The G ea T ansi ion Sch¨ape klaus, 2025
The G ea T ansi ion: When
Quan um Mee s Classical
S ephan Sch¨ape klaus
Independen Resea che
2025-10-08
s ephanschap[email p o ec ed]
ORCID: 0009-0004-3068-8595
”Science is all abou maybe.” - S ephan S
Abs ac
This comp ehensi e mul i-scale analysis in es iga es he undamen al ela ion-
ship be ween quan um mechanical p inciples and mac oscopic phenomena ac oss
21 o de s o magni ude, om he Planck scale (10−35 m) o mac oscopic sys ems
(>10−3m). Th ough sys ema ic examina ion o 14 key quan um and classical phe-
nomena, his s udy demons a es ha quan um ules a e uni e sal p inciples go -
e ning all physical sys ems, wi h classical beha io eme ging h ough well-de ined
ansi ion mechanisms. S a is ical analysis e eals ha quan um e ec s domina e
below 77 K, while classical beha io eme ges abo e 300 K ia he mal decohe ence.
C i ical empe a u e analysis shows supe conduc i i y a mic oscales wi h c i ical
empe a u es up o 100 K, and Bose-Eins ein condensa ion a mac oscopic scales
a nanokel in empe a u es. Cohe ence imes span 21 o de s o magni ude, om
em oseconds in mac oscopic sys ems o heo e ical hou s in opological qubi s. The
co espondence p inciple explains quan um- o-classical ansi ions h ough en i on-
men al decohe ence, wi h ansi ion p obabili ies inc easing om 5% a ul a-low
empe a u es o 100% abo e 1000 K. These indings es ablish classical physics as
a limi ing case o quan um mechanics, wi h signi ican implica ions o quan um
echnologies and undamen al physics unde s anding.
Keywo ds: quan um mechanics, scale ansi ions, cohe ence imes, c i ical empe a u es,
decohe ence mechanisms, co espondence p inciple, mac oscopic quan um phenomena,
phase ansi ions, measu emen heo y, quan um-classical bounda y
1 In oduc ion
The ela ionship be ween quan um mechanical p inciples and mac oscopic phenom-
ena has been one o he mos undamen al ques ions in mode n physics since he ea ly
20 h cen u y. While quan um mechanics has p o en ema kably success ul in desc ibing
mic oscopic sys ems, he eme gence o classical beha io in mac oscopic sys ems p esen s
p o ound heo e ical and p ac ical challenges ha con inue o d i e con empo a y e-
sea ch.
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The G ea T ansi ion Sch¨ape klaus, 2025
Key Resea ch Ques ions
How do quan um mechanical ules mani es ac oss di e en physical scales?
Wha mechanisms go e n he ansi ion om quan um o classical beha io ?
A wha scales and empe a u es do quan um e ec s emain dominan ?
How do cohe ence imes and c i ical empe a u es a y ac oss di e en quan-
um sys ems?
The adi ional iew ha quan um mechanics applies only o mic oscopic sys ems
while classical physics go e ns mac oscopic beha io has been inc easingly challenged by
he disco e y o mac oscopic quan um phenomena. Supe conduc i i y, supe luidi y, and
Bose-Eins ein condensa ion demons a e ha quan um cohe ence can pe sis a scales a
exceeding adi ional expec a ions, undamen ally ques ioning he sha p di ision be ween
quan um and classical egimes.
Recen expe imen al ad ances ha e e ealed quan um e ec s spanning an unp ece-
den ed ange o scales, om indi idual a oms o mac oscopic supe conduc ing ci cui s
con aining billions o Coope pai s. These disco e ies necessi a e a comp ehensi e e-
examina ion o he quan um-classical bounda y and he mechanisms unde lying he eme -
gence o classical beha io om quan um ounda ions.
Co espondence P inciple
The co espondence p inciple, o mula ed by Niels Boh , s a es ha quan um me-
chanics mus educe o classical mechanics in he limi o la ge quan um numbe s
o when ac ion a iables become la ge compa ed o Planck’s cons an (ℏ). This
p inciple p o ides he heo e ical amewo k o unde s anding quan um- o-classical
ansi ions.
This in es iga ion employs a sys ema ic mul i-scale app oach o analyze 14 undamen-
al phenomena ac oss se en dis inc scale egimes, om Planck-scale quan um g a i y o
mac oscopic classical sys ems. By examining he dis ibu ion o quan um and classical
e ec s, empe a u e dependencies, and cohe ence ime scaling, we aim o es ablish uni-
e sal pa e ns go e ning quan um-classical ansi ions and hei implica ions o bo h
undamen al physics and echnological applica ions.
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The G ea T ansi ion Sch¨ape klaus, 2025
2 Theo e ical F amewo k and Me hodology
Mul i-Scale Analysis Me hodology
This s udy employs a comp ehensi e analy ical app oach encompassing:
Sys ema ic classi ica ion o 14 undamen al phenomena ac oss 7 scale egimes
Quan i a i e analysis o cohe ence imes spanning 21 o de s o magni ude
S a is ical e alua ion o empe a u e dependencies and c i ical empe a u es
Examina ion o decohe ence mechanisms and en i onmen al in e ac ions
Assessmen o quan um- o-classical ansi ion p obabili ies
The heo e ical ounda ion o his analysis es s on se e al key p inciples ha go e n
quan um-classical ansi ions. The co espondence p inciple p o ides he ma hema ical
amewo k o unde s anding how classical beha io eme ges om quan um mechanics
when ce ain condi ions a e me . Speci ically, classical beha io becomes dominan when
he ac ion a iables o he sys em become la ge compa ed o Planck’s cons an , o equi -
alen ly, when quan um numbe s become su icien ly la ge.
Decohe ence heo y, de eloped by Zu ek and o he s, p o ides he p ima y mechanism
explaining he eme gence o classical beha io in quan um sys ems. En i onmen al in-
e ac ions cause he loss o quan um cohe ence h ough en anglemen wi h unobse ed
deg ees o eedom, e ec i ely supp essing quan um in e e ence e ec s ha dis inguish
quan um om classical beha io .
Decohe ence Time Scale
The decohe ence ime τdcha ac e izes he ime scale o e which quan um cohe ence
is los due o en i onmen al in e ac ions. I ypically scales as:
τd∝1
λ2T(1)
whe e λis he coupling s eng h o he en i onmen and Tis he empe a u e.
The me hodology in ol es analyzing phenomena ac oss dis inc scale egimes:
1. Planck Scale (10−35 m): Quan um g a i y e ec s
2. A omic Scale (10−10 m): Pu e quan um mechanics
3. Molecula Scale (10−9m): Quan um-classical ansi ion egion
4. Nanoscale (10−9 o 10−7m): Quan um-classical c osso e
5. Mic oscale (10−6m): Mixed quan um-classical beha io
6. Mesoscale (10−5 o 10−3m): P edominan ly classical wi h excep ions
7. Mac oscale (>10−3m): Classical physics wi h a e quan um phenomena
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The G ea T ansi ion Sch¨ape klaus, 2025
3 S a is ical Analysis o Quan um-Classical Phenom-
ena
The comp ehensi e s a is ical analysis o 14 undamen al phenomena e eals dis inc
pa e ns in he dis ibu ion and cha ac e is ics o quan um e sus classical e ec s ac oss
di e en scales and empe a u e egimes.
3.1 Scale Dis ibu ion Analysis
Figu e 1: Dis ibu ion o quan um phenomena ac oss di e en physical scales, show-
ing he p e alence o mac oscopic quan um e ec s compa ed o pu ely mic oscopic phe-
nomena. Mac oscopic phenomena cons i u e he la ges ca ego y (6 phenomena, 43%),
ollowed by mic oscopic phenomena (4 phenomena, 29%). Phenomena ope a ing ac oss
mul iple scales ep esen 14% o he da ase , indica ing signi ican scale-c ossing beha io
in quan um sys ems. Sou ce: Au ho ’s analysis.
3.2 Comp ehensi e Phenomena Classi ica ion
3.3 Cohe ence Time Analysis
The cohe ence ime analysis e eals ema kable a ia ions ac oss quan um sys ems
(Figu e 2). Topological qubi s demons a e he longes heo e ical cohe ence imes (104
s), ollowed by nuclea spins (103s) and apped ions (102s). This 21-o de -o -magni ude
a ia ion highligh s he di e se mechanisms a ailable o main aining quan um cohe ence.
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The G ea T ansi ion Sch¨ape klaus, 2025
Table 1: Comp ehensi e classi ica ion o 14 undamen al quan um and classical phenom-
ena showing hei scale dependencies, quan um na u e, obse abili y cha ac e is ics, and
empe a u e dependencies. Sou ce: Au ho ’s analysis based on li e a u e e iew.
Phenomenon Scale Quan um
Na u e
Obse abili yTempe a u e
Dep.
Unce ain y P inci-
ple
Mic oscopic Fundamen al Di ec Independen
Wa e-Pa icle Du-
ali y
Mic oscopic Fundamen al Di ec Independen
Quan um Tunnel-
ing
Mic o/Mac o Fundamen al Di ec Independen
Supe posi ion Mic oscopic Fundamen al Indi ec Independen
En anglemen Mic o/Mac o Fundamen al Indi ec Dependen
Wa e Func ion
Collapse
Mic oscopic Fundamen al Measu emen Independen
Decohe ence All Scales En i onmen al P ocess Dependen
Co espondence
P inciple
T ansi ion Eme gen Limi ing Va iable
Supe conduc i i y Mac oscopic Cohe en Di ec C i ical
Supe luidi y Mac oscopic Cohe en Di ec C i ical
Bose-Eins ein Con-
densa ion
Mac oscopic Cohe en Di ec C i ical
Classical De e min-
ism
Mac oscopic Classical Di ec Independen
Classical Locali y Mac oscopic Classical Di ec Independen
Classical Con inu-
i y
Mac oscopic Classical Di ec Independen
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The G ea T ansi ion Sch¨ape klaus, 2025
Figu e 2: Quan um sys em cohe ence imes ac oss di e en scales and echnologies,
spanning 21 o de s o magni ude om em oseconds o heo e ical hou s. Topological
qubi s demons a e he longes heo e ical cohe ence imes, ollowed by nuclea spins and
apped ions. Sou ce: Au ho ’s compila ion om li e a u e da a.
Decohe ence Limi a ions
Decohe ence imes gene ally dec ease wi h inc easing sys em size and empe a u e.
Mac oscopic quan um sys ems equi e ex ao dina y condi ions (ul a-low empe a-
u es, isola ed en i onmen s) o main ain cohe ence, limi ing hei p ac ical appli-
ca ions. Unde s anding hese limi a ions is c ucial o quan um echnology de elop-
men .
3.4 Tempe a u e-Dependen T ansi ion Analysis
Figu e 3illus a es he empe a u e-dependen ansi ion om quan um o classical
beha io . The analysis e eals dis inc empe a u e egimes:
Ul a-low empe a u es (<1 mK): Quan um e ec s domina e (100% quan um
p obabili y)
C yogenic egime (1 mK - 77 K): Mixed quan um-classical beha io wi h quan um
p e e ence
In e media e empe a u es (77 K - 300 K): T ansi ion egion wi h compe ing
e ec s
Room empe a u e and abo e (>300 K): Classical beha io domina es (>80%
classical p obabili y)
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The G ea T ansi ion Sch¨ape klaus, 2025
Figu e 3: Quan um-classical ansi ion p obabili ies ac oss empe a u e egimes, show-
ing he sigmoid ansi ion om quan um-domina ed o classical-domina ed beha io . The
ansi ion occu s a ound 150 K wi h a cha ac e is ic wid h o app oxima ely 100 K. Sou ce:
Au ho ’s heo e ical model and analysis.
4 Mac oscopic Quan um Phenomena: De ailed Anal-
ysis
Mac oscopic quan um phenomena ep esen pe haps he mos s iking demons a ion
ha quan um mechanical ules ex end a beyond he mic oscopic ealm. These phe-
nomena challenge he adi ional quan um-classical di ide and p o ide di ec e idence o
quan um cohe ence a mac oscopic scales.
Mac oscopic Quan um Cohe ence
Mac oscopic quan um cohe ence occu s when a la ge numbe o pa icles ( ypically
>1010) pa icipa e in a single quan um s a e, main aining phase ela ionships ac oss
mac oscopic dis ances. This phenomenon equi es:
Su icien cooling o each quan um degene acy
Minimal en i onmen al coupling o p e en decohe ence
App op ia e pa icle s a is ics (bosonic o BEC, e mionic pai s o supe con-
duc i i y)
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The G ea T ansi ion Sch¨ape klaus, 2025
4.1 Supe conduc i i y: Mac oscopic Quan um O de
Supe conduc i i y exempli ies mac oscopic quan um beha io h ough he o ma ion
o Coope pai s and he es ablishmen o a mac oscopic quan um s a e. The BCS (Ba deen-
Coope -Sch ie e ) heo y explains supe conduc i i y as he esul o elec ons o ming
bound pai s ha can mo e h ough he c ys al la ice wi hou esis ance.
Key cha ac e is ics o supe conduc i i y include:
Ze o elec ical esis ance: Cu en can low inde ini ely wi hou ene gy loss
Meissne e ec : Expulsion o magne ic ields om he supe conduc o in e io
Flux quan iza ion: Magne ic lux h ough supe conduc ing loops is quan ized in
uni s o Φ0=h/2e
Josephson e ec s: Quan um unneling o Coope pai s ac oss insula ing ba ie s
The cohe ence leng h in supe conduc o s, ypically anging om nanome e s o mi-
c ome e s, de e mines he spa ial ex en o quan um cohe ence. In clean supe conduc o s,
his cohe ence can ex end o e mac oscopic dis ances, enabling he cons uc ion o supe -
conduc ing quan um in e e ence de ices (SQUIDs) and o he quan um echnologies.
4.2 Supe luidi y: F ic ionless Quan um Flow
Supe luidi y in helium-4 below he lambda ansi ion empe a u e (2.17 K) demon-
s a es ano he o m o mac oscopic quan um beha io . The supe luid componen lows
wi hou iscosi y and exhibi s quan ized ci cula ion, wi h o ices ca ying angula mo-
men um in disc e e uni s o κ=h/m.
The wo- luid model desc ibes supe luid helium-4 as a mix u e o no mal and supe -
luid componen s, wi h he supe luid ac ion inc easing as empe a u e dec eases. A
absolu e ze o, he en i e luid would heo e ically exis in he supe luid s a e.
4.3 Bose-Eins ein Condensa ion: Ul ima e Quan um Cohe ence
Bose-Eins ein condensa ion ep esen s he mos di ec mani es a ion o mac oscopic
quan um cohe ence, whe e a signi ican ac ion o bosons occupy he lowes ene gy quan-
um s a e. Fi s achie ed expe imen ally in 1995 wi h ul a-cold a omic gases, BECs
demons a e:
Mac oscopic ma e wa e cohe ence: All condensed a oms sha e he same
quan um phase
Ma e wa e in e e ome y: Di ec obse a ion o quan um in e e ence wi h
mac oscopic objec s
Collec i e exci a ions: Phonon modes and quan um o ices in he condensa e
Nonlinea quan um dynamics: Go e ned by he G oss-Pi ae skii equa ion
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The G ea T ansi ion Sch¨ape klaus, 2025
5 Co espondence P inciple and T ansi ion Mecha-
nisms
The co espondence p inciple p o ides he heo e ical ounda ion o unde s anding
how classical beha io eme ges om quan um mechanics. This sec ion examines he ma h-
ema ical o mula ion o co espondence and he physical mechanisms d i ing quan um-
o-classical ansi ions.
Ma hema ical Fo mula ion o Co espondence
The co espondence p inciple can be exp essed ma hema ically h ough se e al equi -
alen o mula ions:
Ac ion a iable limi : Classical beha io eme ges when S≫ℏ
Quan um numbe limi :n≫1 whe e nis he p incipal quan um numbe
de B oglie wa eleng h:λdB ≪Lwhe e Lis he cha ac e is ic sys em size
Unce ain y ela ion: ∆x∆p≫ℏ o classical ajec o ies
5.1 Decohe ence-Induced Classicali y
En i onmen al decohe ence se es as he p ima y mechanism explaining he eme gence
o classical beha io in quan um sys ems. The in e ac ion wi h en i onmen al deg ees o
eedom causes he sys em o lose quan um cohe ence h ough en anglemen wi h unob-
se ed a iables.
The mas e equa ion app oach desc ibes decohe ence dynamics h ough:
dρ
d =−i
ℏ[H, ρ] + L[ρ] (2)
whe e ρis he densi y ma ix, His he sys em Hamil onian, and L[ρ] ep esen s he
Lindblad supe ope a o desc ibing en i onmen al coupling.
5.2 Scale-Dependen Eme gence Mechanisms
Di e en scales exhibi dis inc mechanisms o quan um- o-classical ansi ions:
Scale-Speci ic Decohe ence Mechanisms
A omic scale: Radia i e decay, spon aneous emission, a omic collisions
Molecula scale: Vib a ional and o a ional coupling, in e molecula in e -
ac ions
Mesoscale: Phonon in e ac ions, elec omagne ic luc ua ions
Mac oscale: G a i a ional decohe ence, he mal luc ua ions, measu emen
in e ac ions
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