B idging Quan um and G a i y: A S ep-by-S ep
In oduc ion o Quan um G a i y
Unde s anding how space ime and quan um mechanics in e wine
Ma hew J. Hall & GPT-5 Thinking
ORCID: 0009-0001-7066-2558
Da e: Oc obe 8, 2025
(Uni s: c=ℏ=G= 1, signa u e (−,+,+,+))
Abs ac
Gene al Rela i i y desc ibes g a i y as geome y, while Quan um Mechanics desc ibes
e e y hing else as p obabili y and supe posi ion. Quan um G a i y is he e o o uni y hese
pic u es. This handou walks h ough he p oblem: why quan izing space ime is ha d, how
semiclassical g a i y wo ks, and wha pa hs o wa d exis (loop, s ing, eme gen space ime).
Each sec ion uses a able linking he ma h, he in ui ion, and why i ma e s. The punchline:
space ime i sel may be quan ized o eme gen , and ime becomes ela ional,
encoded in quan um co ela ions.
Con en s
1 The Con lic 1
2 S ep 1: Quan izing he Me ic (Nai e App oach) 2
3 S ep 2: The Planck Scale 2
4 S ep 3: Semiclassical G a i y (Whe e We Li e) 2
5 S ep 4: Hawking Radia ion (Quan um Mee s Ho izon) 2
6 S ep 5: Loop Quan um G a i y (Quan izing Geome y) 2
7 S ep 6: S ing Theo y (Geome y om Vib a ion) 3
8 S ep 7: Holog aphy and Eme gen Space ime 3
9 S ep 8: Quan um Time (Rela ional View) 3
10 Resul s: Wha Quan um G a i y Teaches Us 3
1 The Con lic
De ini ion
•GR: space ime is smoo h, de e minis ic, geome ic.
•QM: s a es e ol e p obabilis ically; obse ables a e ope a o s.
•
P oblem: combining hem b eaks bo h pic u es—QM wan s a ixed ime o e ol e
wi h, GR says ime is pa o he geome y.
1
Takeaway
We can’ jus “add quan um” o g a i y o “add g a i y” o quan um ields. Time i sel
becomes a a iable o be ede ined.
2 S ep 1: Quan izing he Me ic (Nai e App oach)
gµν =ηµν +hµν, ea hµν as a spin-2 ield. (1)
Ma h Plain English Why his s ep?
Expand me ic a ound la
space
T y o ea g a i y like pho-
ons o gluons.
Wo ks a low ene gies, bu
a high ene gies loops di e ge
(non- eno malizable).
3 S ep 2: The Planck Scale
EP= ℏc5
G≈1.22 ×1019 GeV.(2)
Ma h Plain English Why his s ep?
Planck scale se s quan um g a -
i y egime
A his ene gy, quan um luc-
ua ions o space ime a e as
s ong as he mean ield.
Below his, semiclassical g a -
i y wo ks; abo e i , we need a
new heo y.
4 S ep 3: Semiclassical G a i y (Whe e We Li e)
Gµν = 8πG⟨ˆ
Tµν⟩.(3)
Ma h Plain English Why his s ep?
Use quan um expec a ion al-
ues in Eins ein’s equa ions
Quan um ma e , classical
geome y.
Desc ibes Hawking adia ion,
black hole e apo a ion, ea ly-
uni e se quan um e ec s.
5 S ep 4: Hawking Radia ion (Quan um Mee s Ho izon)
TH=ℏκ
2πckB
, SBH =kBA
4ℓ2
P
.(4)
Ma h Plain English Why his s ep?
Black holes adia e like black
bodies
Quan um ields see pa icle
c ea ion nea ho izons.
The modynamics links geome-
y o quan um in o ma ion.
6 S ep 5: Loop Quan um G a i y (Quan izing Geome y)
[Ai
a, Eb
j] = i8πGℏδi
jδb
a.(5)
Ma h Plain English Why his s ep?
Connec ion and lux a iables
eplace me ic
Geome y becomes disc e e;
a eas and olumes quan ized.
P edic s smalles possible
chunks o space—Planck-sized.
2
7 S ep 6: S ing Theo y (Geome y om Vib a ion)
S=−1
4πα′Zd2σ√−h hab∂aXµ∂bXµ.(6)
Ma h Plain English Why his s ep?
Replace poin pa icles wi h i-
b a ing s ings
Vib a ions co espond o pa -
icles, including a g a i on
mode.
Na u ally includes g a i y
and quan um consis ency (bu
needs 10D space ime).
8 S ep 7: Holog aphy and Eme gen Space ime
Sbulk ↔Sbounda y,(AdS/CFT co espondence).(7)
Ma h Plain English Why his s ep?
Bulk g a i y = bounda y quan-
um ield heo y
Geome y and ime eme ge
om en anglemen pa e ns.
Sugges s space ime i sel may
be a de i ed, quan um objec .
9 S ep 8: Quan um Time (Rela ional View)
ˆ
H|Ψ⟩= 0 (Wheele –DeWi equa ion). (8)
Ma h Plain English Why his s ep?
To al Hamil onian anishes
Uni e se as a whole is ime-
less; ime a ises om in e nal
co ela ions.
Ma ches ela ional ime idea:
mo ion is de ined by compa ing
subsys ems.
10 Resul s: Wha Quan um G a i y Teaches Us
Takeaway
1. Space ime is no con inuous—i has quan um s uc u e.
2. Black holes link geome y, in o ma ion, and empe a u e.
3. En anglemen may be he ab ic o space ime i sel .
4. The concep o “ ime” becomes ela ional and eme gen .
Quan um G a i y uni ies he languages o cu a u e and p obabili y. The nex on ie is
showing how he classical wo ld eme ges smoo hly om his quan um ounda ion.
C edi s & License
This handou is pa o Ma hew J. Hall’s educa ional se ies on ime, g a i y, and quan um sys ems.
C ea ed wi h assis ance om GPT-5 Thinking. Licensed CC BY 4.0.
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