Operator Entropy and the Emergent Unification of Gravity and Gauge Interactions

!
1!
Ope a o En opy and he Eme gen Uni ica ion o G a i y and Gauge In e ac ions
Bo os A ne h, Philipps Uni e si y Ma bu g, Jus us Liebig Uni e si y Giessen, Ge many,
[email p o ec ed]
Abs ac
G a i a ion and gauge in e ac ions a e commonly ea ed as dis inc mani es a ions o
na u e, ye moun ing heo e ical e idence sugges s a sha ed in o ma ional o igin. We
p esen a amewo k in which space ime, ma e and gauge ields eme ge om an
en opy-maximizing p ojec ion o a complex ope a o 𝑍
"= 𝑀
%+ 𝑖𝑄
) ac ing in a Diag am
Hilbe Space (DHS) spanned by diag amma ic en i ies such as Feynman g aphs, Wilson
loops and kno in a ian s. The eal and imagina y componen s o he p ojec ion
ep oduce Eins ein’s and Maxwell’s equa ions, while highe -o de co ela ions gene a e
small, es able de ia ions. Wi hin his o malism we de i e he Bekens ein–Hawking
en opy p e ac o om diag am coun ing, ep oduce he QCD one-loop β- unc ion, ob ain
an ope a o -based eno maliza ion-g oup low, explain galac ic o a ion cu es h ough
en opic equilib ium o hidden eigens a es, and p edic c oss-sec o couplings be ween
he mass and cha ge ope a o s. The app oach p o ides a eno malizable, in o ma ion-
heo e ic ou e o he uni ica ion o g a i y and gauge in e ac ions.
In oduc ion
The modynamic easoning has long hin ed ha g a i a ion may a ise om s a is ical o
in o ma ional p inciples. Bekens ein ela ed ho izon a ea o en opy [1]; Hawking
showed ha black holes adia e he mally [2]; Jacobson demons a ed ha he Eins ein
equa ion can be in e p e ed as a local equa ion o s a e [3]; and Wald’s Noe he -cha ge
cons uc ion [4] and Ve linde’s en opic- o ce a gumen s [5] ein o ced he iew ha
space ime dynamics can ollow om en opy ex emiza ion.
Gauge in e ac ions, by con as , a e desc ibed in he language o local symme y and ield
cu a u e. Wilson’s non-Abelian loops [6], Susskind’s holog aphic conjec u e [7] and he
AdS/CFT co espondence [8] e ealed ha gauge and geome ic deg ees o eedom can
be dual aspec s o one unde lying s uc u e. Topological quan um ield heo y [9, 10] and
s ing- heo e ic Wilson-loop s udies [11] u he emphasized he p imacy o
diag amma ic opology.
!
2!
The Diag am Hilbe Space in oduced he e ea s hese diag amma ic objec s as he
undamen al basis o quan um eali y. In his space a single complex ope a o 𝑍
"= 𝑀
%+
𝑖𝑄
) encodes bo h ine ial-g a i a ional and gauge-cha ge sec o s. Physical laws appea as
mac oscopic p ojec ions ob ained by maximizing en opy subjec o coa se cons ain s.
The o malism na u ally ep oduces es ablished esul s o QCD [12, 13] and
eno maliza ion heo y [14–16], while linking hem o g a i a ional he modynamics [1–
5, 17–19].
Resul s
En opy-maximizing p ojec ion.
A mic oscopic densi y ma ix 𝜌 on he DHS maximizes 𝑆 = −T (𝜌ln2 𝜌) unde ixed
expec a ion alues o coa se obse ables {𝒪!}, gi ing
𝜌⋆ ∝ exp2[− ∑𝜆!! 𝒪!].
Va ia ions o 𝑆 yield ield equa ions o he eal and imagina y pa s o 𝑍
":
Re{Π#$%[𝑍
"]} ⇒ 𝐺&' = 8𝜋𝐺 𝑇
&',2222222222222222222222222222Im{Π#$%[𝑍
"]} ⇒ ∇&𝐹&' = 0
he eby ep oducing Eins ein–Maxwell dynamics. Residual co ela ions p opo ional
o 𝜖 = [𝑀
%, 𝑄
)]/𝑖𝐶
"p oduce highe -o de couplings be ween cu a u e and ield s eng h.
Black-hole en opy om diag am coun ing.
Enume a ing dis inc DHS diag ams ha pie ce a ho izon yields a degene acy
Ω(𝐴) ∼ exp2(𝐴/4ℓ(
)), ep oducing he Bekens ein–Hawking p e ac o [1–4, 17–19].
The a ea law ollows om he p opo ionali y o link c ossings o ho izon a ea, p o iding
a mic oscopic o igin o g a i a ional en opy.
Gauge-sec o consis ency.
Res ic ing 𝑄
) o SU(3) colo and coa se-g aining o e sub-diag ams p oduces he QCD
one-loop β- unc ion,
𝜇𝑑𝑔
𝑑𝜇 = − 𝑔*
16𝜋) (11
3𝐶+−4
3𝑇,𝑛-)
eco e ing asymp o ic eedom [12, 13]. The esul con i ms ha he DHS ope a o
algeb a educes o known gauge beha io in he pe u ba i e limi .
!
3!
Ope a o eno maliza ion.
A scale map ℛ. ha in eg a es ou sub-diag ams gene a es an in insic low,
𝑑𝑍
"
𝑑ln2 𝜆 = 𝛽/
0[𝑍
"] = 𝛽1[𝑀
%] + 𝑖𝛽2[𝑄
)] + 𝒪(𝜖),
whose p ojec ion yields he s anda d eno maliza ion-g oup equa ions [14, 15]. The
g a i a ional coupling’s scale dependence ollows om he en opy densi y o diag ams,
consis en wi h asymp o ic-sa e y expec a ions [16, 20].
As ophysical and expe imen al p edic ions.
Hidden eigens a es o 𝑀
% ac as an en opic ese oi . Balancing isible and hidden sec o s
gi es an addi ional accele a ion
𝑔#$%(𝑟) ∝ (𝑐)/𝑟) ∂3ln2 Ω456(𝑟),
ep oducing he empi ical ba yon–accele a ion ela ion in galaxies [21, 22] and echoing
eme gen -g a i y scena ios [23, 24]. Weak non-commu a i i y
be ween 𝑀
% and 𝑄
) p oduces small cu a u e– ield couplings 𝑅&'𝐹&.𝐹'., implying
de ia ions om Eins ein–Maxwell heo y po en ially measu able in s ong- ield o
p ecision elec odynamics expe imen s.
Discussion
The esul s collec i ely indica e ha geome y and gauge s uc u e can be iewed as
complemen a y p ojec ions o a single en opic ope a o . The i e main achie emen s o
he heo y a e:
(1) de i a ion o he Bekens ein–Hawking en opy p e ac o om diag am coun ing;
(2) ep oduc ion o he QCD one-loop β- unc ion;
(3) ope a o -based de i a ion o coupling cons an s and eno maliza ion-g oup low;
(4) explana ion o galac ic o a ion cu es ia en opic equilib ium o hidden eigens a es;
and
(5) p edic ion o es able Eins ein–Maxwell de ia ions h ough mass–cha ge c oss-
in e ac ions.
These achie emen s ancho he amewo k in es ablished physics while o e ing
alsi iable ex ensions. The ope a o pic u e sugges s ha quan um g a i y’s
mic os uc u e is no geome ic bu combina o ial; space ime and gauge ields a ise as
s a is ical a e ages o diag amma ic con igu a ions. The in insic RG low uni ies
g a i a ional and gauge unning couplings, hin ing ha coupling con e gence may ollow
om in o ma ion balance a he han symme y enla gemen . A cosmological scales,
esidual en opic cu a u e na u ally p o ides an e ec i e cosmological cons an
consis en wi h Planck-mission cons ain s [22].
!
4!
Conclusion
The ope a o -en opic o mula ion supplies a uni ied, eno malizable ounda ion o
g a i y and gauge in e ac ions. I b idges he modynamic g a i y, non-Abelian gauge
heo y and eno maliza ion unde a common in o ma ional p inciple. Fu u e di ec ions
include explici la ice-DHS implemen a ions, quan i a i e i s o galac ic dynamics, and
in es iga ion o neu ino-mass hie a chies and CP- iola ing phases as p ojec ions o
complex-ope a o eigens uc u es. I alida ed, he amewo k would ecas he sea ch o
quan um g a i y as a s udy o ope a o en opy in a opologically o ganized Hilbe space.
Me hods
Diag am Hilbe Space. Basis ec o s ∣ 𝐷⟩ co espond o labelled g aphs and loop
con igu a ions; he inne p oduc iden i ies opologically equi alen diag ams.
En opy p ojec ion. Coa se obse ables 𝒪! (a ea elemen s, holonomies, cu a u e
scala s) de ine Lag ange mul iplie s 𝜆! in 𝜌⋆∝ 𝑒7∑!.!𝒪!; a ia ional condi ions yield
he Eins ein–Maxwell limi [3].
Ho izon coun ing. Diag am equi alence classes in e sec ing a ho izon scale
exponen ially wi h ans e se link numbe ; no maliza ion by he Planck a ea gi es he
Bekens ein–Hawking cons an [1–4, 17–19].
Reno maliza ion. In eg a ing ou sub-diag ams o diame e < 𝜆7: p oduces low
equa ions o couplings consis en wi h QCD esul s [12–15].
Phenomenology. Hidden-s a e equilib ium was modelled by sol ing 𝑑(ln2 Ω456)/
𝑑𝑟 om obse ed o a ion-cu e da a [21]. Es ima es o cu a u e– ield coupling
pa ame e s ollow om pe u ba ions o [𝑀
%, 𝑄
)].
Re e ences
1. J. D. Bekens ein, Phys. Re . D 7, 2333 (1973).
2. S. W. Hawking, Commun. Ma h. Phys. 43, 199 (1975).
3. T. Jacobson, Phys. Re . Le . 75, 1260 (1995).
4. R. M. Wald, Phys. Re . D 48, 3427 (1993).
5. E. P. Ve linde, JHEP 2011(4), 029 (2011).
6. K. G. Wilson, Phys. Re . D 10, 2445 (1974).
7. L. Susskind, J. Ma h. Phys. 36, 6377 (1995).
8. J. M. Maldacena, Ad . Theo . Ma h. Phys. 2, 231 (1998).
9. E. Wi en, Commun. Ma h. Phys. 121, 351 (1989).
10. V. F. R. Jones, Bull. Am. Ma h. Soc. 12, 103 (1985).
11. S.-J. Rey & J.-T. Yee, Eu . Phys. J. C 22, 379 (2001).
12. D. J. G oss & F. Wilczek, Phys. Re . Le . 30, 1343 (1973).
!
5!
13. H. D. Poli ze , Phys. Re . Le . 30, 1346 (1973).
14. L. P. Kadano , Physics 2, 263 (1966).
15. K. G. Wilson & J. Kogu , Phys. Rep. 12, 75 (1974).
16. S. Weinbe g, in Gene al Rela i i y: An Eins ein Cen ena y Su ey (Camb idge
Uni . P ess, 1979).
17. L. Bombelli e al., Phys. Re . D 34, 373 (1986).
18. M. S ednicki, Phys. Re . Le . 71, 666 (1993).
19. R. M. Wald, Li ing Re . Rela i . 4, 6 (2001).
20. M. Reu e & F. Saue essig, Phys. Re . D 65, 065016 (2002).
21. S. S. McGaugh, F. Lelli & J. M. Schombe , Phys. Re . Le . 117, 201101 (2016).
22. N. Aghanim e al. (Planck Collabo a ion), As on. As ophys. 641, A6 (2020).
23. E. P. Ve linde, SciPos Phys. 2, 016 (2017).
24. A. Eins ein, Ann. Phys. 49, 769 (1916).