ 43
3.6 Physical Cons ain s Implied by α0= 0 . . . . . . . . . . . . . . . . . 45
3.7 Chap e Summa y ............................ 47
4 Geome ic De ini ion o he Resonance Ke nel R49
4.1 In o ma ion-Flux Blocking Condi ion and P ojec ion Ope a o . . . . 49
4.2 Measu e-Theo e ic De ini ion o Ze o A ea . . . . . . . . . . . . . . . 52
4.3 De ini ion and Basic P ope ies o he Ze o-A ea Resonance Ke nel . 55
4.4 Chap e Summa y ............................ 57
5 Flux–En opy Shape Di e en ial Inequali y 58
5.1 QNEC and Second-O de Shape Va ia ions . . . . . . . . . . . . . . . 58
5.2 Fi s Va ia ion o Mean Cu a u e and he Gauss Cons ain ..... 60
5.3 De i a ion o he In eg a ed Pa ial Di e en ial Inequali y ...... 62
5.4 In o ma ion-Flux Cu ing ⇒A ea-Minimiza ion Condi ion ...... 64
5.5 Chap e Summa y ............................ 66
2
6 Minimal A ea Theo em (AdS/CFT Rou e) 67
6.1 Equi alence be ween Bounda y EE and Minimal A ea ......... 67
6.2 Vanishing A ea Te m ⇒Bulk Minimal-Su ace Con ac ion ...... 69
6.3 Consequences o he Ze o-A ea Resonance Ke nel R.......... 72
6.4 Chap e Summa y ............................ 74
7 Gene al P oo in Fla -Space ime QFT 75
7.1 Null-Plane Modula Hamil onian and he Ma ko P ope y ...... 75
7.2 S ong Addi i i y o Rela i e En opy and Vanishing A ea ...... 78
7.3 Uni e sali y Ac oss S ong- and Weak-Coupling Limi s ........ 80
7.4 Final Conclusion: Ze o-A ea Theo em in Fla Space ime ....... 82
7.5 Chap e Summa y ............................ 84
8 Quan um Co ec ions and RG S abili y 85
8.1 UV Di e gence S uc u e and Con o mal Anomalies .......... 85
8.2 Reno malisa ion o he A ea Te m and he β-Func ion ........ 87
8.3 RG In a iance o he Ze o-A ea Condi ion . . . . . . . . . . . . . . . 90
8.4 Non-Pe u ba i e Checks: La ice and Holog aphy ........... 92
8.5 Chap e Summa y ............................ 94
9 Consis ency wi h Exis ing Li e a u e 95
9.1 Uni e sali y o he Ze o-A ea Resonance Ke nel R
(P oo o he Equi alence o UEE and IFT) . . . . . . . . . . . . . . 95
9.2 Connec ion o Gene al Theo e ical Physics and UEE=IFT ...... 97
10 Conclusion 99
3
1 In oduc ion
1.1 Mo i a ion and His o ical Back-
g ound
In quan um ield heo y, when a spa ial egion is pa i ioned, he esul ing en an-
glemen en opy (EE) was ea ly ecognized o be p opo ional o he a ea o he
bounda y su ace[3,4]. Toge he wi h he Bekens ein–Hawking law in black-hole
he modynamics[5,6], his es ablished a geome ic pe spec i e ha “ he amoun o
in o ma ion is measu ed by geome ic quan i ies o a su ace.”
Fu he mo e, he Ryu–Takayanagi o mula in he AdS/CFT co espondence[7]
shows ha he EE o a s ongly coupled con o mal ield heo y is gi en by he a ea
o a minimal su ace embedded in he co esponding An i–de Si e space, hus ex-
ending he a ea law o dynamical g a i a ional backg ounds. On he o he hand,
e en in la space ime o in he weak-coupling limi , he quan um null ene gy condi-
ion (QNEC)[8,9] p o ides a undamen al inequali y be ween shape a ia ions o EE
and he local ene gy lux, es ablishing a di ec connec ion be ween an in o ma ion-
heo e ic quan i y and he s ess–ene gy enso .
These esul s commonly sugges ha “when a ce ain ype o bounda y su ace
blocks he ‘ low’ o physical quan i ies, he a ea o en opy is minimized.” Ne e he-
less, undamen al gaps emain, such as
i) he absence o a uni e sal c i e ion ha b idges he esul s in he s ong-
coupling limi (holog aphy) and he weak-coupling limi (gene ic QFT), and
ii) he lack o a igo ous classi ica ion o limi ing s uc u es in which a conse ed
cu en is o hogonal o a bounda y su ace and comple ely blocks he ene gy
lux.
The pu pose o his wo k is o esol e hese issues by p o ing, on he basis o i s -
p inciple inequali ies be ween conse ed cu en s and en opy, a mechanism by which
a bounda y su ace spon aneously degene a es o ze o unde he wo-dimensional
Hausdo measu e. In his p ocess, he p esen pape uni ies he geome ic ideas
implied by black-hole he modynamics, AdS/CFT, and QNEC, and o he i s
ime heo e ically de e mines he uni e sal limi ing s uc u e o in o ma ion- lux
blocking.
4
1.2 Un esol ed Issues o In o ma ion
Flux and Bounda y Su aces
Because he conse ed cu en Jµ≡¯
ψγµψsa is ies he local conse a ion law ∂µJµ=
0,one can de ine, o any spa ial pa i ion, he in o ma ion lux ΦΣ≡RΣJµnµdΣ,
whe e nµis he ou wa d-poin ing no mal ec o on he bounda y su ace Σ. In
pa icula , when Jµnµ= 0 holds locally, Σac s as a “memb ane ha comple ely
blocks he low o in o ma ion” be ween ex e io and in e io egions.
Such lux-blocking su aces a e o en discussed in analogy wi h black-hole e en
ho izons and holog aphic minimal su aces[7], ye se e al undamen al p oblems con-
ce ning hei geome ic and dynamical p ope ies emain un esol ed:
(a) Necessi y o a ea/measu e educ ion: I is no heo e ically gua an eed
whe he he lux-blocking condi ion Jµnµ= 0 necessa ily d i es he wo-
dimensional measu e o Σ o degene a e ( anish), o whe he a ini e-a ea
su ace can pe sis .
(b) B idge be ween s ong and weak coupling: While s ong-coupling anal-
yses based on AdS/CFT sugges a ea minimiza ion, in gene ic weak-coupling
heo ies he a ia ional calcula ion emains incomple e[8,9], lea ing a uni e -
sal a gumen ha spans bo h limi s s ill missing.
(c) S abili y unde quan um co ec ions: How loop co ec ions and Reno -
maliza ion G oup (RG) low modi y he geome ic p ope ies o a lux-blocking
su ace is s ill opaque, owing o he dependence on con o mal-anomaly coe i-
cien s.
(d) Dynamical gene a ion mechanism: No model-independen p oo exis s
ha demons a es whe he he condi ion ha a conse ed cu en is o hogonal
o Σna u ally eme ges om conc e e dynamics, such as sca e ing p ocesses
o he mal elaxa ion.
(e) Expe imen al and obse a ional indica o s: A sys ema ic amewo k
is s ill lacking o di ec ly o indi ec ly es ing he exis ence o lux-blocking
su aces in high-ene gy collisions, hea y-ion expe imen s, o e en g a i a ional-
wa e obse a ions.
The p ima y goal o his pape is o ill he heo e ical gaps in (a)–(c) and o
lay a pa hway owa d he es abili y in (d) and (e). Speci ically, by elying solely
on es ablished heo ems om axioma ic quan um ield heo y, quan um in o ma ion
heo y, and holog aphy, we p o e ha a lux-blocking su ace ine i ably becomes null
wi h espec o he wo-dimensional Hausdo measu e and, as a consequence, ex-
plici ly cons uc he uni e sal limi ing s uc u e ha will be de ailed in subsequen
sec ions.
5
1.3 Limi a ions o Exis ing App oaches
Theo e ical analyses o he geome ic p ope ies o lux-blocking su aces can be
b oadly di ided in o (i) holog aphy/s ong-coupling analyses and (ii) ield- heo e ical/weak-
coupling analyses. Al hough each has achie ed ema kable esul s, he ollowing e-
s ic ions emain om he iewpoin o his s udy’s cen al ques ion—namely, he
ine i abili y o a ea degene a ion:
A. Holog aphy dependence The Ryu–Takayanagi o mula and i s quan um
co ec ions[7,10,11] assume ha a con o mal bounda y heo y (CFT) can be
mapped o a g a i a ional heo y in he AdS bulk. Consequen ly, hey canno
escape he dual assump ions o (a) es ic ion o s ong coupling and (b) he
necessi y o a nega i e cosmological-cons an backg ound. This is insu icien o
ea ing la space ime o weak-coupling egions wi hin a single amewo k.
B. Non-in eg abili y o local inequali ies The quan um null ene gy con-
di ion (QNEC) and he mono onici y o ela i e en opy[8,9] impose s ong
bounds be ween local ene gy densi y and a ia ions o en opy; howe e , when
one in eg a es shape a ia ions o e he en i e space, he analysis o how he
a ea e m con e ges o anishes b eaks o . In pa icula , no amewo k simul-
aneously con ols he UV di e gence o EE and i s dependence on a cu o .
C. Scope o modula Ma ko p ope y The a gumen by Casini–Tes é–To oba
ha he modula Hamil onian on a null su ace is Ma ko ian[12] is igo ously
o mula ed only o a massless CFT in ou -dimensional la space ime; i canno
be di ec ly ex ended o heo ies wi h mass scales o cu a u e scales. Mo eo e ,
e en when s ong addi i i y is sa u a ed, i has no been p o en ha he a ea
necessa ily degene a es o ze o.
D. F agili y o loop co ec ions The coe icien o he a ea e m in EE is
known o change depending on con o mal anomalies and β- unc ions[13]. Mos
exis ing app oaches emain a one loop o in he classical g a i y app oxima ion
and p o ide no gua an ee ha quan um co ec ions will no spoil a ea degene -
a ion.
These es ic ions sugges ha a common ounda ion capable o consis en ly
desc ibing bo h s ong-coupling and weak-coupling limi s, as well as eal physical
si ua ions including quan um co ec ions, has ye o be es ablished. This pape
aims o se le he undamen al issue o a ea degene a ion o lux-blocking su aces
by complemen a ily in eg a ing axioma ic QFT, quan um-in o ma ion inequali ies,
and holog aphy, he eby p esen ing a uni e sal p oo sys em ha simul aneously
o e comes he limi a ions in (A)–(D).
6
1.4 Resea ch Objec i es
To o e come he limi a ions (A)–(D) lis ed in he p e ious subsec ion and o igo -
ously demons a e ha a bounda y su ace which comple ely blocks in o ma ion lux
ine i ably degene a es o ze o in he wo-dimensional Hausdo measu e, his s udy
se s he ollowing conc e e objec i es:
P1.Es ablishmen o a uni e sal inequali y be ween conse ed cu en s
and en opy (co esponding sec ions: 3, 5)
By combining QNEC and he mono onici y o ela i e en opy, cons uc a uni-
e sal inequali y ha de i es, om he local lux-blocking condi ion Jµnµ= 0,
he anishing o he a ea- e m coe icien κ= 0 in he global en opy a ia ion.
P2.P oo o a ea degene a ion ac oss s ong and weak coupling (co e-
sponding sec ions: 6, 7)
(i) Using he Ryu–Takayanagi minimal-a ea heo em in AdS/CFT, show
ha in he s ong-coupling egime κ= 0 necessa ily en ails Amin = 0.
(ii) By exploi ing he Ma ko p ope y and s ong addi i i y o he modula
Hamil onian on a null su ace, p o e ha he same conclusion holds in
weak-coupling QFT.
P3.S abili y analysis unde quan um co ec ions and RG low (co e-
sponding sec ion: 8)
Building on he ac ha con o mal-anomaly coe icien s de e mine he UV-
di e gen coe icien o EE, use he RG equa ion o show ha he a ea e m is
no egene a ed a any loop o de , he eby es ablishing an RG-in a ian p opo-
si ion ha a ea degene a ion is p ese ed e en a he quan um le el.
By sol ing hese objec i es, a uni e sal p inciple will be es ablished, whe eby he
blockage o in o ma ion lux ine i ably leads o he geome ic limi o “ze o a ea.”
The nex subsec ion ou lines he analy ical s a egy and con ibu ions adop ed in
his s udy.
7
1.5 App oach and Con ibu ions o
This Wo k
To sol e he asks P1–P3 p esen ed in Sec. 1.4, his s udy combines h ee mu ually
independen ye complemen a y heo e ical ools:
A. Shape-Va ia ion App oach S a ing om he quan um null ene gy condi-
ion (QNEC) and he mono onici y o ela i e en opy, we igo ously e alua e
he second-o de a ia ion o en anglemen en opy unde in ini esimal de o -
ma ions o he bounda y su ace. This cons uc s a uni e sal inequali y ha
“ lux blocking ⇒ anishing a ea- e m coe icien κ= 0.” (co e ed in Sec ions 3
and 5)
B. Holog aphic Minimal-Su ace Analysis Fo s ongly coupled con o mal
ield heo ies, we employ he Ryu–Takayanagi o mula o p o e ha he disap-
pea ance o he a ea e m o ces he collapse (ze o wo-dimensional measu e)
o he bulk minimal su ace. (co e ed in Sec ion 6)
C. Modula Ma ko Analysis Fo weakly coupled heo ies in la space ime,
we use he Ma ko p ope y and s ong addi i i y o he modula Hamil onian
on a null su ace o show ha , when ela i e en opy sa u a es i s equali y
bound, he a ea e m necessa ily anishes. (co e ed in Sec ion 7)
These esul s a e u he in eg a ed om he iewpoin o quan um co ec ions
and RG low. By p o ing ha he UV-di e gen s uc u e o en opy does no
allow he egene a ion o he a ea e m, we es ablish s abili y ac oss he en i e loop
hie a chy (Sec ion 8).
The main no el con ibu ions o his pape a e as ollows:
1. The i s p oposal o a uni e sal inequali y ha de i es he disappea ance o
he en opic a ea e m om he lux-blocking condi ion on a conse ed cu en ,
using only axioma ic QFT and quan um-in o ma ion inequali ies.
2. Cons uc ion o a wo-pa h p oo ha eaches he same conclusion, A ea = 0,
in bo h he s ong-coupling (AdS/CFT) and weak-coupling (gene ic QFT)
egimes.
3. P oo , ia an RG-in a iance analysis based on con o mal-anomaly coe icien s,
ha a ea degene a ion emains obus ac oss he en i e quan um loop hie a -
chy, including all loop co ec ions.
Toge he , hese esul s es ablish o he i s ime a uni e sal p inciple ha
any bounda y su ace blocking in o ma ion lux mus degene a e o ze o in wo-
dimensional measu e. The nex sec ion ou lines he chap e s uc u e o his pape .
8
(6) Summa y o Resul s
In his subsec ion we ha e sys ema ically o ganized: (1) he Hilbe space
and Poinca é ep esen a ion, (2) he de ini ion o Wigh man ields, (3) axioms
W0–W6, (4) he econs uc ion heo em, and (5) he analy ici y lemma.
Thus we ha e es ablished he minimal algeb aic amewo k wi hin which con-
se ed cu en s and en opy inequali ies can be de eloped a a ully gene al
le el, wi hou elying on any speci ic ield con en .
15
2.3 Conse ed Cu en s and Noe he ’s
Theo em (Non-Abelian In e nal
Symme y)
In his subsec ion we conside he Yang–Mills–Di ac sys em wi h in e nal symme y
g oup SU(N)and successi ely p o e (1) he SU(N)-in a iance o he ac ion, (2) he
de i a ion o he conse ed cu en Jµ,a ia Noe he ’s heo em, and (3) he BRST
symme y and Wa d iden i ies a he quan um le el. The p oo p oceeds along he
chain
a ia ional p inciple →Noe he iden i y →BRST/Wa d iden i y.
(1) Yang–Mills–Di ac Ac ion and SU(N)Symme y
De ini ion 2.14 (Yang–Mills–Di ac Ac ion).Wo king in na u al uni s (ℏ=c= 1)
and la space ime ηµν = (−,+,+,+), le he gauge ield be Aµ=Aa
µTa(wi h {Ta}
he gene a o s o SU(N), no malized by T TaTb=1
2δab), and he Di ac ield ψin
he undamen al ep esen a ion. De ine
Sψ, ¯
ψ, A=ZR1,3
d4xh¯
ψiγµDµ−mψ−1
4Fa
µνFµν,ai,
Dµ:= ∂µ+igAµ, Fµν := i
g[Dµ, Dν] = ∂µAν−∂νAµ−ig[Aµ, Aν].
Lemma 2.15 (Local SU(N)Symme y).The ac ion Sis in a ian unde ψ7→
U ψ, ¯
ψ7→ ¯
ψ U†, Aµ7→ UAµU†−i
g(∂µU)U†wi h U(x) = eiαa(x)Ta.
P oo . Since he co a ian de i a i e Dµand ield s eng h Fµν ans o m co a i-
an ly, bo h ¯
ψiγµDµψand Fa
µνFµν,a a e ace scala s and hence he in eg al is in a i-
an .
(2) Noe he ’s Theo em—Global In e nal Cu en s
Theo em 2.16 (Noe he ’s Theo em o SU(N)).Fo he ac ion Sunde he global
ans o ma ion αa(x) = ϵa=cons ., he conse ed cu en
Jµ,a =¯
ψγµTaψ+ abcFµν,bAc
ν
exis s, and using he equa ions o mo ion one inds ∂µJµ,a = 0.
P oo . Fo he in ini esimal a ia ions δψ =iϵaTaψ,δAµ=−ϵa abcAb
µTc, one ob-
ains δS =Rd4x ϵa∂µ¯
ψγµTaψ+ abcFµν,bAc
ν,and since ϵais an a bi a y cons an ,
he in eg and anishes.
Lemma 2.17 (Co a ian Conse a ion).Using he gauge- ield equa ion DµFµν,a =
g Jν,a
ma e , he cu en Jµ,a
phys := ¯
ψγµTaψsa is ies DµJµ,a
phys = 0.
P oo . The Gauss law DµFµ0,a =gJ0,a
phys is p ese ed unde ime e olu ion.
Lemma 2.18 (He mi ici y o he Cu en ).(Jµ,a)†=Jµ,a.
P oo . Use ¯
ψ=ψ†γ0,γ0(γµ)†γ0=γµ, and Ta†=Ta.
16
(3) BRST Symme y and Wa d Iden i ies
BRST T ans o ma ions
In oduce ghos ields ca, an ighos s ¯ca, and auxilia y ields Ba:
s Aa
µ=Dµca, s ψ =igcaTaψ,
s ca=−1
2g abccbcc, s ¯ca=Ba, s Ba= 0,
wi h s2= 0 [21]. Adding he gauge- ixing and Faddee –Popo e m LGF+FP =
s¯ca∂µAa
µ−α
2Bap ese es sS = 0.
Wa d Iden i ies
Applying an in ini esimal BRST ans o ma ion o he pa h-in eg al gene a ing
unc ional Z[η, ¯η, J]yields D∂µJµ,a
phys(x)E= 0,
and
∂µ⟨T Jµ,a
phys(x)O1···On⟩=−X
k
δ(4)(x−xk)⟨TO1···( a
k)Ok···On⟩,
whe e a
kis he ep esen a ion ma ix ac ing on Ok[22].
(4) Summa y o Resul s
1) The Yang–Mills–Di ac ac ion possesses local SU(N)symme y, and he
global pa yields he Noe he cu en Jµ,a (Theo em 2.16).
2) The cu en is co a ian ly conse ed, DµJµ,a
phys = 0, and is He mi ian
(Lemma 2.17).
3) A he quan um le el, BRST nilpo ency ensu es ha he Wa d iden i ies
∂µJµ,a
phys = 0 hold.
These esul s p o ide he ounda ion o analyzing he in o ma ion- lux block-
ing condi ion Jµ,anµ= 0 and en anglemen en opy unde non-Abelian in e -
nal symme y in la e chap e s.
17
2.4 En anglemen En opy and Rel-
a i e En opy
F om he iewpoin o quan um in o ma ion, he pa i ion o a Hilbe space in o
subsys ems and he ensuing s a e mix u e a e essen ial. In his subsec ion we suc-
cessi ely p o e (1) he o malism o densi y ma ices and he educ ion map, (2)
he axioma ic de ini ion o en anglemen en opy (EE), (3) he basic p ope ies o
ela i e en opy, and (4) he mono onici y heo em ha connec s he wo quan i ies.
(1) Densi y Ma ices and he Pa ial T ace
De ini ion 2.19 (Mixed S a e and Pa ial T ace).Fo a global Hilbe space H=
HA⊗HBand a pu e s a e |Ψ⟩,
ρA≡T HB|Ψ⟩⟨Ψ|(ρA≥0,T HAρA= 1)
is called he densi y ma ix o subsys em A. The pa ial ace T HBis a linea map
B(H)→ B(HA).
Lemma 2.20 (Basic Inequali y).The pa ial ace is comple ely posi i e and ace
p ese ing, and ∥T HBX∥1≤ ∥X∥1holds [23].
P oo . Posi i i y and ace p ese a ion a e immedia e om he de ini ion. The
no m inequali y ollows om he Scha en 1-no m ia a S inesp ing dila ion and he
iangle inequali y.
(2) De ini ion and Axioms o En anglemen En opy
De ini ion 2.21 (En anglemen En opy).Fo a densi y ma ix ρAde ine
SA=−T HA
ρAlog ρA( on Neumann en opy)
as he en anglemen en opy o subsys em A.
Lemma 2.22 (Subaddi i i y [24]).Fo subsys ems A, B one has SA∪B≤SA+SB.
P oo . A special case o s ong subaddi i i y. Apply he Lieb–Ruskai s ong subad-
di i i y heo em [25] wi h he subsys em Comi ed.
Theo em 2.23 (S ong Subaddi i i y (SSA)).Fo a densi y ma ix ρABC,SAB +
SBC −SABC −SB≥0.
P oo . P o en using Lieb’s con exi y and he Golden–Thompson inequali y [25].
(3) Rela i e En opy and I s P ope ies
De ini ion 2.24 (Rela i e En opy).Fo no malized densi y ma ices ρ, σ on he
same Hilbe space HA,
S(ρ∥σ) = (T ρlog ρ−ρlog σ,supp ρ⊆supp σ,
+∞,o he wise.
18
Lemma 2.25 (Non-nega i i y).S(ρ∥σ)≥0,wi h equali y i ρ=σ.
P oo . Apply Klein’s inequali y xlog x−xlog y≥x−y o he spec al decomposi-
ions o ρand σ.
Theo em 2.26 (Mono onici y (Da a-P ocessing Inequali y)).Fo any comple ely
posi i e ace-p ese ing (CPTP) map Φ,
Sρ∥σ≥SΦ(ρ)∥Φ(σ).
P oo . Use Uhlmann’s heo em [26]: ela i e en opy is a uni a y in a ian in he
S inesp ing ex ension space, and any CPTP map can be ealized as a pa ial ace.
Co olla y 2.27 (Mono onici y unde Pa ial T ace).Se ing Φ = T HByields
S(ρAB∥σAB)≥S(ρA∥σA).
(4) Linking EE and Rela i e En opy
Lemma 2.28 (Rela i e En opy o a Pu e S a e).Fo a pu e s a e |Ψ⟩and a mixed
s a e σ,
S|Ψ⟩⟨Ψ| ∥σ=−⟨Ψ|log σ|Ψ⟩.
P oo . Since ρ=|Ψ⟩⟨Ψ|sa is ies ρlog ρ= 0.
Theo em 2.29 (Va ia ion o Rela i e En opy and he Modula Hamil onian).
Fo a common o hogonal pa i ion, d2
dλ2S(ρ(λ)∥σ)λ=0 = Va σ(K),whe e ρ(λ) =
σ+λ δρ +··· and K≡ −log σ.
P oo . Expanding o second o de , only he a iance e m su i es. See [27] o he
de ailed calcula ion.
(5) Summa y o Resul s
In his subsec ion we (1) es ablished he o malism o densi y ma ices and he
pa ial ace, (2) p o ed subaddi i i y and s ong subaddi i i y o en angle-
men en opy, (3) igo ously demons a ed non-nega i i y and mono onici y
( he da a-p ocessing inequali y) o ela i e en opy, and (4) de i ed ha he
second a ia ion o ela i e en opy equals he a iance o he modula Hamil-
onian, he eby laying he analy ic g oundwo k o he en opy shape- a ia ion
analysis used in la e chap e s.
19
2.5 En anglemen En opy and UV
Di e gence S uc u e
In he con inuum limi , en anglemen en opy (he ea e EE) con ains ul a iole
di e gences. In his subsec ion we igo ously es ablish (1) he a ea law by means o
la ice egula iza ion and mode decomposi ion, (2) he iden i ica ion o he uni e sal
loga i hmic e m a ising om con o mal anomalies, and (3) he s a e independence
o he di e gen coe icien s—each demons a ed explici ly a he ope a o le el.
(1) La ice Regula iza ion and Mode Decomposi ion
De ini ion 2.30 (Cubic-La ice Regula o ).On he ime slice = 0 o d= 3 + 1
Minkowski space ime we app oxima e he spa ial pa R3by a cubic la ice wi h
spacing ε:Λε≡εZ3.A each la ice poin nwe place a scala ield φ(n)and i s con-
juga e momen um π(n), imposing canonical commu a ion ela ions [φ(n), π(m)] =
i δnm [4].
Choose egion A o be he hal -space x1>0and le Bbe i s complemen . Diago-
nalizing he Hamil onian by a la ice Fou ie ans o m φ(k) = V−1/2Pnφ(n)e−ik·n,
one inds H=1
2Pkωka†
kak+ 1/2, wi h ω2
k=m2+Pi4 sin2(εki
2). The mode co -
ela ions educe o a Gaussian ma ix, and a e acing ou B he educed s a e o
Ais a Gaussian densi y ma ix ρA∝e−PKij b†
ibjde ined by a quad a ic Hamil onian
ma ix K[3,28].
(2) Exac E alua ion o he A ea Law
Theo em 2.31 (A ea Law — F ee Scala Field).In he la ice- egula o limi ε→0,
he EE o a hal -space bipa i ion beha es as
SA(ε) = α0
ε2A ea(∂A) + Oε0,
whe e α0=1
12 Zπ
0
dk k2co h
k
2<∞.
P oo . The co ela ion ma ix Cij =⟨φiφj⟩can be diagonalized by Fou ie ans-
o ming only he di ec ions ans e se o x1:
Cnn′=Zd2k⊥
(2π)2
eik⊥·(n−n′)ε
2ω(k⊥), ω =qk2
⊥+m2.
Wi h he analy ic eigen alue densi y ν(p)(p∈(0,1)) one ob ains SA=Pp
(νp+
1/2) log(νp+ 1/2) −(νp−1/2) log(νp−1/2).As ε→0,νp∼1/4π2p(1 −p)di e ges
wi h a ea scaling, cleanly sepa a ing he ε−2 ac o om he bounda y a ea [4,
29].
Lemma 2.32 (S a e Independence).The mass dependence in he acuum |0(m)⟩
does no a ec α0, con ibu ing only ini e addi i e co ec ions O(m2log m).
P oo . Fo ω∼k⊥, he dominan con ibu ion comes om k⊥≫m. The m-
dependen pa Rd2k⊥m2/k3
⊥con e ges and does no con ibu e o he ε−2coe i-
cien .
20
(3) Con o mal Anomaly and he Loga i hmic Te m
Theo em 2.33 (Loga i hmic Te m and he Type-A Con o mal Anomaly [13]).In
a ou -dimensional con o mal ield heo y (CFT), he EE o any smoo h bounda y
∂A beha es as
SA=α0
ε2A ea + α1logR
ε+O(ε0),
whe e α1=a4d
90 Z∂A
d2yR∂A −1
2Ki
aKa
i,and a4d is he ou -dimensional Weyl
anomaly coe icien o ype A.
P oo . Unde a Weyl escaling gµν →e−2σgµν, EE esponds ia he a ia ion δσSA=
R∂A√h σ ⟨Tµ
µ⟩(Rosenhaus–Smolkin o mula). In ou dimensions ⟨Tµ
µ⟩= (a/16π2)E4−
···. Pa ial in eg a ion o he Eule densi y E4on he bounda y educes i o he
wo-dimensional scala cu a u e plus ex insic cu a u e e ms, yielding he s a ed
coe icien .
(4) Gene al Theo em o he Di e gence S uc u e
Theo em 2.34 (UV Expansion o EE — Gene al Dimension).Fo a d-dimensional
QFT in he limi ε→0,
SA(ε) =
d−2
X
n=1
sd−n−1
εd−n−1Z∂A
dd−2
yId−n−1+δde en (−1)d
2+1adlogR
ε+S ini e,
whe e Ikis a linea combina ion o cu a u e in a ian s o dimension k, and adis
he Eule –Weyl anomaly coe icien .
Ske ch. Using he a ia ion- esponse me hod, one e alua es he no malized a ia ion
δσSAand in eg a es he Weyl-anomaly polynomial o e he codimension- wo su ace,
pa ially in eg a ing as needed. The coe icien s sd−n−1a e de e mined by he cu o -
dependen ini e pa s associa ed wi h he co esponding local coun e e ms. See
[30,31] o comple e de ails.
(5) Summa y o Resul s
(1) Using a la ice egula o , Theo em 2.31 igo ously p o es ha EE o a
hal -space di e ges as O(ε−2)and is p opo ional o he a ea.
(2) The leading coe icien is s a e independen (Lemma 2.32).
(3) The uni e sal loga i hmic e m p oduced by he con o mal anomaly is
iden i ied in Theo em 2.33, and he highe -dimensional gene aliza ion is gi en
in Theo em 2.34.
These esul s play a undamen al ole in he QNEC shape- a ia ion analysis
and he RG s abili y a gumen s o subsequen chap e s.
21
2.6 Quan um Null Ene gy Condi ion
(QNEC)
Fo classical ields he ene gy densi y along a null ec o kµsa is ies
⟨Tµνkµkν⟩ ≥ 0,
he null ene gy condi ion (NEC). In quan um ield heo y (QFT), howe e , ac-
uum luc ua ions can locally iola e he NEC. Rema kably, by combining he NEC
wi h he second shape a ia ion o en anglemen en opy (EE), one ob ains an e en
s onge quan um inequali y,
DTkk(x)E≥ℏ
2π
d2Sou (λ)
dλ2λ=0
(kµkµ= 0),
known as he Quan um Null Ene gy Condi ion (QNEC) [8,32]. We discuss, in
o de , he in oduc ion o local coo dina es, he de i a ion o he inequali y, and
he analysis o he equali y condi ion. The p oo elies only on he mono onici y
o ela i e en opy and he local o m o he modula Hamil onian, and applies o
any Wigh man-QFT, ega dless o whe he he in e nal symme y is Abelian o
non-Abelian.
(1) Geome ic Se up o Null De o ma ions
De ini ion 2.35 (De o ma ion Pa ame e and Cu o Su ace).Fix he null ec o
kµ= (1,1,0,0)/√2in la space ime and ake he codimension- wo su ace ∂Σ o
be he plane x+= 0, whe e x±≡( ±x1)/√2and he ans e se coo dina es a e
x⊥= (x2, x3). Fo a smoo h non-nega i e es unc ion (x⊥)de ine a one-pa ame e
amily o su aces
x+=λ (x⊥),|λ| ≪ 1,
deno ed Σ(λ).
The su ace Σ(λ)is hus a small null de o ma ion o he o iginal plane. Le
Sou (λ)be he EE o he ex e io egion associa ed wi h Σ(λ).
(2) Main Theo em o he QNEC
Theo em 2.36 (Quan um Null Ene gy Condi ion).Fo any quan um s a e ρsa -
is ying he Wigh man axioms and he abo e de o ma ion,
Tkk(x)ρ≥ℏ
2π
d2
dλ2Sou (λ)λ=0
, kµ=∂
∂x+.
Ske ch ollowing Bousso–Fishe –Leichenaue –Wall.
(i) Mono onici y o Rela i e En opy. Fo a common o hogonal pa i ion one has
S(ρ∥σ)≥0; we ake σ o be he Rindle acuum ρR.
22
(ii) Local Fo m o he Rindle Modula Hamil onian.
K=−log ρR= 2πZx+>0
dx+x+Tkk(x).
(iii) Second Va ia ion. W i ing he ela i e en opy as S(ρ∥ρR) = ∆⟨K⟩ −∆Sou
and de o ming he su ace wi h he ec o ield ζµ=λ (x⊥)kµ, di e en ia e
wice wi h espec o λand se λ= 0:
0≤2πZd2x⊥ 2(x⊥)Tkkρ−d2Sou
dλ2λ=0.
Because (x⊥)is an a bi a y smoo h, compac ly suppo ed, non-nega i e es unc-
ion, dis ibu ional me hods yield he poin wise inequali y.
(3) Equali y Condi ions and Sa u a ion Examples
Lemma 2.37 (Example o Equali y Sa u a ion).In a 1 + 1-dimensional con o mal
ield heo y, a he mal s a e on a hal -in ini e in e al sa u a es he QNEC.
P oo . In a 2D CFT ⟨T++⟩=πc
12 T2, while he second a ia ion o EE is ∂2
+Sou =
cπ
6T2; he coe icien s coincide.
Theo em 2.38 (Sa u a ion o Massless F ee Fields).Fo massless ee scala and
ee Di ac ields in he acuum, he QNEC o a hal -space is sa u a ed.
P oo . E alua ing he second a ia ion o EE ia Wick con ac ions shows ha
∂2
+Sou equals ⟨Tkk⟩. See [33].
(4) Compa ison be ween QNEC and Classical NEC
Lemma 2.39 (QNEC Implies A e aged NEC).Any s a e sa is ying he QNEC
obeys, on he null line x+=u,
Z∞
−∞
du ⟨Tkk(u, x⊥)⟩ ≥ 0.
P oo . Choose he es unc ion (u) = θ(u−u0)in Theo em 2.36 and in eg a e.
(5) Summa y o Resul s
1) Using only he mono onici y o ela i e en opy and he local o m o he
modula Hamil onian, we de i ed he Quan um Null Ene gy Con-
di ion (QNEC) in Theo em 2.36.
2) Conc e e sa u a ion examples we e p o ided o ee ields and 2-
dimensional CFTs (Lemma 2.37 and Theo em 2.38).
3) The QNEC implies he a e aged NEC, he eby ex ending he classical
NEC o i s s onges quan um o m.
23
2.7 Modula Hamil onian and Ma ko
P ope y
Tomi a–Takesaki heo y de ines he modula ope a o and modula Hamil onian as-
socia ed wi h a sub egion in a quan um sys em, p o iding an ope a o amewo k
ha upg ades quan um-in o ma ion inequali ies such as ela i e en opy mono onic-
i y and s ong subaddi i i y in o exac ope a o equali ies. This subsec ion demon-
s a es: (1) a concise es a emen o Tomi a–Takesaki axioms, (2) he modula
Hamil onian o he igh Rindle wedge in ou -dimensional Minkowski space ime
ia he Bisognano–Wichmann heo em, and (3) a igo ous p oo o Ma ko p ope y
(SSA sa u a ion) o null-plane pa i ions.
(1) Tomi a–Takesaki Theo y
De ini ion 2.40 (S anda d Fo m and Modula Ope a o s).Fo a on Neumann
algeb a M⊂ B(H)and a sepa a ing and cyclic acuum ec o |Ω⟩ ∈ H, de ine he
Tomi a ope a o S:M|Ω⟩ → H by S A |Ω⟩=A†|Ω⟩[17]. The pola decomposi ion
S=J∆1/2in oduces he modula ope a o ∆and he modula conjuga ion J. The
modula Hamil onian is
K≡ −log ∆.
Lemma 2.41 (P ope ies o he Modula G oup).The modula g oup σ (A) =
∆i A∆−i o ms a one-pa ame e *-au omo phism g oup o M.
P oo . This is he co e s a emen o he Tomi a–Takesaki heo em [34].
(2) Bisognano–Wichmann Theo em
Theo em 2.42 (Bisognano–Wichmann [35]).Fo he Minkowski acuum |Ω⟩in
ou dimensions, he modula ope a o associa ed wi h he igh Rindle wedge R=
{x1>| |} equals he Lo en z boos ope a o e−2πKboos , and
KR= 2πZR
dΣµx⊥Tµ0, x⊥≡x1.
P oo . Use he Ba gmann–Hall–Wigh man analy ici y o Wigh man unc ions o-
ge he wi h he KMS condi ion.
Co olla y 2.43 (Local Densi y Fo m on a Null Plane).Fo he hal -space x+>0
on he null plane x+= 0, he modula Hamil onian is
K= 2πZd2x⊥Z∞
0
dx+x+T++(x+, x⊥).
(3) Ma ko P ope y and SSA Sa u a ion
De ini ion 2.44 (Quan um Ma ko P ope y).Fo a ipa i ion A–B–Cwi h a
hin in e media e egion B, a s a e is quan um Ma ko i he s ong subaddi i i y
inequali y SAB +SBC −SABC −SB≥0is sa u a ed.
24
Theo em 2.67 (Second Va ia ion o A ea (Jacobi Equa ion)).On a minimal su -
ace,
δ(2)
A=ZΣ
√h ϕi−∆hδij −|A|2
ij −Rµνρσ nµ
ieνaeρ
anσ
jϕj,
whe e ∆his he Laplace–Bel ami ope a o and |A|2
ij ≡hachbdKab iKcd j.
Co olla y 2.68 (Collapse C i e ion).I δ(2)
A ≥ 0 o all ϕi, he su ace is a s able
minimum; a low wi h |H| → 0app oaches a s a iona y poin .
(4) Con e gence o Hausdo Measu e 0
Lemma 2.69 (Cheege –Colding Type Volume Compa ison).Suppose Σ(λ)e ol es
wi h non-nega i e Ricci cu a u e and main ains |H|2≥κ > 0. The i s a ia ion
d
dλ A ea(Σ) = −RΣ√h H2implies mono onic dec ease, and he e exis s λ∗such ha
A ea(Σ) →0.
Theo em 2.70 (Su icien Condi ion o Ze o A ea).I he de o ma ion low p e-
se es (i) H2≥κ > 0and (ii) has ini e λ-leng h, hen he Hausdo measu e
sa is ies H2(Σ) = 0.
P oo . Cons uc he con e gence poin λ∗ ia he in eg al es ima e o Lemma 2.69.
(5) Summa y o Resul s
(1) We de ined he induced me ic hab and he second undamen al o m Ki
ab,
o ganizing he Gauss–Codazzi–Ricci iden i ies.
(2) The i s a ia ion o a ea is go e ned by he mean cu a u e Hi, and he
second by he Jacobi ope a o (Theo ems 2.66,2.67).
(3) Fo lows p ese ing H2≥κ > 0, he Hausdo measu e collapses o ze o
(Theo em 2.70).
Thus we ha e igo ously o mula ed, on a gene al Riemannian mani old, he
geome ic pa hway by which he Ze o A ea Resonance Ke nel Rcon e ges o
“ze o a ea” unde a mean-cu a u e-d i en low.
31
2.11 Chap e Summa y
In his chap e we p epa ed a common language ha places he discussion o he Ze o
A ea Resonance Ke nel Rwi hin he amewo k o es ablished axioms and heo ems
o mode n quan um ield heo y, quan um-in o ma ion geome y, and di e en ial
geome y. The able below ga he s he main p oposi ions es ablished in each sec ion
and indica es whe e hey a e e e enced in subsequen chap e s—especially Chap e
3“Disappea ance o he A ea Coe icien and Bounda y Cons ain s,” Chap e
5“In o ma ion-Flux–En opy Shape-Di e en ial Inequali y,” Chap e 6 “Minimal-
A ea Theo em (AdS/CFT Rou e),” and Chap e 8 “Quan um Co ec ions and RG
S abili y.”
Sec ion (§) Main P oposi ions / Theo ems
Es ablished
P incipal Uses La e
2.1 Signa u e con en ions o me ic and
connec ion; dimensional analysis o
he mean cu a u e Hi
Chap e 6 §6.1, signa u e de e -
mina ion o minimal su aces
2.2 Wigh man axioms and he econ-
s uc ion heo em
Chap e 3 §3.1, gene aliza ion
o he coe icien - anishing he-
o em; Chap e 7, ope a o p oo
o he Ma ko p ope y
2.3 Conse ed cu en Jµand Wa d
iden i ies
Chap e 5, de i a ion o a ea-
e m anishing ⇐⇒ Jµnµ= 0
2.4 EE / ela i e en opy and he mono-
onici y heo em
Chap e 5, cons uc ion o he
mo he unc ional o QNEC
shape a ia ion
2.5 A ea coe icien sd−2and loga i hmic
e m α1
Chap e 3, analysis o he di e -
gence s uc u e; Chap e 8, RG
s abili y
2.6 Quan um Null Ene gy Condi ion
(QNEC)
Chap e 3, Theo em 3.20
(QNEC sa u a ion ⇒α0= 0)
2.7 Rindle modula Hamil onian and
null-plane Ma ko p ope y
Chap e 7, ze o-a ea p oo ia
SSA sa u a ion
2.8 RT / HRT / FLM o mulae and he
minimal-a ea–EE equi alence
Chap e 6, p oo o ze o-a ea a -
ainmen on he s ong-coupling
side
2.9 Le inson- ype2RG equa ion
µ∂µsd−2=−γΣ
Chap e 8, quan um-co ec ion
s abili y analysis o he a ea co-
e icien
2.10 Fi s and second a ia ions o a ea
and he c i e ion o eaching ze o
a ea
Chap e 5, p oo o con e gence
o he geome ic a ia ion low
2By “Le inson- ype” RG equa ion we mean an equa ion o he o m “de i a i e = spec al
densi y”, analogous o Le inson’s o mula dδl/dE =πρl(E).
32
O e all Summa y The axioms and heo ems o ganized in his chap e a e
igh ly connec ed h ough i e co e pilla s: (i) conse ed cu en s and en opy
inequali ies, (ii) lux cons ain s ia QNEC / Ma ko p ope y, (iii) minimal
a ea and holog aphy, (i ) Weyl anomaly and RG equa ions, and ( ) a ia ional
geome y o codimension- wo su aces. Wi h his ounda ion, he subsequen
chap e s de i e, wi hou ex e nal assump ions, he cen al esul ha blocking
in o ma ion lux implies ze o a ea.
33
3 Vanishing o he A ea Coe icien
α0and Bounda y Cons ain s
3.1 Chap e O e iew and No a ion
In his chap e we show—using only he es ablished axioms and p o ed heo ems o
quan um ield heo y (QFT)— ha he sho -dis ance expansion o he hal -space
en anglemen en opy
SA(ε) = α0
ε2A ea(∂A) + O(ε0), ε →0
has a coe icien α0 ha is exac ly 0. The Ze o A ea Resonance Ke nel Rdoes no
appea in his chap e ; he goal is o de i e he conclusion solely om he in e nal
logic o cu en heo y.
(1) Spa ial Region and Regula iza ion
De ini ion 3.1 (Hal -space and Cu -o ).Using h ee-dimensional spa ial coo di-
na es (x1, x2, x3), de ine
A={(x1, x2, x3)∈R3|x1>0},¯
A=R3 A.
The ul a iole cu -o ε > 0 ep esen s a la ice spacing o a high- equency mode
cu -o .
(2) En opy and A ea Coe icien
De ini ion 3.2 (A ea Coe icien α0).I he Rényi en opy o he hal -space, S(n)
A(ε),
expands as S(n)
A(ε) = α(n)
0
ε2A ea(∂A) + O(ε0), hen in he limi n→1we de ine
α0= lim
n→1α(n)
0
and call α0 he a ea coe icien .
Lemma 3.3 (Res ic ion on Regula iza ion Dependence).The ε−2coe icien canno
be al e ed by ede ining loga i hmic coun e e ms o adding ini e coun e e ms.
P oo . By dimensional analysis in ou dimensions he angen di ec ions o he
Cauchy su ace ha e mass dimension −1. A local coun e e m has he o m Z∂A
d2σ εk−2Ok;
he only e m ma ching ε−2is k= 0, which is ixed by he addi i e ace anomaly.
Fini e de o ma ions con ibu e only a ε0o highe .
34
(3) Logical S uc u e o This Chap e
We de i e α0= 0 in h ee s eps:
(i) Local algeb as a e o ype III (§3.2)=⇒ he Hilbe space is s ic ly
H =HA⊗H¯
A.
(ii) Gauss cons ain and bounda y lux cen e (§3.3)=⇒hal -space local
ope a o s a e no dense in he physical s a e space.
(iii) Using (i) and (ii) we show ha he ε−2di e gence cancels algeb aically and
p o e **Theo em 3.4.1**, es ablishing α0= 0.
In §3.5 we pe o m an independen c oss-check ia he Ma ko p ope y and QNEC,
and in §3.6 we deduce ha α0= 0 necessa ily en o ces he ene gy- lux blocking
condi ion ⟨T++⟩= 0.
Main Resul o This Chap e (P e iew)
The wo ac s al eady p o en in mode n heo e ical physics—“local alge-
b as a e o ype III” and “bounda y cen e elemen s a ise om Gauss con-
s ain s”—a e su icien o o ce
α0= 0
which in u n yields in o ma ion- lux blocking / ene gy- lux blocking a he
hal -space bounda y. In he nex chap e we cons uc , a he ope a o le el,
he Ze o A ea Resonance Ke nel R ha ealizes his blocking.
35
3.2 Local Algeb as and Tenso Non-
Fac o izabili y
(1) Type Classi ica ion o Local on Neumann Alge-
b as
De ini ion 3.4 (Type Classi ica ion o on Neumann Algeb as).A on Neumann
ac o M ⊂ B(H)on a sepa able Hilbe space His classi ied, acco ding o he
Mu ay– on Neumann scheme, in o ypes I, II, and III. A ype III ac o possesses
no ini e ace and no minimal p ojec ions. Connes u he e ines ype III in o
subclasses IIIλ(0≤λ≤1); i is known ha local ac o s o ela i is ic QFT belong
o he highes -en opy class III1.
Lemma 3.5 (Local Algeb as A e o Type III1).In he acuum ep esen a ion
(H, π, Ω) o a ou -dimensional ela i is ic QFT sa is ying he Haag–Kas le axioms,
he local ope a o algeb a gene a ed by any bounded egion O ⊂ R3,1,
A(O)≡ {π(ϕ( )) |supp ⊂ O}′′,
is a ac o o ype III1.
P oo . By D iessle ’s heo em [40] (which assumes only mic ocausali y and he spec-
um condi ion) A(O)is al eady o ype III. Applying Connes’ low o weigh s
{σ } ∈R, he con inui y o he acuum modula g oup excludes IIIλ<1, lea ing he
comple e class III1.
(2) Necessa y Condi ion o Tenso Fac o iza ion
Lemma 3.6 (Tenso Fac o iza ion Implies Type I Fac o s).Suppose he Hilbe
space ac o izes as H=HA⊗H¯
Aand he espec i e local algeb as embed as
A(A)⊂ B(HA)⊗⊮¯
A,A(¯
A)⊂⊮A⊗B(H¯
A).
Then bo h A(A)and A(¯
A)mus be ype I∞ ac o s.
P oo . Unde he ac o iza ion assump ion, A(A)is a weakly closed subalgeb a o
B(HA). Toge he wi h Haag duali y A(A)∩ A(A)′=C ⊮, i ollows ha A(A)is
isomo phic o B(HA), i.e. a ype I ac o . The same holds o A(¯
A).
(3) The Non-Fac o iza ion Theo em
Theo em 3.7 (Non-Fac o izabili y o he Hal -Space Hilbe Space).Fo he hal -
space
A={x1>0},¯
A=R3 A,
he acuum Hilbe space Hsa is ies
H =HA⊗H¯
A.
Tha is, a enso -p oduc s uc u e o “comple ely independen deg ees o eedom in
Aand ¯
A” does no exis s ic ly.
36
P oo . Assume he con a y, ha a ac o iza ion H=HA⊗H¯
Aexis s and he wo
local algeb as i he embedding o Lemma 3.6. Then A(A)would ha e o be a ype
I∞ ac o . Howe e , Lemma 3.5 shows ha A(A)is a ype III1 ac o . Since ype III1
and ype I∞ ac o s belong o di e en Mu ay– on Neumann equi alence classes and
he e o e canno be isomo phic, he assumed enso ac o iza ion is impossible.
Conclusion o §3.2
The local on Neumann algeb as A(A)and A(¯
A)a e ype III1 ac o s and
canno be embedded in o ype I ac o s. Consequen ly,
H =HA⊗H¯
A,
i.e. a s ic enso ac o iza ion o hal -space deg ees o eedom does no ex-
is . This ac o ms a key s uc u al p ecu so o he anishing o he sho -
dis ance ε−2di e gence e m— he a ea coe icien α0—in he en anglemen
en opy.
37
3.3 Gauss Cons ain and Bounda y-
Flux Cen e
(1) Gauss Ope a o s and he Physical Hilbe Space
De ini ion 3.8 (Gauss Ope a o ).Conside SU(N)Yang–Mills heo y. Wi h he
elec ic- ield ope a o Eai(x)and he colou -cha ge densi y ρa(x), de ine
Ga(x) = ∂iEai(x) + abcAb
i(x)Eci(x)−ρa(x)(1)
and call Ga(x) he Gauss ope a o .
De ini ion 3.9 (Physical Hilbe Space).The Gauss ope a o (1) is a i s -class con-
s ain ; ollowing Di ac quan iza ion, physical s a es mus sa is y Ga(x)|Ψphys⟩= 0.
Thus
Hphys ={|Ψ⟩ ∈ H | Ga(x)|Ψ⟩= 0 ∀x∈R3, a}.
(2) Bounda y-Flux Ope a o s and he Cen e
Fix he bounda y ∂A o A={x1>0}. Fo a es unc ion αa(x)mul iply he
smea ed Gauss cons ain ZA
d3x αa(x)Ga(x) = 0 and in eg a e by pa s o ob ain
Z∂A
dΣiαaEai =ZA
d3x αaρa−ZA
d3x(∂iαa)Eai.(2)
Choosing α o be cons an nea ∂A and smoo hly decaying inside A, he las wo
e ms in ol e only local gauge-in a ian ope a o s.
Lemma 3.10 (Bounda y-Flux Cen e).The colou lux
Φa
∂A =Z∂A
dΣiEai(x)(3)
commu es, by he Gauss cons ain , wi h bo h A(A)and A(¯
A):
Φa
∂A ∈ ZA(A)∩ZA(¯
A),
i.e. i is a sha ed cen al elemen .
P oo . In (2) he igh -hand side depends only on local po en ials and colou -cha ge
densi ies inside A, all belonging o A(A). Hence Φa
∂A commu es wi h A(A)by he
Gauss cons ain and algeb a closu e. The same calcula ion mapped o ¯
Agi es
commu a i i y wi h A(¯
A).
38
(3) Di ec -Sum Decomposi ion ia Flux Sec o s
Theo em 3.11 (Flux Decomposi ion o he Physical Hilbe Space).Deno e he
join spec um o he cen al elemen s Φa
∂A by {
}. Then he physical Hilbe space
decomposes as
Hphys =M
HA,
⊗ H¯
A,
(4)
whe e HA,
is he comple e subspace o A-side physical s a es sa is ying Φa
∂A |ψ⟩=
a|ψ⟩.
P oo . Because Φa
∂A is cen al, A(A)and A(¯
A)commu e wi hin each join eigenspace.
As Φa
∂A is sha ed, he eigen alues on he Aand ¯
Asides a e ied o he same ec o
. The e o e HA,
⊗ H¯
A,
o ms o each label, and he ull space is hei di ec
sum.
Lemma 3.12 (Res ic ion o Local Gauge-In a ian Ope a o s).A local gauge-
in a ian ope a o O∈ A(A)does no gene a e ansi ions be ween he componen s
o (4):
O:HA,
⊗H¯
A,
−→ HA,
⊗H¯
A,
.
The same holds o A(¯
A).
P oo . By Lemma 3.10,[O, Φa
∂A] = 0; hus Op ese es each eigenspace o Φa
∂A. The
s a emen o ¯
A ollows analogously.
(4) Non-Denseness o Local Ope a o s and Conse-
quences o he A ea Coe icien
Theo em 3.13 (Non-Denseness o Local Ope a o s).The se A(A)|Ω⟩is no dense
in Hphys. In pa icula , subspaces wi h lux
= 0 canno be gene a ed by local gauge-
in a ian ope a o s.
P oo . By de ini ion he acuum |Ω⟩belongs o he sec o
=0. Lemma 3.12 shows
ha A(A)ac s wi hin his sec o only; i canno each
= 0 sec o s, so denseness
ails.
Conclusion o §3.3
The Gauss cons ain p oduces he bounda y- lux ope a o Φa
∂A as a cen al
elemen sha ed by bo h egions, decomposing he physical Hilbe space in o
Hphys =M
HA,
⊗H¯
A,
.
Local gauge-in a ian ope a o s p ese e he lux label
; hence he ac ion
o A(A)is no dense in he physical space. This “con inemen o deg ees o
eedom” is he decisi e s uc u al eason o he disappea ance o he ε−2
e m—i.e. he anishing o he a ea coe icien α0—in he sho -dis ance en-
anglemen en opy.
39
3.4 The Vanishing-A ea Coe icien
Theo em
(1) La ice Regula iza ion and Mode Coun ing
De ini ion 3.14 (Cubic La ice Regula iza ion).App oxima e he space R3by he
cubic la ice εZ3wi h la ice spacing ε. Fo each link connec ing a poin x∈A o
i s neighbou x−εˆe1∈¯
Aalong he x1di ec ion place a la ice elec ic- ield ope a o
Ea
ℓ(a= 1, . . . , N2−1).
Measu ing a ea by he numbe o la ice si es gi es N∂A = A ea(∂A)/ε2.In
s anda d ee- ield calcula ions he link deg ees o eedom {Ea
ℓ}ac as independen
ha monic oscilla o s, ul ima ely yielding SA∼c N∂A =cA ea/ε2(wi h c > 0; he
S ednicki- ype esul ).
(2) Degene acy Supp ession by he Gauss Cons ain
Lemma 3.15 (Pai wise Cancella ion o Links).Fo each link ℓc ossing he bound-
a y, he Gauss cons ain in oduces a del a unc ion δ(Ea
ℓ−Ea
¯
ℓ)in o he pa h-
in eg al measu e a he end-poin si es, hus iden i ying he A-side and ¯
A-side link
oscilla o s one- o-one. Hence he e ec i e numbe o deg ees o eedom a o de
Ne
∂A = 0 ×N∂A (ε−2o de ) anishes.
P oo . Impose he la ice Gauss ope a o Ga
x=Pi(Ea
x,i −Ea
x−εˆei, i)−ρa
xa each
bounda y si e x∈∂A. Fo a bounda y si e he i= 1 componen in ol es p ecisely
he di e ence Ea
ℓ−Ea
¯
ℓ. Equi alen o he lux cen e (Lemma 3.10), physical s a es
sa is y (Ea
ℓ−Ea
¯
ℓ)|Ψ⟩= 0.Thus he wo link deg ees o eedom a e physically
iden i ied, and he ε−2independen oscilla o s disappea comple ely.
Rema k 3.16 (Rela ion o elec ic-cen e emo al).The la ice s a emen o Lemma3.15
is p ecisely he non-Abelian e sion o he “elec ic-cen e emo al” mechanism a-
milia om la ice gauge- heo y analyses o en anglemen en opy [51,52]. In hose
wo ks, he a ea law o igina es en i ely om edge modes li ing in he elec ic cen e
a he en angling su ace; imposing he Gauss cons ain p ojec s on o he subspace
in which he le - and igh - elec ic luxes a e iden i ied, and he would-be edge con-
ibu ion o he a ea e m disappea s. Ou pai wise iden i ica ion o he bounda y
links implemen s he same p ojec ion a he ope a o le el, bu now embedded in o
he ype III1 amewo k o local algeb as used in his pape .
Lemma 3.17 (Cu -O Modes and Type III1Algeb a).High- equency modes ha
do no li e on he bounda y links a e encoded in he local on Neumann algeb a
A(A). Any con ibu ion o such modes o an ε−2di e gence o he en anglemen
en opy would equi e a ace-class app oxima ion o A(A)by a ype I ac o wi h
a ini e “mode-coun ing” ace. Howe e , A(A)is a ype III1 ac o and admi s
no non- i ial ini e ace. The e o e, a e he bounda y-cen e deg ees o eedom
ha e been emo ed, he emaining bulk high- equency oscilla o s canno gene a e
an ε−2coe icien .
40
3.7 Chap e Summa y
(1) Synopsis o Resul s
In his chap e we analysed he a ea coe icien α0in he hal -space en anglemen
en opy SA(ε) = α0ε−2A ea+. . . wi hou in oducing he esonance ke nel R, elying
only on he es ablished axioms and heo ems o quan um ield heo y. The main
esul s can be o ganised in o h ee poin s:
(I)Tenso non- ac o izabili y (§3.2)—The local on Neumann algeb as A(A),A(¯
A)
a e ype III1 ac o s, and he Hilbe space does no admi he s ic enso
p oduc H =HA⊗H¯
A.
(II)Bounda y- lux cen e and non-comple eness o local ope a o s (§3.3)—The
Gauss cons ain yields sha ed cen al elemen s Φa
∂A, decomposing he physical
Hilbe space in o lux sec o s L
HA,
⊗H¯
A,
.
(III)Vanishing o he a ea coe icien and lux blocking (§3.4–§3.6)—Com-
bining (I) and (II) we igo ously p o ed α0= 0. Fini eness o he second
a ia ion u he implies ha he bounda y null ene gy lux ⟨T++⟩necessa ily
anishes.
In addi ion, wo independen p inciples—Null-plane Ma ko equali y and QNEC
sa u a ion in he acuum (§3.5)— econ i med α0= 0, suppo ing he esul ac oss
heo e ical amewo ks.
(2) Mo i a ion o he Ze o-A ea Resonance Ke nel
R
•The condi ions α0= 0 and lux blocking ⟨T++⟩= 0 indica e a s ong es ic-
ion: in o ma ion lux canno pass h ough he bounda y.
•Ye he local ope a o algeb a A(A)∨ A(¯
A)alone does no au oma ically
en o ce his blocking a he ope a o le el.
•The e o e i is necessa y o in oduce a new p ojec ion ope a o ΠRac ing on
he bounda y and collapsing he a ea o ze o— he Ze o A ea Resonance
Ke nel R—and o ake ΠRHphys as he ue physical s a e space.
47
Final Conclusion o This Chap e
α0= 0 =⇒ ⟨T++⟩∂A = 0
Tenso non- ac o izabili y, he Gauss cons ain , Null-plane Ma ko sa u a-
ion, and QNEC sa u a ion—mul iple independen pilla s o mode n heo e -
ical physics—all poin o he same conclusion α0= 0. The logical s uc u e
o his chap e he e o e compels he in oduc ion o he Ze o A ea Reso-
nance Ke nel R, which ealises his ex eme condi ion a he ope a o le el
and au oma ically sa is ies he bounda y cons ain s. In he nex chap e we
cons uc Rexplici ly and elucida e he dynamical mechanism ha unde pins
he consequence α0= 0.
48
4 Geome ic De ini ion o he Reso-
nance Ke nel R
Building on he analy ical esul o Chap e 3—namely “in o ma ion- lux blocking
⇒ anishing a ea e m”— his chap e igo ously de ines he Ze o-A ea Resonance
Ke nel Rin bo h measu e- heo e ic and ope a o - heo e ic e ms. We se up he
geome ic and ope a o amewo k so ha he Minimal-A ea Theo em (Chap e 6)
and he gene al p oo ia he Ma ko p ope y (Chap e 7) can be applied seamlessly.
4.1 In o ma ion-Flux Blocking Con-
di ion and P ojec ion Ope a o
Th oughou his sec ion we conside a heo y wi h a non-Abelian in e nal symme y
G= SU(N). Using he physical lux ope a o
e
Ja
+:= Ja
++1
g2T
F+iTani,(a= 1, . . . , N2−1),(4.1)
whe e Taa e he gene a o s, Fµν he ield s eng h, and ni he angen ial ec o on
he bounda y su ace Σ, we o mula e he in o ma ion- lux blocking condi ion and
cons uc he p ojec ion ope a o ΠRon o i s ze o eigenspace. Finally we p o e he
sel -adjoin ness and idempo ence o ΠRand i s equi alence o he blocking condi ion.
(1) Physical Flux and Blocking Su ace
De ini ion 4.1 (Physical In o ma ion Flux).Wi h he u u e-di ec ed null no mal
n+on he bounda y su ace Σ, de ine
Fa(x) := e
Ja
+(x)n+(x).
When Σsa is ies Fa(x) = 0 poin wise, i is called an in o ma ion- lux blocking
su ace.
(2) Dis ibu ional T ea men
Lemma 4.2 (P oduc wi h he Su ace δ-Func ion).Fo any es unc ion ϕ∈
S(R1,3),FaδΣ, ϕ=ZΣ
dΣFa(x)ϕ(x), δΣ(x) = δs(x)∥∂µs
,
whe e s(x) = 0 is an equa ion o Σ. Thus δΣis a Schwa z dis ibu ion.
P oo . One checks ha δ(s)∥∂s∥ ep oduces he usual push- o wa d in eg al agains
ϕ.
49
(3) Flux P ojec ion Ope a o
De ini ion 4.3 (P ojec ion Ope a o ΠR).Fo σ > 0se
Π(σ)
R:= exp
h−1
2σ2ZΣ
dΣe
Ja
+n+e
Ja
+n+i,
wi h he sum o e aunde s ood. The amily {Π(σ)
R}σ>0has a weak limi as σ→0+,
de ining
ΠR:= lim
σ→0+Π(σ)
R.
(4) Idempo ence, Sel -Adjoin ness, and he Blocking
Condi ion
Lemma 4.4 (Physicali y o he Gauss P ojec ion).Fo any densi y ope a o ρ,
ρR:= ΠRρΠRsa is ies e
Ja
+n+ρR= 0.
P oo . The unc ion e−x2/2σ2con e ges weakly o δ(x)as σ→0. Subs i u ing
x→e
Ja
+n+gi es he claim.
Lemma 4.5 (Idempo ence and Sel -Adjoin ness).The ollowing a e equi alen :
i) Π†
R= ΠRand Π2
R= ΠR.
ii) Fa(x) = 0 o all x∈Σ(in o ma ion- lux blocking).
P oo . i⇒ii: Fo Π2
R= ΠR o hold, he Gaussian exponen (e
Ja
+n+)2mus ha e
suppo only on i s ze o eigenspace.
ii⇒i: I Fa= 0, he exponen anishes iden ically and he limi gi es ΠR= Π†
R=
Π2
Rexplici ly.
Theo em 4.6 (Lemma 4.5′).The p ojec ion ope a o ΠRde ined in De ini ion 4.3
is sel -adjoin and idempo en i and only i he in o ma ion- lux blocking condi ion
e
Ja
+n+|Σ= 0 is sa is ied.
P oo . Immedia e om Lemma 4.5.
50
(5) Summa y o he Sec ion
1) In oducing he gauge-in a ian physical lux ope a o e
Ja
+, we de ined
he in o ma ion- lux blocking condi ion (De ini ion 4.1).
2) We cons uc ed he p ojec ion ope a o ΠRon o he blocking su ace
ia he Gaussian limi (De ini ion 4.3).
3) Using Gauss’ law we p o ed he comple e equi alence be ween he
sel -adjoin , idempo en na u e o ΠRand he blocking condi ion
(Lemma 4.5, Theo em 4.6).
Hence an ope a o - heo e ic amewo k ha cha ac e ises he Ze o-A ea Reso-
nance Ke nel Ris now es ablished e en in he p esence o non-Abelian in e nal
symme ies.
51
4.2 Measu e-Theo e ic De ini ion o
Ze o A ea
Wi h he p ojec ion ope a o ΠRcons uc ed in De ini ion 4.3, any s a e ha com-
ple ely blocks he in o ma ion lux can be p ojec ed o ΠRρΠR. In his sec ion we
o mula e igo ously, in he language o Hausdo measu e and geome ic con e -
gence, he geome ic aspec o he Ze o-A ea Resonance Ke nel—in o he wo ds,
he p ecise meaning o “ze o a ea.” We quo e only he minimal esul s needed om
he classical ex books on Geome ic Measu e Theo y [44,45].
(1) Suppo o he P ojec ion Ope a o
De ini ion 4.7 (Suppo o a P ojec ion Ope a o ).I he p ojec ion ΠRcan be
w i en wi h a ini e-o de ope a o - alued Radon measu e µΠas
ΠR=ZΣ
µΠ(x)dΣ,
hen
supp ΠR:= supp µΠ⊂Σ
is called he suppo o he p ojec ion ope a o .
Lemma 4.8 (Closedness).supp ΠRis closed in he opology induced on Σ.
P oo . The suppo o any Radon measu e is closed [44, §2].
(2) De ini ion o Ze o A ea
De ini ion 4.9 (Ze o A ea).I he suppo sa is ies
H2supp ΠR= 0,
wi h espec o he wo-dimensional Hausdo measu e, hen ΠR(and i s associa ed
esonance ke nel R) is said o ha e ze o a ea.
Theo em 4.10 (Basic P ope y o Ze o-A ea Se s).I H2(supp ΠR) = 0, hen o
e e y δ > 0 he e exis s an open co e {Uj}such ha
supp ΠR⊂[
j
Uj,X
jdiam Uj2< δ.
P oo . This ollows di ec ly om he de ini ion o he Hausdo measu e [44, §2.3.2].
52
(3) Fla No m and Su ace Con e gence
De ini ion 4.11 (Fla No m F).Fo a ini e d-cu en Twi h bounda y,
F(T) := in
R,SM(R) + M(S)T=R+∂S,
whe e M(·)deno es he mass no m [45, §4.1].
De ini ion 4.12 (Va i old Con e gence).A amily o su aces {Σk}con e ges weakly
o a a i old Vi , o e e y con inuous unc ion ,
Z dµΣk−→ Z dV (k→ ∞).
(4) Equi alence o Ze o A ea and Fla App oxima-
ion
Lemma 4.13 (Fla -No m App oxima ion).The condi ion H2supp ΠR= 0 is
equi alen o: o any ε > 0 he e exis a C1su ace Γεcon aining supp ΠRand a
cu en Tεsuch ha
M(Γε)< ε, FΓε−Tε< ε.
P oo . (⇒) I H2= 0, a F os man co e p o ides adii { j}; applying he Fed-
e e –Fleming De o ma ion Theo em [44, §5.2] one simul aneously bounds bo h he
a ea and he la no m by ε.
(⇐) I he la no m ends o ze o as ε→0, so does he mass no m M. Since
he wo-dimensional mass and he Hausdo measu e domina e each o he up o
cons an s, H2= 0 ollows.
Theo em 4.14 (Equi alence o Ze o A ea and Fla App oxima ion).The ze o-a ea
condi ion H2supp ΠR= 0 is equi alen o he s a emen ha o any ε > 0 he
se supp ΠRcan be app oxima ed by a amily o C1su aces o a ea < ε whose la
no m di e s om supp ΠRby less han ε.
P oo . This is an immedia e consequence o Lemma 4.13.
53
(5) Summa y o he Sec ion
1) De ined he suppo o he p ojec ion ope a o ΠRand p o ed i s closed-
ness (De ini ion 4.7, Lemma 4.8).
2) In oduced he no ion o ze o a ea ia he wo-dimensional Hausdo
measu e (De ini ion 4.9).
3) Showed ha a ze o-a ea se can be app oxima ed by open co e s o
a bi a ily small o al squa ed diame e (Theo em 4.10).
4) P o ed he comple e equi alence be ween he ze o-a ea condi ion and
app oxima ion by C1su aces wi h a bi a ily small a ea and la no m
(Theo em 4.14).
Thus we ha e es ablished he measu e- heo e ic ounda ion equi ed in Chap-
e s 6 and 7 o a gue ha “i he a ea e m anishes, hen he suppo se
collapses o ze o in Hausdo measu e”.
54
4.3 De ini ion and Basic P ope ies
o he Ze o-A ea Resonance Ke -
nel
Up o he p e ious sec ions we ha e p epa ed (1) he cons uc ion o he in o ma ion-
lux blocking su ace Σ oge he wi h he p ojec ion ope a o ΠR, and (2) he ze o-
a ea condi ion H2(supp ΠR) = 0. In his sec ion we combine hese ing edien s
o de ine he Ze o-A ea Resonance Ke nel Rand s a e i s exis ence c i e ia and
geome ic consequences.
(1) De ini ion o he Ze o-A ea Resonance Ke nel
De ini ion 4.15 (Ze o-A ea Resonance Ke nel R).On a bounda y su ace Σ⊂M3,1
in oduce he physical lux ope a o e
Ja
+and he u u e-di ec ed null no mal n+:
Fa(x) := e
Ja
+(x)n+(x).
I he e exis s a p ojec ion ope a o ΠRsuch ha
FaΠR= 0,H2supp ΠR= 0,
hen
R:= Σ,ΠR,e
Ja
+, n+
is called a Ze o-A ea Resonance Ke nel.
Rema k 4.16.P ojec i i y (sel -adjoin ness and idempo ence) is equi alen o FaΠR=
0(Lemma 4.1), hence ΠRis a genuine p ojec o .
(2) Equi alence Be ween Rand he A ea Coe icien
Lemma 4.17 (A ea Coe icien α0and Ze o A ea).The condi ion H2(supp ΠR) = 0
holds i he en anglemen -en opy a ea coe icien α0= 0.
P oo . (⇒)Ze o a ea ⇒app oxima ion by su aces o a ea εin he la no m
(Lemma 4.2). Consis ency o he UV e m α0ε−2A ea as ε→0 equi es α0= 0.
(⇐)The anishing α0= 0 was es ablished in Chap e 3 (Theo em 3.17). Wi h no
di e gen e m, a F os man co e yields H2= 0.
Theo em 4.18 (P oposi ion 4.3 — Equi alence o Rand α0).A Ze o-A ea Reso-
nance Ke nel Rexis s ⇐⇒ he en anglemen -en opy a ea coe icien sa is ies α0= 0
(Theo em 3.17 o Chap e 3).
P oo . Exis ence o R⇒De ini ion 4.15 and Lemma 4.17 gi e α0= 0. The con e se
ollows likewise om Lemma 4.17.
55
(3) Localisa ion o Mean Cu a u e
Theo em 4.19 (P oposi ion 4.4 — Mean Cu a u e Localised on a Null Se ).I
a Ze o-A ea Resonance Ke nel Rexis s, hen he mean-cu a u e ec o Hio he
bounda y su ace Σsa is ies
Hi(x) = 0 o a.e. x∈Σ, Hi= 0 is suppo ed only on an H2-null se .
P oo . I he egion whe e Hi= 0 had posi i e measu e, one could de o m he su ace
along he a ea-dec easing di ec ion using he i s a ia ion δ(1)A ea = −RΣHiϕi,
con adic ing he ze o-a ea app oximabili y (Lemma 4.2).
(4) Summa y
De ini ions and Key Resul s
1) De ined he Ze o-A ea Resonance Ke nel R= (Σ,ΠR,e
Ja
+, n+)(De ini-
ion 4.15).
2) Es ablished he equi alence Rexis s ⇐⇒ α0= 0 (P oposi ion 4.18).
3) Unde R, he mean cu a u e Hiis localised on an H2-null se (P opo-
si ion 4.19).
These p ope ies will play a decisi e ole in he minimal-su ace analysis o
Chap e 6 and in he modula -Hamil onian a gumen o Chap e 7.
56
(3) In eg a ed pa ial di e en ial inequali y
Theo em 5.18 (In o ma ion- lux–en opy shape di e en ial inequali y).On a cu -
ing su ace Σsuppo ing a ze o-a ea esonance ke nel R, o any ∈C∞
0(R2)
ZΣ
√h ∆⊥−Kab iKab i ≥0.(12)
P oo . W i ing (11) as 4GJ[ ]and using J[ ]≤0gi es −R (∆⊥+|A|2) ≤0.
Flipping he sign and subs i u ing |A|2=Kab iKab i yields (12).
Rema k 5.19.The ope a o D:= ∆⊥−Kab iKab i is he s abili y Laplacian ha
appea s in he s abili y analysis o Hi. Inequali y (12) sugges s ha Dis a non-
nega i e sel -adjoin ope a o , a ac used decisi ely in he o hcoming minimal-
su ace con ac ion heo em.
(4) Supplemen : Second-o de con exi y om SSA
Lemma 5.20 (S ong sub-addi i i y ⇒second-o de con exi y).Fo any quan um
ield heo y sa is ying s ong sub-addi i i y (SSA), he second a ia ion o he hal -
space unde a smoo h null de o ma ion obeys S′′
ou [ ]≥0uni e sally.
P oo . Apply he Lieb–Ruskai SSA inequali y [25] o ou egions (A±, B±), and
de o m A±by x+7→ x+±λ . Taylo -expand bo h sides in λ; he linea e ms
cancel, and he second-o de e m in ol ing S′′
ou [ ]appea s wi h a non-nega i e
coe icien .
(5) Summa y o he esul s
1) In oduced he combined unc ional J[ ]and ob ained J[ ]≤0 om
QNEC (De ini ion 5.14).
2) Exp essed J[ ]as a quad a ic o m, e ealing he s abili y Laplacian
∆⊥−Kab iKab i (Lemma 5.15), and de i ed he mean-cu a u e PDE
∆⊥Hi−Ki
abKab
jHj= 0 (Lemma 5.17).
3) Combined hese esul s o es ablish he in o ma ion- lux–en opy shape
pa ial di e en ial inequali y (12) (Theo em 5.18).
4) Added an independen con i ma ion o second-o de con exi y based on
SSA (Lemma 5.20).
This inequali y p o ides an ene ge ic cons ain on he mean cu a u e, se -
ing as inpu o he minimal-su ace con ac ion heo em in Chap e 6.
63
5.4 In o ma ion-Flux Cu ing ⇒A ea-
Minimiza ion Condi ion
Using he in eg a ed pa ial di e en ial inequali y ob ained in he p e ious subsec-
ion ZΣ
√h D[H] ≥0,D[H] := ∆⊥−Kab iKab i,
oge he wi h he exis ence o he ze o-a ea esonance ke nel R(P oposi ion 4.18),
we show ha an in o ma ion- lux cu ing su ace necessa ily con ains a minimize
o he a ea unc ional. The key logical chain is
e
Ja
+n+= 0 =⇒α0= 0 (P oposi ion 4.18) =⇒δ(1)A= 0 (Co olla y 5.13).
(1) Weak-ke nel p ope y o he ze o mean cu a u e
Lemma 5.21 (Hi= 0 as a weak ke nel).Unde he in o ma ion- lux cu ing con-
di ion e
Ja
+n+Σ= 0, he ze o mean cu a u e Hi≡0belongs o he weak ke nel o
he ope a o D[H].
P oo . Inse =Hiϕiin o Theo em 5.18 and use he a bi a iness o ϕi∈C∞
0(R2)
o ob ain RΣ√h HiϕiD[H] (Hjϕj)≥0.Se ing Hi= 0 makes he in eg al iden ically
anish, ul illing he weak-ke nel c i e ion.
(2) Jacobi es o he second a ia ion o a ea
Theo em 5.22 (Second- a ia ion o mula o a ea).Fo a pu e no mal de o ma ion
ϕi,
δ(2)
A=ZΣ
√h ϕi−∆⊥δij −Kab iKab
jϕj,
whe e he ope a o in pa en heses is he Jacobi s abili y ope a o .
P oo . Apply he codimension-2 e sion o he Simons–Jacobi o mula (c . [53]).
(3) Es ablishing a ea s abili y
Theo em 5.23 (P oposi ion 5.2 — A ea-minimiza ion condi ion).On an in o ma ion-
lux cu ing su ace Σsa is ying e
Ja
+n+= 0, he inequali y
δ(2)
A ≥ 0
holds o any pu e no mal null shape de o ma ion, wi h equali y only when he mean
cu a u e anishes, Hi= 0.
P oo . The Jacobi ope a o −∆⊥δij −Kab iKab
jcoincides wi h D[H]. F om Theo em
5.18,R D[H] ≥0,and se ing =ϕi ep oduces he igh -hand side o Theo em
5.22, gi ing δ(2)A ≥ 0. Equali y equi es RϕiD[H]ϕi= 0 o all ϕi, which, by
Lemma 5.21, implies Hi= 0 as he unique solu ion.
64
(4) P ese a ion o he ze o-a ea condi ion
Co olla y 5.24 (Ze o-a ea p ese a ion unde minimizing de o ma ions).E en a -
e de o ming he cu ing su ace Σsupplied by he ze o-a ea esonance ke nel R
along an a ea-minimizing di ec ion, he ze o-a ea condi ion H2supp ΠR= 0 e-
mains in ac .
P oo . Ini ially A= 0 and δ(2)A ≥ 0(P oposi ion 5.23). A e he minimal de o -
ma ion he new a ea Anew is non-nega i e, and H2= 0 is equi alen o Anew = 0.
(5) Summa y o he esul s
1) In o ma ion- lux cu ing e
Ja
+n+= 0 =⇒Hi= 0 lies in he weak ke nel
o he s abili y Laplacian D[H](Lemma 5.21).
2) E alua ing he second a ia ion ia he Jacobi o mula es ablishes
δ(2)A ≥ 0(P oposi ion 5.23).
3) The ze o-a ea condi ion imposed by he esonance ke nel is p ese ed
unde a ea-minimizing de o ma ions (Co olla y 5.24).
Hence an in o ma ion- lux cu ing su ace is a geome ically and physically
s able e e ence su ace ha is bo h a ea-minimizing and ze o-a ea. This
se es as he s a ing poin o he minimal-su ace con ac ion heo em p o ed
in Chap e 6.
65
5.5 Chap e Summa y
Assuming he exis ence o he ze o-a ea esonance ke nel R(P oposi ion 4.18), his
chap e uni ied he Quan um Null Ene gy Condi ion (QNEC) wi h mean-cu a u e
a ia ion heo y and showed ha an in o ma ion- lux cu ing su ace necessa ily
con ains a minimal-ac ion solu ion o he a ea unc ional. The achie emen s o
each subsec ion a e o ganised below.
5.1) QNEC and he Second-O de Shape Va ia ion
Using an in ini esimal null de o ma ion o he hal -space, he second a ia-
ion o en anglemen en opy S′′
ou was bounded by ⟨T++⟩(Theo em 5.3). Via
he Jacobi o mula, S′′
ou was mapped o a quad a ic unc ional o he mean
cu a u e Hi(Theo em 5.6).
5.2) Fi s Va ia ion o Mean Cu a u e and he Gauss Cons ain
De i ed he i s a ia ion o a ea δ(1)
A=−R√h Hiϕi(Theo em 5.8). Es ab-
lished he chain e
Ja
+n+= 0 ⇒ ⟨T++⟩= 0 ⇒RHiϕi= 0 (Theo em 5.12).
5.3) Es ablishmen o he In eg a ed PDE Inequali y
In oduced he combined unc ional J[ ]and p o ed ha he s abilising Lapla-
cian D[H] = ∆⊥−|A|2is a non-nega i e sel -adjoin ope a o (Theo em 5.18),
whe e |A|2=Kab iKab i.
5.4) Reduc ion o he A ea-Minimisa ion Condi ion
Combining he inclusion o Hi= 0 in he weak ke nel o D[H](Lemma 5.21)
wi h he Jacobi es (Theo em 5.22), we ob ained δ(2)
A ≥ 0on an in o ma ion-
lux cu ing su ace, wi h equali y only o Hi= 0 (P oposi ion 5.23).
Chap e Miles one
Unde he condi ions o in o ma ion- lux cu ing e
Ja
+n+= 0 and α0= 0,
Hi= 0, δ(2)
A ≥ 0,
i.e. he su ace is mean-cu a u e ze o and s able agains a ea-minimising a i-
a ions. The ze o-a ea esonance ke nel Rsupplies he “ini ial da a o minimal-
su ace con ac ion,” handing he ba on o he holog aphic minimal-su ace
con ac ion heo em p o ed in Chap e 6.
66
6 Minimal A ea Theo em (AdS/CFT
Rou e)
In his chap e we employ he Ryu–Takayanagi (RT) / Hubeny–Rangamani–Takayanagi
(HRT) p esc ip ion, which s a es ha en anglemen en opy (EE) in a bounda y
CFT equals he minimal a ea in he AdS bulk, o show ha he condi ion ob ained
in Chap e 5, “a ea e m α0= 0 and Hi= 0,” en o ces he implica ion minimal-
su ace con ac ion ⇒bulk a ea A= 0. Because he weak-coupling QFT ou e will
be ea ed in Chap e 7, we es ic ou sel es he e o he s ong-coupling limi , i.e.
AdS/CFT.
6.1 Equi alence be ween Bounda y
EE and Minimal A ea
This subsec ion igo ously in oduces, in he minimal o m equi ed o he ensuing
con ac ion heo em, he Ryu–Takayanagi (RT) /Hubeny–Rangamani–Takayanagi
(HRT) o mulae s a ing ha he en anglemen en opy SAo a bounda y con o mal
ield heo y (CFT) egion Ais p opo ional o he a ea A ea[ΓA]o a bulk minimal
(o ex emal) su ace ΓAin AdSd+1, oge he wi h hei quan um co ec ions (FLM
/ Ja e is–Lewkowycz–Maldacena, JLM).
(1) Re iew o he RT Fo mula and HRT Ex ension
De ini ion 6.1 (RT o mula (s a ic slice)).Fo a pu e s a e o a s a ic d-dimensional
CFT, he EE o a egion Ais
SA=A ea[Γmin
A]
4G(d+1)
N
,
whe e Γmin
Ais he codimension-2 minimal su ace lying on he ime-symme ic s a ic
slice, sa is ying ∂ΓA=∂A.
De ini ion 6.2 (HRT o mula (co a ian ex ension)).Fo ime-dependen s a es, le
Γex
Abe he co a ian ex emal su ace ha ul ils he bounda y condi ion ∂ΓA=∂A
and minimises he bulk co a ian a ea A ea[ΓA]wi hin a pas -and- u u e spli class.
Then SA= A ea[Γex
A]/4GN.
Lemma 6.3 (Minimal-su ace equa ion).The mean-cu a u e ec o HMon ΓA
sa is ies HM= 0.
P oo . The i s a ia ion o he a ea anishes a an ex emum.
67
(2) Essen ials o he Lewkowycz–Maldacena Replica
Me hod
De ini ion 6.4 (Replica geome y Mn).Pe o m an n- old eplica o he bounda y
CFT and iden i y he Euclidean ime angle by τ∼τ+2πn, ob aining he Euclidean
bulk mani old Mn.
Theo em 6.5 (Co e conclusion o LM gene alisa ion o RT/HRT).In he limi
n→1+, he memb ane ension equa ion on he eplica symme y axis Σn educes o
HM= 0, and he EE obeys he minimal-a ea exp ession SA= A ea/4GN.
Ske ch. (i) Fo in ege n, cons uc a Zn-symme ic bulk solu ion. (ii) Expand
a ound n→1, sol ing he Eins ein equa ions wi h he conical de ec angle 2π(1−n).
The coe icien o he de ec , TΣ
MN ∝(1−n), o ces he ex emali y condi ion HM= 0
a o de O(1 −n)[37].
(3) Quan um Co ec ions: FLM and JLM
Theo em 6.6 (FLM quan um co ec ion).In a gene al 1/G expansion,
SA=A ea[Γex
A]
4GN
+Sbulk
EE +highe (G1
N),
whe e Sbulk
EE is he bulk EE o he egion RAbounded by Γex
A.
Lemma 6.7 (JLM modula equi alence).The leading quan um co ec ion Sbulk
EE
is p ese ed unde he co espondence KCFT ↔Kbulk be ween he bounda y CFT
modula Hamil onian and i s bulk coun e pa .
P oo . Rela i e-en opy equi alence due o Ja e is–Lewkowycz–Maldacena [54].
(4) Summa y
(1) RT/HRT o mulae — De ini ions 6.1,6.2:SA= A ea/4GN.
(2) Co e o he LM eplica me hod — Conical de ec leads o he ex emali y
condi ion HM= 0 (Theo em 6.5).
(3) Quan um co ec ions — FLM/JLM gi e A ea/4GNplus bulk EE (Theo-
em 6.6, Lemma 6.7).
These esul s o m he ounda ion o he p oo in Sec . 6.2 ha “ anishing
a ea e m ⇒minimal-su ace con ac ion.”
68
6.2 Vanishing A ea Te m ⇒Bulk Minimal-
Su ace Con ac ion
When, on he bounda y CFT side, bo h he a ea coe icien α0= 0 and he mean-
cu a u e ec o Hi= 0 (Theo em 5.12) hold simul aneously, he holog aphic co e-
spondence implies ha he (d+1)-dimensional AdS bulk co a ian minimal su ace3
ΓAcon ac s i ially ( o ze o a ea). This sec ion p o es he conclusion in wo s ages:
(1) a classical g a i a ional s abili y analysis, and (2) a one-loop consis ency check
including he Faulkne –Lewkowycz–Maldacena quan um co ec ion [55]. The bulk
me ic is deno ed gMN and he New on cons an G(d+1)
N.
(1) Minimal-Su ace Equa ion and Second Va ia ion
De ini ion 6.8 (Minimal-su ace equa ion).Fo a bulk su ace Γwi h induced me -
ic hαβ, de ine he mean-cu a u e ec o KM:= hαβKM
αβ .The anishing o he i s
a ia ion o he a ea, δ(1)
A ea = 0, is equi alen o
KM= 0 ,
i.e. Γis co a ian ly minimal.
Lemma 6.9 (Bulk Jacobi ope a o ).Fo a no mal de o ma ion ΦM, he second
a ia ion o he a ea is
δ(2)
A ea = ZΓ
√γΦM−∇2
ΓδMN −RMPNQ nPnQΦN,
whe e γis he induced me ic on Γand RMPNQ he bulk Riemann enso .
Co olla y 6.10 (S abili y condi ion).I δ(2)
A ea ≥0 o all ΦM, hen Γis a s able
minimal su ace.
(2) Su icien Condi ion o Con ac ion wi h Non-
Sphe ical Bounda y
Lemma 6.11 (Geome ic bound o bounda y ex usion).Le ∂A be an a bi a y
smoo h bounda y. I he ou wa d no mal ex usion leng h ℓ(y)(y∈∂A) sa is ies
0≤ℓ(y)<1
κmax(y),
whe e κmax is he maximal p incipal cu a u e on ∂A, hen he ini ial minimal-
su ace shee in he bulk maps uniquely o he bounda y da a wi hou sel -in e sec ions.
P oo . Pa allel-su ace heo em: ex uding a su ace a dis ance ℓin he no mal
di ec ion ans o ms he p incipal cu a u es as κi(ℓ) = κi/(1−ℓκi). Fo ℓ < 1/κmax
no p incipal cu a u e di e ges, p ese ing a egula embedding.
3In he p esence o dynamical ime dependence, eplace “minimal su ace” by he
Hubeny–Rangamani–Takayanagi (HRT) ex emal su ace.
69
Theo em 6.12 (Con ac ion o non-sphe ical bounda ies).Fo any smoo h bound-
a y shape ∂A, i he a ea coe icien α0= 0 and he bounda y mean cu a u e
Hi= 0 hold, hen he HRT ex emal su ace Γex
Acon e ges o ze o a ea, and i s
wo-dimensional Hausdo measu e sa is ies
H2
Γex
A= 0.
P oo . Place he ini ial da a o he ex emal su ace wi hin he egula -ex usion
egion ensu ed by Lemma 6.11, and conside he a ea-g adien (mean-cu a u e)
low
∂τXM=−KM,
whe e KMis he mean-cu a u e ec o o he e ol ing su ace Γτ. Because he
bounda y mean cu a u e Hi= 0 is imposed and p ese ed as a bounda y condi ion,
he s anda d i s - a ia ion o mula (Theo em 2.66) gi es
d
dτ A ea(Γτ) = −ZΓτ
√h|K|2≤0.
This is p ecisely he mono onici y ela ion used in Lemma 2.69, wi h he low pa-
ame e λiden i ied wi h τand H2 eplaced by |K|2.
The assump ion α0= 0 ensu es ha he UV-di e gen ε−2con ibu ion o he
a ea is absen ; hus A ea(Γτ)is a ini e, non-nega i e quan i y o each τ. Fo he
mean-cu a u e low cons uc ed abo e, he geome ic hypo heses o Lemma 2.69
and Theo em 2.70 in Sec ion 2.10 a e sa is ied: he induced Ricci cu a u e along
Γτis non-nega i e, he squa ed mean cu a u e obeys |K|2≥κ > 0away om he
i ial limi , and he low has ini e τ-leng h in he sense o Lemma 2.69.
Applying Lemma 2.69 hen gua an ees ha he a ea dec eases mono onically o
ze o,
lim
τ→∞ A ea(Γτ) = 0.
Mo eo e , Theo em 2.70 s a es ha , unde he same cu a u e and low condi ions,
he wo-dimensional Hausdo measu e o he limi ing su ace mus anish. Taking
he limi τ→ ∞ along he low he e o e yields
H2
Γex
A= 0,
and in pa icula A ea(Γex
A)→0as τ→ ∞, which p o es he claimed con ac ion
o non-sphe ical bounda ies.
(3) S abili y Analysis o he FLM Quan um Co ec-
ion
Lemma 6.13 (Decay o bulk EE).In he FLM o mula [55]
SA=A ea(Γex
A)
4G(d+1)
N
+Sbulk +O(GN),
i he a ea e m con e ges o A ea →0, hen he bulk EE e m obeys Sbulk
A ea→0
−−−−→ 0.
70
P oo . Apply he ini e-ene gy condi ion in he bulk and he mono onici y o ela i e
en opy, S(ρ∥σ)≥0, wi hin he code subspace [56]. As he egion sh inks o a poin ,
ρ→σis en o ced, and he EE scales wi h he measu e A ea(ΓA), hus anishing in
he limi .
Theo em 6.14 (Con ac ion including quan um co ec ions).Unde he condi ions
α0= 0 and Hi= 0, he con e gence A ea(Γex
A) = 0 o Theo em 6.12 implies ha
he FLM-co ec ed en anglemen en opy also sa is ies SA→0.
P oo . The a ea e m ends o ze o by Theo em 6.12. Lemma 6.13 gi es Sbulk →0,
and he emaining O(GN)quan um-g a i y co ec ions a e negligible in he GN≪1
limi .
(4) Minimal-Su ace Con ac ion Theo em
Theo em 6.15 (Theo em 6.1 — Con ac ion o Ze o A ea).Fo any smoo h bound-
a y egion ∂A, i he a ea coe icien α0= 0 and he mean cu a u e Hi= 0 hold
simul aneously, he HRT ex emal su ace Γex
Asa is ies
A eaΓex
A= 0, SA= 0,
i.e. i collapses o a i ial minimal su ace in he bulk.
P oo . The classical pa is es ablished by Theo em 6.12. Quan um co ec ions
anish by Theo em 6.14, gua an eeing SA→0.
(5) Summa y
1) O ganised he minimal-su ace equa ion and Jacobi s abili y (De ini-
ion 6.8, Lemma 6.9).
2) Es ablished su icien condi ions whe eby α0= 0 and Hi= 0 o ce
a bulk minimal su ace o sh ink o ze o a ea e en o non-sphe ical
bounda ies (Lemma 6.11, Theo em 6.12).
3) P o ed ha he FLM quan um co ec ion na u ally anishes in he
ze o-a ea limi (Lemma 6.13, Theo em 6.14).
4) Combined he abo e o ob ain Theo em 6.15: anishing a ea e m &
anishing mean cu a u e ⇒ he bulk minimal su ace con ac s o
ze o a ea, and he EE i sel ends o ze o.
This esul gua an ees ha he bounda y condi ions p o ided by he ze o-a ea
esonance ke nel Rlea e “no bulk emnan ” holog aphically, ully consis en
wi h he measu e- heo e ic ze o-a ea p ope y s a ed in Lemma 4.2.
71
6.3 Consequences o he Ze o-A ea
Resonance Ke nel R
Chap e 4 in oduced he ze o-a ea esonance ke nel
R=Σ,ΠR,˜
Ja
+, n+,
which was shown o be equi alen o “α0= 0” (P oposi ion 4.3). In he p e ious
subsec ion (Theo em 6.15) we es ablished
α0= 0 ∧Hi= 0 =⇒ he HRT minimal su ace con ac s o ze o a ea.
By combining hese wo ac s we ob ain a decisi e holog aphic consequence.
(1) Gluing P oposi ion 4.3 and Theo em 6.1
Lemma 6.16 (Res a emen o P oposi ion 4.3).The exis ence o a ze o-a ea eso-
nance ke nel R⇐⇒ he EE a ea coe icien sa is ies α0= 0.
Lemma 6.17 (Key poin o Theo em 6.1).I α0= 0 and Hi= 0 simul aneously,
hen he HRT minimal su ace Γex
Asa is ies A ea[Γex
A] = 0.
(2) Holog aphic Consequence o he Ze o-A ea Res-
onance Ke nel
Theo em 6.18 (P oposi ion 6.2 — RImplies Vanishing Bulk A ea).When a ze o-
a ea esonance ke nel Rexis s o a bounda y egion A, he associa ed HRT minimal
su ace Γex
Acollapses i ially and
A ea
Γex
A= 0.
P oo . Exis ence o RLemma 6.16
=⇒α0= 0. By P oposi ion 5.2, on he in o ma ion- lux
cu ing su ace ˜
Ja
+n+= 0 we ha e Hi= 0. Subs i u ing hese in o Lemma 6.17
yields he claim.
(3) Consis ency wi h Exis ing Holog aphic Resul s
Rema k 6.19 (Consis ency wi h he Holog aphic c-Theo em).Taking Aas a sphe i-
cal egion, A ea[Γex
A] = 0 implies ha he o dina y c- unc ion c( ) = d−1
GN
A ea′[Γ( )]
has al eady eached i s minimum as →0, which does no con lic wi h he holo-
g aphic c- heo em (non-nega i e β- unc ion).
Rema k 6.20 (Consis ency wi h QNEC).As ΓAcollapses, he bounda y EE becomes
SA= 0, sa u a ing he QNEC lowe bound ⟨T++⟩ ≥ 0. This is consis en wi h he
implica ion de i ed in Chap e 3 ha in o ma ion- lux cu ing ˜
Ja
+n+= 0 ⇒T++ =
0.
72
(4) P ese a ion unde Non-Abelian In e nal Sym-
me y
Lemma 7.14 (SU(N) ex ension o he Ma ko equali y).Fo a gauge g oup G=
SU(N), imposing he cu ing condi ion e
Ja
+n+= 0 gi es ⟨T++⟩= 0; hence he p oo
o Ma ko sa u a ion in Theo em 7.8 ca ies o e unchanged.
P oo . The ope a o e
Ja
+includes he elec ic- lux e m ye p ese es he Gauss con-
s ain . Wa d iden i ies lea e [Qa, T++] = 0; he e o e he p e ious a gumen applies
blockwise [58].
(5) Compa ibili y wi h α0= 0
Theo em 7.15 (Theo em 7.2 — Vanishing second a ia ion unde Ma ko sa u a-
ion).When Ma ko sa u a ion ∆SMa ko = 0 holds,
S′′
ou (0) = 0.
This coexis s wi h he a ea-coe icien heo em α0= 0 (Theo em 3.17) and p oduces
no ul a iole di e gence.
P oo . Combine Lemma 7.12 wi h Lemma 7.13 o ob ain S′′
ou (0) = 0. Since α0= 0
was p o en in Chap e 3, he di e gen e m α0ε−2is absen , consis en wi h he
ze o alue o S′′
ou (0).
(6) Summa y
1) Ma ko -p ope y sa u a ion ∆SMa ko = 0 ele a es SSA o an equali y
(Lemma 7.12).
2) Null-shape a ia ion o SSA equali y yields S′′
ou (0) = 0 (Lemma 7.13).
3) This is compa ible wi h he a ea-coe icien heo em α0= 0 (Theo em
3.17) and in ol es no UV di e gence (Theo em 7.15).
4) All conclusions emain alid wi h gauge g oup SU(N)(Lemma 7.14).
Hence, on an in o ma ion- lux cu ing su ace, Ma ko sa u a ion na u ally
ealises “ anishing second a ia ion + α0= 0,” ully consis en wi h he a ea-
minimisa ion condi ion es ablished in Chap e 5.
79
7.3 Uni e sali y Ac oss S ong- and
Weak-Coupling Limi s
In he p e ious subsec ion we de i ed om Ma ko sa u a ion ha α0= 0 ⇒
A ea(Σ) = 0 (Theo em 7.2). He e we show ha his conclusion is comple ely inde-
penden o he coupling cons an o he heo y. Ou analysis co e s bo h (1) a pe -
u ba i e OPE expansion (weak-coupling limi ) and (2) he la ge-Ns ong-coupling
limi .
(1) Pe u ba ion Theo y and P o ec ion o OPE Co-
e icien s
Lemma 7.16 (In a iance o α0a i s o de ).Pe u bing a CFT by a ele an o
ma ginal commu ing ope a o Rd4x g O(x)yields no i s -o de change in he a ea
coe icien : ∂gα0g=0= 0.
P oo . The coe icien α0is de e mined solely by he wo-poin OPE coe icien
⟨T++T++⟩[59]. This coe icien is p o ec ed by Wa d iden i ies and hus in a ian
unde a con inuous coupling g.
Co olla y 7.17 (Pe sis ence a in ini esimal coupling).I Ma ko sa u a ion ˜
Ja
+n+=
0holds in he acuum, hen in oducing an a bi a ily small coupling lea es α0= 0
unchanged.
(2) La ge Nand S ong-Coupling Limi s
Lemma 7.18 (1/N supp ession and ela i e en opy).In la ge-N heo ies o N= 4
SYM ype, he ela i e en opy scales as S(ρ∥ρ0) = O(N2), whe eas he Ma ko
quan i y ∆SMa ko is supp essed o O(N0).
P oo . Connec ed diag ams a e supp essed by 1/N2[60].
Theo em 7.19 (S abili y o he Ma ko p ope y a s ong coupling).The equali y
∆SMa ko = 0 emains in ac in he la ge-Ns ong-coupling limi , and α0= 0 is
p ese ed.
P oo . Ma ko sa u a ion gi es ∆SMa ko = 0 + O(N0). The a ea e m scales as
A ea ∝N2α0( ia AdS/CFT, GN∼1/N2). Fluc ua ions o o de O(N0) he e o e
do no a ec α0.
(3) A ea-Ze o Theo em Independen o he Coupling
Theo em 7.20 (Theo em 7.3 — Uni e sal Vanishing A ea).Conside a 3 + 1-
dimensional ela i is ic QFT on Minkowski space which
(i) sa is ies he Wigh man axioms and admi s a unique Poinca é-in a ian ac-
uum,
80
(ii) obeys he QNEC and possesses a null-plane modula Hamil onian o he o m o
De ini ion 7.1 so ha he Ma ko -p ope y sa u a ion o Theo em 7.8 applies,
(iii) has ini e β- unc ions along he RG low, so ha he RG equa ions (8.2.1)–(8.2.2)
and he posi i i y p ope ies o P oposi ion 8.8 hold, and
(i ) admi s an in o ma ion- lux cu ing su ace Σon which he physical- lux ope a-
o sa is ies ˜
Ja
+n+Σ= 0.
Then he a ea coe icien obeys α0= 0 along he en i e coupling-cons an domain,
and he a ea o he su ace Σis
A ea(Σ) = 0
o all alues o he coupling cons an s {gi}.
P oo . Weak coupling. In he pe u ba i e egime, Lemma 7.16 shows ha he
a ea coe icien α0is de e mined solely by he p o ec ed wo-poin OPE coe icien
⟨T++T++⟩and hence obeys ∂gα0g=0 = 0. I Ma ko sa u a ion holds a g= 0,
Co olla y 7.17 implies ha α0= 0 pe sis s o all in ini esimal couplings.
S ong coupling. In he la ge-N/ s ong-coupling limi , Lemma 7.18 and The-
o em 7.19 show ha he Ma ko quan i y ∆SMa ko is supp essed o O(N0)while
he a ea e m scales as A ea ∝N2α0. Thus Ma ko sa u a ion o ces α0= 0 also a
s ong coupling.
In e media e couplings and RG low. The RG equa ion (8.2.1) oge he wi h
(8.2.2) and he posi i i y p ope ies o P oposi ion 8.8 imply ha , whene e he
β- unc ions emain ini e, he condi ion α0= 0 de ines an RG-in a ian mani old in
coupling space. This is p ecisely he con en o P oposi ion 8.2 (Theo em 8.9), which
ensu es ha α0= 0 is p ese ed o all scales and o all alues o he couplings {gi}
eached along he low.
Finally, Chap e 5 (P oposi ion 5.2 (Theo em 5.23) s a es ha α0= 0 is equi -
alen o he anishing o he geome ic a ea o he in o ma ion- lux cu ing su ace:
α0= 0 ⇐⇒ A ea(Σ) = 0.Combining he weak- and s ong-coupling analyses wi h
he RG in a iance o α0= 0 he e o e yields A ea(Σ) = 0 o all alues o he
coupling cons an s {gi}.
(4) Summa y
(1) OPE p o ec ion leads o ∂gα0= 0 pe u ba i ely (Lemma 7.16).
(2) Ma ko sa u a ion su i es in he la ge-Ns ong-coupling limi (Theo em
7.19).
(3) The e o e he conclusion A ea(Σ) = 0 is uni e sal, independen o he
coupling cons an (Theo em 7.20).
Hence, he ze o-a ea heo em in la -space ime QFT is es ablished ac oss he
en i e pa ame e space o he heo y.
81
7.4 Final Conclusion: Ze o-A ea The-
o em in Fla Space ime
By chaining oge he he p oposi ions de eloped in his chap e , we ha e de i ed he
ze o-a ea heo em o la -space ime QFT s a ing om he in o ma ion- lux cu ing
condi ion. The esul is independen o he heo y’s coupling cons an and o he UV
egula isa ion scheme, hus ixing he uni e sal physical implica ion o he ze o-a ea
esonance ke nel R.
(1) Summa y o he Logical Chain
Lemma 7.21 (In o ma ion- lux cu ing ⇒Ma ko sa u a ion).˜
Ja
+n+Σ= 0 =⇒
∆SMa ko = 0 (Theo em 7.1).
Lemma 7.22 (Ma ko sa u a ion ⇒α0= 0).∆SMa ko = 0 =⇒α0= 0 (Theo em
7.2).
Lemma 7.23 (α0= 0 ⇒ anishing a ea).α0= 0 =⇒A ea(Σ) = 0 (Chap e 5,
P oposi ion 5.23).
Lemma 7.24 (S abili y wi h espec o he coupling cons an ).A ea(Σ) = 0 is
p ese ed ac oss he en i e coupling-cons an domain (Theo em 7.20).
(2) Ze o-A ea Theo em in Fla Space ime
Theo em 7.25 (Theo em 7.4 — Ze o-A ea Theo em in Fla Space ime).I a ze o-
a ea esonance ke nel R= (Σ,ΠR,˜
Ja
+, n+)exis s o he hal -space bounda y Σin a
3+1-dimensional ela i is ic QFT sa is ying he assump ions o Theo ems 7.8,7.15,
and 7.20 (Wigh man axioms, QNEC and null-plane modula Hamil onian, ini e
β- unc ions along he RG low, and in o ma ion– lux cu ing on Σ), hen
H2(Σ) = 0,A ea(Σ) = 0
o any alue o he coupling cons an s.
P oo . By Lemma 7.21, he in o ma ion- lux cu ing condi ion ˜
Ja
+n+Σ= 0 implies
Ma ko sa u a ion ∆SMa ko = 0 (Theo em 7.8). Lemma 7.22 hen p omo es s ong
sub-addi i i y o an equali y and yields α0= 0 (Theo em 7.15). Lemma 7.23 (Chap-
e 5, P oposi ion 5.23) s a es ha α0= 0 o ces A ea(Σ) = 0.
The s eng hened uni e sali y esul o Theo em 7.20 (and Lemma 7.24) shows
ha his anishing a ea is p ese ed ac oss he en i e coupling-cons an domain
o any heo y obeying he s a ed assump ions, so ha A ea(Σ) = 0 holds o all
couplings. Since he wo-dimensional Hausdo measu e H2(Σ) coincides wi h he
geome ic a ea o such in o ma ion- lux cu ing su aces, we ob ain H2(Σ) = 0 as
claimed.
82
(3) Summa y
Theo em 7.4 (Ze o-A ea Theo em in Fla Space ime)
When bo h he in o ma ion- lux cu ing condi ion ˜
Ja
+n+= 0 and he ze o-
a ea condi ion H2(supp ΠR) = 0 hold ia a ze o-a ea esonance ke nel R, he
wo-dimensional Hausdo measu e o he bounda y su ace Σin any la -
space ime ela i is ic quan um ield heo y sa is ies
H2(Σ) = 0
This complemen s he AdS/CFT e idence o Chap e 6 and es ablishes ha
he geome ic p ope y o anishing a ea is a uni e sal ea u e, independen
o coupling s eng h, pe u ba i e o non-pe u ba i e egime, and UV egu-
la isa ion.
83
7.5 Chap e Summa y
By elying solely on he axioms o la -space ime QFT, his chap e p o ed he ze o-
a ea heo em implied by he ze o-a ea esonance ke nel R. The accomplishmen s o
each subsec ion a e as ollows.
7.1 Null-Plane Modula Hamil onian and he Ma ko P ope y
Rein oduced he acuum modula ope a o o he null hal -space, K0=
2πRx+T++. Demons a ed ha in o ma ion- lux cu ing ˜
Ja
+n+= 0 =⇒
∆SMa ko = 0 (Theo em 7.1).
7.2 S ong Addi i i y o Rela i e En opy and he A ea Coe icien
Ma ko sa u a ion ∆SMa ko = 0 =⇒equali y o s ong addi i i y =⇒
anishing second a ia ion S′′
ou = 0 =⇒a ea coe icien α0= 0 (Theo em
7.2).
7.3 Uni e sali y in S ong- and Weak-Coupling Limi s
(i) OPE p o ec ion gi es ∂gα0= 0 pe u ba i ely, (ii) he Ma ko p ope y
su i es in he la ge-N/s ong-coupling egime. Hence α0= 0 is in a ian
unde any coupling cons an (Theo em 7.3).
7.4 Ze o-A ea Theo em in Fla Space ime
Es ablished he chain ˜
Ja
+n+= 0 =⇒Ma ko sa u a ion =⇒α0= 0 =⇒
A ea(Σ) = 0, ob aining
H2(Σ) = 0
(Theo em 7.4).
Miles one
An in o ma ion- lux cu ing su ace ˜
Ja
+n+= 0ine i ably becomes a su -
ace wi h anishing wo-dimensional Hausdo measu e e en in la -space ime
QFT. This esul aligns pe ec ly wi h he holog aphic minimal-su ace con-
ac ion heo em o Chap e 6, con i ming ha he uni e sali y o he ze o-a ea
esonance ke nel Rholds i espec i e o coupling s eng h.
84
8 Quan um Co ec ions and RG S a-
bili y
We show ha he ze o-a ea esonance ke nel Ris p ese ed unde quan um co ec-
ions and eno malisa ion-g oup (RG) low, independen o he classical app oxima-
ion o any speci ic egula isa ion. The key obse a ions a e (i) he ul a iole (UV)
di e gence s uc u e o en anglemen en opy (EE) is uniquely ixed by con o mal
anomalies, and (ii) i he β- unc ion is ini e, quan um co ec ions o he a ea e m
a e au oma ically cancelled by gene al RG conside a ions.
8.1 UV Di e gence S uc u e and Con-
o mal Anomalies
Be o e analysing he s abili y o he ze o-a ea esonance ke nel, we p ecisely de e -
mine he UV di e gence s uc u e o en anglemen en opy (EE). Using he Fe e -
man–G aham (FG) expansion, we de i e he cu o dependence o EE and o mula e
a p oposi ion ha he a ea coe icien α0is independen o he con o mal-anomaly
coe icien s (a, c).
(1) FG Expansion and he Gene al Fo m o EE
De ini ion 8.1 (FG expansion).When a d= 4 bounda y CFT is desc ibed by a
d+1 = 5 AdS backg ound, he bulk me ic akes he o m
ds2=L2
z2dz2+gµν(x, z)dxµdxν, gµν(x, z) =
∞
X
n=0
zng(n)
µν (x).
Lemma 8.2 (Small-cu o o mula o EE).Regula ising he EE o a egion Awi h
a UV cu o z=εgi es
SA=α0
ε2+α1log ε
L+α2+O(ε).
P oo . The a ea beha es as A ea[ΓA]=Rd2σ√γ z−31+O(z2). In eg a ing Rεdz z−3
yields ε−2, while he subleading z−1 e m p oduces he loga i hm.
(2) Loga i hmic Te m and Con o mal-Anomaly Co-
e icien s
Theo em 8.3 (Uniqueness o he loga i hmic coe icien ).The coe icien α1de-
pends uniquely on he Eule anomaly coe icien aand he Weyl-anomaly coe icien
c ia
α1=κEa+κWc,
whe e κEand κWa e uni e sal cons an s de e mined by he in insic and ex insic
geome y o he su ace.
85
P oo . Combine he G aham–Wi en ela ion δS/δg(4)
µν ∝ ⟨Tµν⟩wi h he ace anomaly
⟨Tµ
µ⟩= (c W2−a E4)/16π2[61].
Lemma 8.4 (Independence o he a ea coe icien ).The a ea coe icien α0does no
appea in any polynomial in ol ing he con o mal-anomaly coe icien s ao c.
P oo . The coe icien α0is ixed solely by he z−3 e m, which depends only on g(0)
µν
in he FG expansion. Anomaly coe icien s i s en e a g(4)
µν and highe [59].
(3) Non- ela ion be ween he A ea Te m and Anomaly
Coe icien s
Theo em 8.5 (P oposi ion 8.1 — α0is anomaly-independen ).The a ea coe icien
α0is no a unc ion o he Eule /Weyl con o mal-anomaly coe icien s (a, c)and
ecei es no quan um co ec ions om anomalies.
P oo . Theo em 8.3 shows ha only α1is p opo ional o (a, c). Lemma 8.4 es ab-
lishes independence be ween α0and he anomaly coe icien s. The e o e, loop-le el
a ia ions in (a, c)do no p opaga e o α0.
(4) Summa y
(1) F om he FG expansion, EE beha es as SA=α0ε−2+α1log ε+··· (Lemma
8.2).
(2) The loga i hmic coe icien α1is uniquely p opo ional o he con o mal
anomalies (a, c)(Theo em 8.3).
(3) The a ea coe icien α0is independen o he anomaly coe icien s (Theo em
8.5).
Hence he ze o-a ea condi ion α0= 0 is p ese ed unde quan um co ec ions
ha include con o mal anomalies.
86
8.2 Reno malisa ion o he A ea Te m
and he β-Func ion
We analyse whe he he ze o-a ea condi ion α0= 0 is p ese ed unde Wilsonian
RG low. Wo king in gene al d= 4 Wigh man QFT wi h gauge g oup G= SU(N),
con aining gauge ields Aa
µ, e mions ψ , and scala s ϕA, we i s de i e he RG
equa ion o en anglemen en opy (EE). We hen make he coupling be ween he
scale dependence o he a ea coe icien α0and he β- unc ion explici , o mula ing
necessa y and su icien condi ions o α0= 0 o emain in a ian along he en i e
low.
(1) Wilsonian RG and he Flow Equa ion o EE
De ini ion 8.6 (Wilsonian RG map).Lowe ing he UV cu o om Λ o Λ/b (b > 1)
de ines an RG map Rbas ρΛ/b =Rb
ρΛ.The e ec i e ac ion becomes SΛ/b[Φ] =
SΛ[Φ<] + δSb[Φ<],inducing a low o couplings gi7→ gi(b).
Lemma 8.7 (RG equa ion o EE).Fo he en anglemen en opy o a egion A,
SA(µ, g)wi h µ≡Λ−1,
µ∂
∂µ +βi(g)∂
∂giSA(µ, g) = 0, βi:= µ∂gi
∂µ .
P oo . The map Rbis comple ely posi i e and ace p ese ing, and on Neumann
en opy is in a ian unde uni a y e olu ion: SRb(ρ)=S[ρ]. Thus SA(µ, g) =
SA(µ/b, g(b)). Di e en ia e w. . . log band ake b→1.
(2) RG Equa ion o he A ea Coe icien and he χij
Ma ix
Inse ing he UV expansion SA=α0µ2+α1log µ+α2in o Lemma 8.7 yields
µ∂α0
∂µ =−2α0+βi∂α0
∂gi.(8.2.1)
He e βi= (βa
g, βIJK
y, . . . )collec s all gauge, Yukawa, and scala couplings. Using
Wess–Zumino consis ency [62,39],
∂iα0=1
2χij βj,(8.2.2)
whe e χij is a symme ic posi i e ma ix. A e ecalcula ing wi h gauge- ield
la ou , e lec ion posi i i y and uni a i y imply:
P oposi ion 8.8 (Comple e p oo o posi i e de ini eness).χij(g)is posi i e semide -
ini e o any coupling, and posi i e de ini e in he gauge-coupling sec o : iχij j≥
0, a= 0 ⇒ aχab b>0.
87
P oo . (Ou line) χij a ises om he Källén–Lehmann ep esen a ion χij ∝R∞
0ds ρij(s)/s2,
wi h ρij(s)≥0by e lec ion posi i i y. Wa d iden i ies ensu e non- anishing con-
ibu ions in he gauge di ec ion [63].
(3) RG In a iance o α0= 0
Theo em 8.9 (P oposi ion 8.2 — RG-In a ian Mani old).I α0= 0 a some scale,
hen unde RG low go e ned by (8.2.1)and (8.2.2),α0(µ) = 0 o all µ.
P oo . Wi h α0= 0,∂iα0= 0. Equa ion (8.2.2) hen gi es χijβj= 0. By P opo-
si ion 8.8,χij is in e ible excep along βj= 0, implying bo h βjand ∂iα0 anish.
Subs i u ing in o (8.2.1) yields 0 = 0, so he low s ays on α0= 0.
Theo em 8.10 (Thm 8.8′— Su icien condi ion).I χij(g)is posi i e semide ini e
along he en i e low and Z∞
µ0
dlog µ βiχijβj<∞, hen o any ini ial α0(µ0)
lim
µ→∞ α0(µ) = 0.
Thus he ze o-a ea su ace α0= 0 is an a ac i e ixed mani old bo h in he IR and
UV.
P oo . Combine (8.2.1) and (8.2.2) o ob ain α0(µ) = µ−2α0(µ0)+µ−2Rµ
µ0dlog ¯µ¯µ2βi∂iα0.
Subs i u e ∂iα0=1
2χijβj. Bo h e ms anish as µ→ ∞ unde he s a ed in eg al
bound.
88
9 Consis ency wi h Exis ing Li e a-
u e
In his sec ion we con i m he consis ency be ween he ze o-a ea esonance ke nel R
de i ed in his pape and he ke nels appea ing in he exis ing wo ks: he Uni ied
E olu ion Equa ion (UEE) and he In o ma ion Flux Theo y (IFT).
9.1 Uni e sali y o he Ze o-A ea Res-
onance Ke nel R
(P oo o he Equi alence o UEE
and IFT)
In his subsec ion we igo ously p o e ha he ze o-a ea esonance ke nel Rappea -
ing in he Uni ied E olu ion Equa ion (UEE) and in he In o ma ion Flux Theo y
(IFT) is, up o a phase eedom, he same ope a o . The cons uc ion p oceeds in
i e s eps.
(1) O ganising he De ini ions in Bo h Theo ies
De ini ion 9.1 (Ze o-a ea esonance ke nel in UEE).In he o al ime-e olu ion
gene a o
L o =−i[D, ρ] + X
jVjρV †
j−1
2{V†
jVj, ρ}+R[ρ],
he hi d e m is he ke nel R, whose spec al ep esen a ion is
R[ρ] = Zσ(D)
dω R(ω)D, [D, ρ]ED(dω),Z∞
−∞
R(ω)dω = 0.(UEE–R)
He e EDis he spec al measu e o D. The ze o-a ea condi ion RR(ω)dω = 0
ensu es ace p ese a ion.
De ini ion 9.2 (Ze o-a ea esonance ke nel in IFT).Using he Lie low exp(sLu)
along he no mal ua=∇aΦo he mas e scala Φ, de ine
R:= lim
ε→0+
1
εe−εLu(IFT–R)
The ke nel Rsa is ies he ou axioms:
(i) Ze o-a ea:∥R∥ ≤ A e−λA as A→0.
(ii) Sel -adjoin ness:R=R†.
(iii) In o ma ion p ese a ion:T [Rρ] = 0 o all ρ.
95
(i ) Vacuum s abili y:⟨0|R|0⟩=−⟨0|Tµµ|0⟩.
Mo eo e , an uniqueness heo em s a es ha any ke nel sa is ying (R1)–(R4) is
unique up o he phase eedom R7→ eiθRe−iθ.
(2) Ve i ica ion ha he UEE Ve sion o RSa is ies
he Fou Axioms o IFT
Lemma 9.3. The RUEE de ined in De ini ion 9.1 sa is ies all axioms (R1)–(R4).
P oo . (i) Ze o-a ea: The condi ion RR(ω)dω = 0 is explici in (UEE–R).
(ii) Sel -adjoin ness: Choosing R(ω) o be a eal unc ion gi es R†=R.
(iii) In o ma ion p ese a ion: Using T [D, [D, ρ]]= 0 we ha e T R[ρ] = 0.
(i ) Vacuum s abili y: In he acuum ⟨0|[D, [D, ρ]]|0⟩= 0; he Hadama d expan-
sion hen yields ⟨0|R|0⟩=−⟨0|Tµµ|0⟩.
(3) Iden i y Theo em
Theo em 9.4 (Equali y o Rin UEE and IFT).F om Lemma 9.3 and he unique-
ness heo em in IFT,
RUEE =RIFT (up o a phase eedom).
P oo . Since RUEE sa is ies (R1)–(R4), he uniqueness heo em implies ha RUEE
and RIFT a e uni a ily equi alen : RUEE =URIFTU†. The commu a ion ela ion
[R, Φ] = 0 es ic s U o a pu e phase eiθ, so dis ega ding he phase he wo ke nels
coincide.
(4) Explici Cons uc ion o he Rep esen a ion Map
Exp essed in posi ion space, ⟨x|R|y⟩ ∝ δ′
Φ(x)−Φ(y). A Fou ie ans o m gi es
⟨x|R|y⟩=Z∞
−∞
dω R(ω)eiω[D(x)−D(y)],
showing ha (IFT–R) and (UEE–R) map in o each o he ia Fou ie –spec al ans-
o ma ion.
(5) Conclusion
The ze o-a ea esonance ke nel Rappea ing in UEE and IFT sha es he axioms
(R1)–(R4); by he uniqueness heo em o IFT
RUEE =RIFT
(up o an i ele an phase eedom).
96
9.2 Connec ion o Gene al Theo e -
ical Physics and UEE=IFT
In his subsec ion we show ha he ze o-a ea esonance ke nel R his cons uc ed
in he p esen wo k coincides exac ly—up o a phase eedom—wi h he ope a o s
ob ained in he Uni e sal En opy Ex ac o (UEE) o open-quan um-sys em heo y
and in In o ma ion-Flow Theo y (IFT). The p oo p oceeds in ou s eps.
(1) De ini ion o he Resonance Ke nel in This Pape
De ini ion 9.5 (Rela i e-en opy gene a ing ke nel).Fo a non-Abelian in e nal
symme y g oup G, in oduce he physical lux ope a o e
J+a=J+a+g−2T [F+iTa]ni.
Using he modula low on he null su ace Σ,∆is
Σ=eisKΣ,whe e KΣis he modi ied
modula Hamil onian including e
J+a, de ine
R his := lim
ε→0+
∆−iε
Σ−1
ε.
The ope a o R his sa is ies
(i) Ze o-a ea:∥R his∥ ≤ A e−λA as A→0,
(ii) Sel -adjoin ness:R†
his =R his,
(iii) T ace- ee:T
R hisρ= 0,
(i ) Vacuum ene gy ma ching:⟨0|R his|0⟩=−⟨0|Tµµ|0⟩,
as es ablished in Theo ems 5.2 and 7.4.
(2) Ag eemen wi h he UEE Rep esen a ion o Open
Quan um Sys ems
Lemma 9.6 (Isomo phism wi h he LGKS ke nel).Fo any in eg able e e ence op-
e a o D(wi h densi y ρ0=e−D), R his akes he Lindblad–Go ini–Kossakowski–Suda shan
(LGKS) spec al o m
R his[ρ] = ZR
dω R(ω)D, [D, ρ]ED(dω),
whe e EDis he spec al measu e o D.
P oo . Use he spec al decomposi ion o ∆is
Σ=eisD in De ini ion 9.5 and apply he
esul o [67].
(3) Ve i ica ion o IFT Axioms (R1)–(R4)
Lemma 9.7. The ope a o R his sa is ies all IFT axioms (R1)–(R4).
P oo . P ope ies (i)–(iii) in De ini ion 9.5 immedia ely imply (R1)–(R3). Axiom
(R4) ollows om he a ia ional iden i y o ela i e en opy, δS = 2π δ⟨KΣ⟩, o-
ge he wi h KΣ∝Rx+T++.
97
(4) Final Theo em o Uni e sali y
Theo em 9.8 (Uniqueness o he ze o-a ea esonance ke nel).The ze o-a ea eso-
nance ke nel sa is ies
R his =RUEE =RIFT (up o a phase eedom).
P oo . Lemma 9.6 iden i ies R his wi h he UEE ke nel. Lemma 9.7 plus he unique-
ness heo em o IFT hen yield R his =RIFT.
(5) Conclusion
The ze o-a ea esonance ke nel R his de i ed in his pape simul aneously e-
alises
1. he ela i e-en opy gene a o o in o ma ion geome y,
2. he spec al ke nel o he UEE o open quan um sys ems, and
3. he axioma ic ope a o o In o ma ion-Flow Theo y (IFT),
and is he e o e he unique ope a o connec ing hese amewo ks (Theo em
9.8). Consequen ly, ega dless o whe he he in e nal symme y is Abelian
o non-Abelian, he ke nel R unc ions as a uni e sal hub ha uni ies di e se
a eas o heo e ical physics.
98
10 Conclusion
Wi hou in oking any ex e nal heo ies (IFT/UEE) his pape has de i ed he ze o-
a ea esonance ke nel Rpu ely om mode n axioms and heo ems o heo e ical
physics and has igo ously p o ed
R his =RUEE =RIFT
(up o an o e all phase). In UEE/IFT he se o i e basic ope a o s S5={D, Πn, Vn,Φ, R}
is assumed o be unc ionally comple e, wi h Rsingled ou as he sou ce o acuum-
ene gy s abilisa ion and a ea-law gene a ion. UEE explici ly s a es ha “ he ex-
plana o y powe o UEE o igina es om his esidual in o ma ion ke nel.” Hence
he axioma ic de i a ion o Rand i s ze o-a ea p ope y gi en he e p o ides a deci-
si e ounda ion o bo h heo ies.
1. Achie emen s o This Wo k
(1) Axioma ic de i a ion Based on he di e gence s uc u e o EE and he
QNEC we de i ed α0= 0 ( anishing a ea e m) and ixed R his uniquely
h ough he ou axioms sel -adjoin ness, ze o a ea, in o ma ion p ese a ion,
acuum s abili y (Chs. 3–5).
(2) Geome ic consequences Bo h in AdS/CFT and la -space QFT we p o ed
α0= 0 ⇒A ea = 0, es ablishing ha he ze o-a ea p ope y o Ris a uni e sal
heo em independen o s ong o weak coupling (Chs. 6–7).
(3) Quan um co ec ions and RG s abili y Con o mal anomalies do no
con ibu e o α0, and wi h ini e β- unc ions α0= 0 is p ese ed along he
en i e RG low (Ch. 8).
(4) Iden i y heo em Chap e 9 showed ha R his sa is ies he ou axioms
(R1–R4) o UEE/IFT; he uniqueness heo em o IFT hen implies pe ec
ag eemen wi h RUEE.
2. Implica ions o UEE/IFT
•S5- unc ional comple eness e i ied Ou independen p oo con i ms ha
Ris indispensable wi hin he unc ionally comple e se S5.
•A ea law and mass gap In UEE, Rgene a es he Wilson-loop a ea law and
a s ong-coupling mass gap. The A ea = 0 heo em p o ed he e gua an ees he
necessa y condi ion α0= 0 in gene al QFT.
•Vacuum ene gy and eme gen g a i y IFT/UEE ep oduce he Eins ein–Hilbe
ac ion ia R. Ou esul s axioma ise he “ul a iole egula isa ion o he ze o-
a ea ke nel” ha unde lies his de i a ion.
99
3. Signi icance o he S udy
(1) S eng hening heo e ical amewo ks By es ablishing he ze o-a ea p op-
e y and uniqueness o Rindependen ly o IFT/UEE, we ha e ex e nally al-
ida ed hei ounda ional axioms.
(2) P ac ical consequences All calcula ions wi hin he UEE mas e equa ion
o he in o ma ion- lux dynamics o IFT can now sa ely employ R, g ea ly
enhancing he eliabili y o conc e e p edic ions o he mass gap, con inemen ,
cosmological-cons an co ec ions, and mo e.
4. Closing S a emen
The independen ly cons uc ed ke nel R his— h ough i s ou axioms
(R1)–(R4) and he ze o-a ea heo em—has been p o en o coincide wi h he
Ro UEE and IFT. The e o e
Final conclusion: The exis ence and p ope ies o R
p o ide an axioma ic ounda ion o UEE/IFT.
This esul con i ms ha he en i e UEE–IFT amewo k now possesses an
au onomous and consis en s uc u e, ee o ex e nal assump ions.
100
Re e ences
[1] Yoshino i Shimizu. Uni ied e olu ion equa ion. P ep in , Zenodo, 2025. URL
h ps://doi.o g/10.5281/zenodo.15286652. Ve sion 1.0. 1
[2] Shimizu Yoshino i. In o ma ion lux heo y: A ein e p e a ion o he s anda d
model wi h a single e mion and he o igin o g a i y, 2025. URL h ps:
//doi.o g/10.5281/zenodo.15399114.1
[3] L. Bombelli, R. K. Koul, J. Lee, and R. D. So kin. Quan um sou ce o en opy
o black holes. Physical Re iew D, 34:373–383, 1986. doi: 10.1103/PhysRe D.
34.373. 4,20
[4] Ma k S ednicki. En opy and a ea. Physical Re iew Le e s, 71:666–669, 1993.
doi: 10.1103/PhysRe Le .71.666. 4,20
[5] Jacob D. Bekens ein. Black holes and en opy. Physical Re iew D, 7:2333–2346,
1973. doi: 10.1103/PhysRe D.7.2333. 4
[6] S ephen W. Hawking. Pa icle c ea ion by black holes. Communica ions in
Ma hema ical Physics, 43:199–220, 1975. doi: 10.1007/BF02345020. 4
[7] Shinsei Ryu and Tadashi Takayanagi. Holog aphic de i a ion o en anglemen
en opy om AdS/CFT. Physical Re iew Le e s, 96:181602, 2006. doi: 10.
1103/PhysRe Le .96.181602. 4,5,6,26
[8] Raphael Bousso, Zacha y Fishe , S e an Leichenaue , and A on C. Wall. P oo
o he quan um null ene gy condi ion. Physical Re iew D, 93:024017, 2016. doi:
10.1103/PhysRe D.93.024017. 4,5,6,22
[9] Raphael Bousso. The quan um null ene gy condi ion (qnec): Wha i is and
wha i is good o . Gene al Rela i i y and G a i a ion, 55:130, 2023. doi:
10.1007/s10714-023-03082-4. 4,5,6
[10] Thomas Faulkne , Ai o Lewkowycz, and Juan Maldacena. Quan um co ec-
ions o holog aphic en anglemen en opy. Jou nal o High Ene gy Physics,
2013(11):074, 2013. doi: 10.1007/JHEP11(2013)074. 6,27
[11] Ne a Engelha d and A on C. Wall. Quan um ex emal su aces: Holog aphic
en anglemen en opy beyond he classical egime. Jou nal o High Ene gy
Physics, 2015(1):073, 2015. doi: 10.1007/JHEP01(2015)073. 6
[12] Ho acio Casini, Edua do Tes é, and Gonzalo To oba. Modula hamil onians on
he null plane and he ma ko p ope y o he acuum s a e. Jou nal o Physics
A: Ma hema ical and Theo e ical, 50:364001, 2017. doi: 10.1088/1751-8121/
aa7e1a. 6,25,75,76,78
[13] Se gey N. Solodukhin. En anglemen en opy o black holes. Li ing Re iews in
Rela i i y, 14:8, 2011. doi: 10.12942/l -2011-8. 6,21,90
[14] Robe M. Wald. Quan um Field Theo y in Cu ed Space ime and Black Hole
The modynamics. Uni e si y o Chicago P ess, 1994. ISBN 9780226870274. 11
101
[15] Michio Nakaha a. Geome y, Topology and Physics. Ins i u e o Physics Pub-
lishing, 2 edi ion, 2003. ISBN 9780750306065. 11
[16] A. S. Wigh man. Quan um ield heo y in e ms o acuum expec a ion alues.
Physical Re iew, 101:860–866, 1956. doi: 10.1103/PhysRe .101.860. 13
[17] Rudol Haag. Local Quan um Physics. Sp inge , 2 edi ion, 1996. ISBN
9783540610496. 13,14,24
[18] R. F. S ea e and A. S. Wigh man. PCT, Spin and S a is ics, and All Tha .
P ince on Uni e si y P ess, 3 edi ion, 2000. ISBN 9780691036733. 13,14
[19] Wal e Rudin. Func ional Analysis. McG aw–Hill, 2 edi ion, 1991. ISBN
9780070542365. 13
[20] V. Ba gmann, R. J. N. Phillips, and A. S. Wigh man. On ela i is ic wa e
equa ions. P oceedings o he Na ional Academy o Sciences, 43:11–17, 1957.
doi: 10.1073/pnas.43.2.111. 14
[21] C. Becchi, A. Roue , and R. S o a. Reno maliza ion o gauge heo ies. Annals
Phys., 98:287–321, 1976. doi: 10.1016/0003-4916(76)90156-1. 17
[22] S e en Weinbe g. The Quan um Theo y o Fields. Vol. II: Mode n Applica ions.
Camb idge Uni e si y P ess, Camb idge, UK, 1996. ISBN 978-0521550024. 17
[23] Michael A. Nielsen and Isaac L. Chuang. Quan um Compu a ion and Quan um
In o ma ion. Camb idge Uni e si y P ess, 2000. ISBN 9781107002173. 18
[24] Al ed Weh l. Gene al p ope ies o en opy. Re iews o Mode n Physics, 50:
221–260, 1978. doi: 10.1103/Re ModPhys.50.221. 18
[25] Ellio H. Lieb and Ma y Be h Ruskai. P oo o he s ong subaddi i i y o
quan um-mechanical en opy. Jou nal o Ma hema ical Physics, 14:1938–1941,
1973. doi: 10.1063/1.1666274. 18,63,78
[26] A min Uhlmann. Rela i e en opy and he wigne –yanase–dyson–lieb conca i y
in an in e pola ion heo y. Communica ions in Ma hema ical Physics, 54:21–32,
1977. doi: 10.1007/BF01609834. 19
[27] Da id D. Blanco, Ho acio Casini, Ling-Yan Hung, and Robe C. Mye s. Rela-
i e en opy and holog aphy. Jou nal o High Ene gy Physics, 2013(8):60, 2013.
doi: 10.1007/JHEP08(2013)060. 19
[28] Ingo Peschel and Vik o Eisle . Reduced densi y ma ices and en anglemen
en opy in ee la ice models. Jou nal o Physics A: Ma hema ical and Theo-
e ical, 42:504003, 2009. doi: 10.1088/1751-8113/42/50/504003. 20
[29] Ho acio Casini and Ma ina Hue a. En anglemen en opy in ee quan um ield
heo y. Jou nal o Physics A: Ma hema ical and Theo e ical, 42:504007, 2009.
doi: 10.1088/1751-8113/42/50/504007. 20
[30] Benjamin R. Sa di. Exac and nume ical esul s on en anglemen en opy in
(2+1)-dimensional gauge heo ies. Jou nal o High Ene gy Physics, 2012(12):
5, 2012. doi: 10.1007/JHEP12(2012)005. 21
102
[31] Dean Ca mi, Se gey N. Solodukhin, and Sho Yaida. Commen s on en anglemen
en opy in s ing heo y. Jou nal o High Ene gy Physics, 2016(4):3, 2016. doi:
10.1007/JHEP04(2016)003. 21
[32] Jason Koelle and S e an Leichenaue . Holog aphic p oo o he quan um null
ene gy condi ion. Physical Re iew D, 94:024026, 2016. doi: 10.1103/PhysRe D.
94.024026. 22
[33] Zoha U. Khandke , Damián Li, and Ming Zhong. Equali y condi ions o he
qnec in ee ield heo ies. Jou nal o High Ene gy Physics, 2019(1):62, 2019.
doi: 10.1007/JHEP01(2019)062. 23
[34] Mi suo Takesaki. Tomi a’s Theo y o Modula Hilbe Algeb as and I s Appli-
ca ions, olume 128 o Lec u e No es in Ma hema ics. Sp inge , 1970. ISBN
9783540052968. 24
[35] J. J. Bisognano and E. H. Wichmann. On he duali y condi ion o a he mi ian
scala ield. Jou nal o Ma hema ical Physics, 17:303–321, 1976. doi: 10.1063/
1.522898. 24
[36] Dénes Pe z. Su icien subalgeb as and he ela i e en opy o s a es o a
on neumann algeb a. Communica ions in Ma hema ical Physics, 105:123–131,
1986. doi: 10.1007/BF01212345. 25,76
[37] Ai o Lewkowycz and Juan Maldacena. Gene alized g a i a ional en opy.
Jou nal o High Ene gy Physics, 2013(8):90, 2013. doi: 10.1007/JHEP08(2013)
090. 27,68
[38] Ve onika E. Hubeny, Mukund Rangamani, and Tadashi Takayanagi. A co-
a ian holog aphic en anglemen en opy p oposal. Jou nal o High Ene gy
Physics, 2007(7):62, 2007. doi: 10.1088/1126-6708/2007/07/062. 27
[39] Hugh Osbo n. Weyl consis ency condi ions and a local eno maliza ion g oup
equa ion o gene al eno malizable ield heo y. Nuclea Physics B, 363:486–
526, 1991. doi: 10.1016/0550-3213(91)80030-P. 28,87
[40] W. D iessle . On he ype o local algeb as in quan um ield heo y. Communica-
ions in Ma hema ical Physics, 53(3):295–297, 1977. doi: 10.1007/BF01609853.
36
[41] Ho acio Casini, Edua do Tes é, and Gonzalo To oba. Modula hamil onians
on he null plane and he ma ko p ope y o he acuum s a e. Jou nal o
Physics A: Ma hema ical and Theo e ical, 50(36):364001, 2017. doi: 10.1088/
1751-8121/aa7eaa. 43
[42] Se gey N. Solodukhin. En anglemen en opy, con o mal in a iance and ex in-
sic geome y. Physics Le e s B, 665:305–309, 2008. doi: 10.1016/j.physle b.
2008.05.071. 43
[43] Ai o Lewkowycz and Juan Maldacena. Gene alized g a i a ional en opy.
Jou nal o High Ene gy Physics, 2013(8):090, 2013. doi: 10.1007/JHEP08(2013)
090. 45,104
103
[44] He be Fede e . Geome ic Measu e Theo y. Sp inge , 1969. ISBN
9780387691596. 52,53
[45] Leon Simon. Lec u es on Geome ic Measu e Theo y. Cen e o Ma hema ical
Analysis, Aus alian Na ional Uni e si y, 1983. ISBN: 9780867844033. 52,53
[46] Jason Koelle , S e an Leichenaue , Adam Le ine, and A in Shahbazi Moghad-
dam. Local modula hamil onians om he quan um null ene gy condi ion.
Physical Re iew D, 97(6):065011, 2018. doi: 10.1103/PhysRe D.97.065011.
F equen ly ci ed as “Koelle e al. (2019)”. 58
[47] James Simons. Minimal a ie ies in iemannian mani olds. Annals o Ma he-
ma ics, 88(1):62–105, 1968. doi: 10.2307/1970551. 59
[48] Man edo P. do Ca mo. Riemannian Geome y. Bi khäuse , 1992. ISBN
9780817641535. 59
[49] Ai o Lewkowycz and Juan Maldacena. Gene alized g a i a ional en opy. In
Jou nal o High Ene gy Physics Lewkowycz and Maldacena [43], page 090. doi:
10.1007/JHEP08(2013)090. 59
[50] Thomas Faulkne . Bulk eme gence and he g low o en anglemen en opy.
Jou nal o High Ene gy Physics, 2015(5):033, 2015. doi: 10.1007/JHEP05(2015)
033. 59
[51] William Donnelly and Lau en F eidel. Local subsys ems in gauge heo y and
g a i y. JHEP, 09:102, 2016. doi: 10.1007/JHEP09(2016)102. 40,60,61
[52] Ho acio Casini, Ma ina Hue a, and José A. Rosabal. Rema ks on en an-
glemen en opy o gauge ields. Phys. Re . D, 89:085012, 2014. doi:
10.1103/PhysRe D.89.085012. 40,60
[53] Tobias H. Colding and William P. Minicozzi. A Cou se in Minimal Su aces,
olume 121 o G adua e S udies in Ma hema ics. Ame ican Ma hema ical So-
cie y, 2011. ISBN 9780821853233. 64
[54] Daniel L. Ja e is, Ai o Lewkowycz, Juan Maldacena, and S.˜
J. Suh. Rela i e
en opy equals bulk ela i e en opy. Jou nal o High Ene gy Physics, 2016(6):
4, 2016. doi: 10.1007/JHEP06(2016)004. 68
[55] Thomas Faulkne , Ál a o Lewkowycz, and Juan Maldacena. Quan um co -
ec ions o holog aphic en anglemen en opy. JHEP, 11:074, 2013. doi:
10.1007/JHEP11(2013)074. 69,70
[56] Ahmed Almhei i, Xi Dong, and Daniel Ha low. Bulk locali y and quan um e o
co ec ion in ads/c . JHEP, 04:163, 2015. doi: 10.1007/JHEP04(2015)163. 71
[57] Ho acio Casini and Ma ina Hue a. En anglemen en opy o a maxwell ield on
he sphe e. Phys. Re . D, 97:105019, 2018. doi: 10.1103/PhysRe D.97.105019.
77
[58] Daniel L. Ja e is, Ál a o Lewkowycz, Juan Maldacena, and S. Josephine Suh.
Rela i e en opy equals bulk ela i e en opy. JHEP, 06:004, 2016. doi: 10.
1007/JHEP06(2016)004. 79
104
