On he quan iza ion o he modynamic
ac ion
Tan Jun Hao
Oc obe 8, 2025
Abs ac
To da e, he e a en’ any eal and alida ed heo y no amewo k, ha explains
quan um g a i y. Cu en leading candida es a e s ing heo y, M- heo y and loop
quan um g a i y, bu hey lack o physical obse a ions ha ag ees wi h hem. In
his wo k, we p opose a di ec connec ion be ween he modynamics, ela i i y and
quan um mechanics h ough a simple ela ion:
T S =ℏ
This equa ion sugges s a na u al b idge be ween he he modynamic low o ime
and quan um ac ion, p o iding a po en ial link be ween en opy and he quan iza-
ion o physical p ocesses.
1 In oduc ion
One o he ha des , and he mos beau i ul goals in heo e ical physics, a e uni ying h ee
majo pilla s in physics, he modynamics, quan um mechanics and ela i i y. Each ield
cap u es a undamen al aspec o na u e: ene gy and hea , unce ain y and disc e eness,
he geome y o space ime. In he cou se o esea ch, we disco e a compac uni ica ion
among hese ields h ough a simple equa ion ha na u ally eme ges om dimensional
easoning and physical symme y.
2 Planck Rela ion and Ene gy-Time Symme y
In quan um mechanics, he Planck ela ion
E =ℏ
de ines he undamen al quan um o ac ion, connec ing ene gy and ime. Fo a pa icle
wi h mass m,
E=mc2
and hus
mc2 =ℏ.
This exp esses how ela i is ic ene gy and ime oge he p oduce he in a ian quan um
o ac ion.
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3 Ex ension o The modynamics
In he modynamics, ene gy can also be exp essed as
E=TS,
whe e Tis empe a u e and Sis en opy. Subs i u ing his in o he quan um ac ion
ela ion E =ℏgi es
TS =ℏ.
This equa ion e eals ha en opy low in ime is quan ized, and he undamen al cons an
o na u e, ℏ, encapsula es bo h he mal and quan um beha io s.
4 Physical In e p e a ion
The ela ion TS =ℏimplies ha any he modynamic p ocess e ol ing o e ime ca ies
a minimum quan um o ac ion. En opy (S) ep esen s he mic oscopic in o ma ion
con en o a sys em, while empe a u e (T) desc ibes he mac oscopic ene gy exchange.
Thus, ime e olu ion o en opy a a gi en empe a u e di ec ly co esponds o quan um
ac ion.
This b idges:
•The modynamics: E=TS
•Quan um mechanics: E =ℏ
•Rela i i y: E=mc2
Combining hese gi es a uni ied pe spec i e ha links he a ow o ime, he mal p o-
cesses, and quan um disc e eness.
5 Applica ion o Schwa zschild Black Hole
To es he alidi y o he ela ion
TS =ℏ,
we apply i o he simples g a i a ional sys em known o exhibi bo h quan um and
he modynamic beha io , he Schwa zschild black hole.
The Hawking empe a u e, Bekens ein–Hawking en opy, and e apo a ion ime o a
non- o a ing, uncha ged black hole o mass Ma e gi en by:
T=ℏc3
8πGMkB
, S =4πkBGM2
ℏc, =5120πG2M3
ℏc4.
Mul iplying hese quan i ies yields:
TS = 2560πG3M4
ℏc6.
Fo he equali y TS =ℏ o hold, he mass mus sa is y:
M=ℏ2c6
2560πG31/4
.
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Subs i u ing he undamen al cons an s
ℏ= 1.054 ×10−34 J s, c = 2.998 ×108m/s, G = 6.674 ×10−11 m3/kg/s2,
we ind
M≈7.6×101kg.
This mass co esponds o a Schwa zschild adius o app oxima ely 1.1×10−25 m—a
scale app oaching he Planck egime.
Rema kably, his sugges s ha he equali y TS =ℏbecomes exac o black holes
nea he quan um–g a i a ional bounda y, whe e he modynamics, quan um mechanics,
and gene al ela i i y con e ge. A la ge masses, TS > ℏ, while a smalle masses,
TS < ℏ, indica ing ha he ela ion may se e as a balance poin be ween he quan um
and g a i a ional domains.
In his iew, he equa ion TS =ℏna u ally iden i ies he Planck-scale limi as a
egion o undamen al equilib ium, whe e he he mal, empo al, and quan um aspec s
o space ime become insepa able.
6 Implica ions o Quan um G a i y
A he Planck scale, we can de ine
TpSp p=ℏ,
sugges ing ha he Planck empe a u e, en opy, and ime a e bound by he same in a i-
an quan um ac ion, connec ing h ee undamen al cons an s a once. This could o m a
he modynamic basis o quan um g a i y, whe e space ime i sel beha es as an e ol ing
he modynamic sys em cons ained by quan um limi s.
7 P edic ions and Physical Implica ions
Any g oundb eaking equa ion o heo y mus p edic a new phenomena, jus as when
Eins ein’s Gene al Rela i i y p edic ed he exis ence o black holes and g a i a ional
wa es. Thus, om his equa ion, se e al p edic ions migh occu o can be es ed:
•The mal Time Dila ion: A e y small scales, he low o ime may depend on
empe a u e and en opy. A sys em wi h highe he mal ene gy could expe ience
a sligh ly al e ed a e o ime e olu ion. This e ec may be es ed using ul acold
a oms o supe conduc ing qubi s.
•En opy-Induced Quan um Shi : I ℏbalances TS , hen changing en opy
may a ec he phase o a quan um sys em. A small a ia ion in en opy could p o-
duce measu able phase shi s, simila o a he modynamic e sion o he Aha ono –Bohm
e ec .
•Planck-Scale Balance: A he Planck scale, he p oduc o empe a u e, en-
opy, and ime emains cons an . Any inc ease in empe a u e o en opy mus
be balanced by a sho e ime in e al. This balance could ep esen a uni e -
sal equilib ium be ween hea , in o ma ion, and quan um ime, possibly unde lying
quan um g a i y.
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8 Discussion
We’ e disco e ed a simple equa ion, connec ing he modynamics, quan um mechanics
and ela i i y, sugges ing when en opy changes wi h ime he esul is a ixed quan um
ac ion. In o he wo ds, ime, hea and quan um beha io a e linked oge he a he same
le el.
I holds a Planck-scaled black holes, whe e g a i a ional and quan um e ec s mee ,
hin ing space ime migh ha e a he mal na u e, a ising om he low o en opy unde
quan um limi s.
In his way, his ela ion poin s o a deepe idea: he smalles ac ion in he uni e se
may come om he na u al balance o ime, empe a u e and en opy, b inging a s ep
close owa ds quan um g a i y.
9 Conclusion
The equa ion
TS =ℏ
encapsula es a minimal and elegan connec ion be ween he modynamics, quan um me-
chanics, and ela i i y. I implies ha he e olu ion o en opy in ime is quan ized, and
ha e e y p ocess — g a i a ional, he mal, o quan um — sha es a uni e sal limi se
by ℏ. This ela ion may se e as a ounda ional link owa d a consis en amewo k o
quan um g a i y.
Re e ences
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891–921 (1905).
[2] M. Planck, On he Law o Dis ibu ion o Ene gy in he No mal Spec um, Annalen
de Physik, 4, 553–563 (1901).
[3] J. D. Bekens ein, Black Holes and En opy, Physical Re iew D, 7, 2333–2346 (1973).
[4] S. W. Hawking, Pa icle C ea ion by Black Holes, Communica ions in Ma hema ical
Physics, 43, 199–220 (1975).
[5] T. Jacobson, The modynamics o Space ime: The Eins ein Equa ion o S a e, Physical
Re iew Le e s, 75, 1260–1263 (1995).
[6] E. Ve linde, On he O igin o G a i y and he Laws o New on, Jou nal o High Ene gy
Physics, 2011, 29 (2011).
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[8] C. Ro elli, The O de o Time, Penguin Books (2017).
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