Blu , Quan a, and he Obse e
A Toy Epis emic Pic u e o Disc e eness
Aleksanda Pe išić
Decembe 2025
Abs ac
We ske ch an epis emic pic u e in which disc e eness (“quan a”) is no p ima ily a
p ope y o he wo ld in i sel , bu a s uc u al consequence o how we access i h ough
limi ed in o ma ion channels and ixed blu . The s a ing poin is a h ee–laye iew: an
un eachable Blank ( eali y in ull), a ious le els o blu (ex ac able in o ma ion a ini e
cos ), and he his o ies we econs uc om blu ed da a. A each ixed blu scale, we can
o ganize beha iou in o ini ely o coun ably many e ec i e s a es; changing blu co esponds
o jumping be ween di e en possible his o ies. We a gue ha quan iza ion, conse ed
“cons an s” and he need o a ixed blu a e igh ly ela ed a his epis emic le el.
The discussion is delibe a ely in o mal. I does no p opose a compe ing physical heo y,
bu a he a way o ead exis ing physics— om chao ic sys ems o quan um expe imen s
and cosmology— h ough he lens o blu , in o ma ion, and he obse e . The unde lying
ma hema ical machine y o blu and ini e amilies is de eloped in mo e echnical no es.
1
A simple abs ac me a– heo em (Sec ion 8) explains why ini e blu – amilies appea in so
many o he wise un ela ed se ings.
1 S a ing om ze o: Blank, blu , and his o ies
We begin wi h a delibe a ely minimal pic u e.
1.1 Blank
A he op si s wha we call Blank:
• he wo ld in i s ull de ail,
•beyond any ini e desc ip ion o compu a ion,
•no a ma hema ical objec we can exhaus i ely w i e down.
Blank is no some hing we aim o model di ec ly; i se es as a eminde ha any heo y we
build is a coa se p ojec ion, no he hing i sel .
1.2 Blu
Be ween Blank and ou heo ies li es blu .
A blu le el is:
•a bound on how much in o ma ion we can ex ac pe uni ime, ene gy, and memo y;
•a choice o wha di e ences we e use o dis inguish;
• he wid h o he obse a ional lens we use on Blank.
1See o ins ance [1,3,4,5].
1
In o mally, hink o blu as:
• he ini e esolu ion o ou ins umen s (spa ial, empo al, spec al),
• he ini e p ecision o ou models ( unca ion, ounding, coa se–g aining),
• he ini e capaci y o ou minds and da a channels.
Di e en blu le els allow us o see di e en s uc u es in he same unde lying Blank. Wi h
in ini e esou ces one could, in p inciple, explo e a bi a ily ine blu s, bu ne e each he Blank
i sel .
1.3 His o ies unde blu
Ahis o y in his pic u e is no an absolu e ajec o y in Blank, bu :
•a compa ible amily o obse a ions a a ixed blu ,
•
oge he wi h a model ha comp esses hese obse a ions in o some hing we call a “wo ld-
line”, a “solu ion”, o a “law”.
Fo a ixed blu le el and a ixed ime window [0
, T
], we can g oup all possible beha iou s
in o a ini e o coun able se o blu his o ies: classes o beha iou s ha a e indis inguishable
wi hin ha blu . The wo k on blu o h ee–body sys ems and Na ie –S okes equa ions i s
his pa e n e y explici ly: on any compac egion and ini e ime in e al, con inui y and
compac ness gi e us ini ely many blu – amilies o ajec o ies a each ixed esolu ion [4,5].
As we e ine blu (lowe he e o ole ance and y o dis inguish mo e), he numbe o
blu –his o ies explodes. In he limi o ze o blu , hey become uncoun ably many, and he
a emp o ack a single sha p his o y becomes uns able in chao ic sys ems.
2 Why disc e eness appea s
2.1 Fini e channel + ixed blu = disc e e s a es
A cen al idea o his no e is ha disc e eness—“quan a” o wha e e we a e measu ing—a ises
whene e we combine:
•a ini e channel: he amoun o in o ma ion pe snapsho is bounded;
•a ixed blu : we commi o no changing he esolu ion as we ollow a his o y.
Unde hese wo condi ions, he ollowing pic u e eme ges.
•
The incoming da a s eam is e ec i ely a sequence o ames ( ime slices) a some maximal
sampling a e ( hink o 60 ames pe second as a me apho ).
•
Wi hin each ame, ou ixed blu pa i ions he con inuum o possible alues in o a ini e
o coun able se o dis inguishable bins.
•
O e ime, a his o y becomes a sequence o isi s o hese bins: a wo d o e a ini e o
coun able alphabe . The in e nal con inuum inside each bin is in isible o us.
The disc e e beha iou is he e o e no assumed; i is a s uc u al consequence o insis ing
on:
• ini e in o ma ion pe ame, and
•a consis en , non–wobbling blu om ame o ame.
2
I we allow he blu o change in ime o om place o place, hen we a e e ec i ely changing
he alphabe along he way. The same unde lying p ocess may hen appea as:
•a single, sha ply de ined s a e a one momen ,
•and a b oad, di use pa e n a ano he .
T acking such a sys em consis en ly becomes ex emely di icul : we a e no longe ollowing one
his o y, bu jumping be ween di e en ep esen a ions o many possible his o ies.
2.2 Quan iza ion as a bookkeeping ule
F om his iewpoin , quan iza ion is a bookkeeping ule en o ced by blu :
•We choose a blu scale ha limi s bo h ou esolu ion and ou in o ma ion budge .
•
A ha scale, he “sensible” obse ables mus come in chunks ha ma ch he blu ; o he wise
we could no ack he sys em cohe en ly.
•
The equal spacing o hese chunks is no a me aphysical necessi y, bu he simples way o
keep he desc ip ion comp essible and he his o ies compa able.
In o he wo ds:
Limi in o ma ion + ixed blu + desi e o cohe en his o ies o ces he appea ance
o quan a, ega dless o he unde lying con inuum.
The de ails will depend on he sys em and he obse ables we choose, bu he logic is he
same whe he we a e coun ing pho ons in a blackbody spec um, momen um cell isi s in a
u bulen low, o allowed con igu a ions in a h ee–body sys em.
3 A sho blu mani es o
Fo la e e e ence, we collec he co e commi men s in a compac o m.
•
Blank. The e is a eali y (Blank) ha we ne e access in ull; e e y heo y is a coa se
p ojec ion.
•
Blu . Access o Blank is always media ed by a blu : limi ed esolu ion, bandwid h, and
memo y. Blu is no noise added on op, i is he mode o access.
•
His o ies. A “his o y” is a compa ible amily o obse a ions a a ixed blu , comp essed
in o a model. Di e en blu le els p oduce di e en e ec i e his o ies.
•
Quan a. Whene e we demand a globally cohe en s o y a ini e in o ma ion cos and
ixed blu , we a e o ced o see he wo ld in disc e e s a es (quan a) whose spacing e lec s
he chosen blu .
•
Ze o–blu limi . The a emp o ake blu all he way o ze o is ma hema ically singula
(especially in chao ic sys ems) and physically unachie able. The igh objec s li e a ini e
blu .
•
Obse e s. Obse e s a e physical channels wi h a na i e blu . Consis ency o physics
ac oss obse e s equi es no only co a iance o he laws, bu a compa ible blu s uc u e.
The es o he no e can be ead as elabo a ing hese bulle s in inc easingly conc e e se ings,
and ying hem back o he unde lying ma hema ical o mula ion o blu [1,2,3].
3
4 Chao ic sys ems as oy models
The blu iewpoin is especially anspa en in chao ic sys ems such as he h ee–body p oblem
and he Na ie –S okes equa ions on bounded domains. In hose se ings, one can p o e p ecise
s a emen s abou ini e blu – amilies and blu –s able egimes [4,5].
4.1 Fini e blu – amilies and swi ching
Fo he h ee–body p oblem, one can show (on a compac non–collision egion in phase space,
o e a ini e ime ho izon [0, T ]) ha :
•
o any ixed blu scales on pa ame e s and s a es, he e exis only ini ely many blu – amilies
o ajec o ies a ha esolu ion;
•
hese amilies can be ealized by ini ely many ep esen a i e ajec o ies, such ha e e y
o he ajec o y is blu –close o one o hem on [0, T ].
Fo Na ie –S okes on a bounded domain (wi h sui able egula i y assump ions), one expec s
an analogous s a emen : o each ixed blu scale and ime window, only ini ely many blu –
amilies a e needed o co e a compac se o ini ial da a, and he o al collec ion o possible
blu his o ies is a mos coun able [5].
The impo an phenomenon is no he ini eness i sel , bu swi ching:
•
The bounda y be ween blu – amilies is ypically ac al in he space o ini ial da a and
pa ame e s.
•
Tiny changes in he sha p s a e (o iny changes in blu ) can mo e a con igu a ion ac oss
such a bounda y.
•
When his happens, he sys em jumps om one blu – amily o ano he : mac oscopically, a
di e en e ec i e his o y is being selec ed.
F om he blu iewpoin , his is whe e unp edic abili y hides:
•
no in he exis ence o in ini ely many s a es a a gi en blu ( he e a e only ini ely o
coun ably many),
•
bu in he sensi i e dependence o he index o he blu s a e on small changes in he sha p
con igu a ion o he blu scale.
Mo al om chaos
•
I we keep he blu ixed and ame, we can alk abou a ini e ca alogue o possible his o ies
and ega d he sys em as eadable.
•
I we allow he blu o wobble o d i , we s a jumping ac oss ac al bounda ies be ween
s a es, and he desc ip ion becomes e a ic e en i he unde lying dynamics is de e minis ic.
This will be ou guiding analogy when we u n o quan um phenomena and cosmology.
5 Quan a as blu ed his o ies
5.1 Blackbody and he in o ma ion budge
Heu is ically, he blackbody adia ion p oblem can be eph ased in his language as ollows:
•We a e ex ac ing in o ma ion abou he ene gy dis ibu ion o modes in a ca i y.
4
•Ou desc ip ion mus i :
– he coa se mac oscopic cons ain s ( empe a u e, ene gy),
– he blu inhe en in ou ins umen s, and
– he ini eness o he in o ma ion we can physically exchange wi h he ca i y.
•
The disc e e Planck spec um can hen be seen as he simples way o package his
in o ma ion in o uni o m quan a ha ma ch he blu .
This does no con adic he usual de i a ion o Planck’s law; i complemen s i wi h an
in o ma ion– heo e ic eading: a pa icula blu and channel s uc u e make ha disc e iza ion
no only possible bu essen ially ine i able.
5.2 Elec on: pa icle, wa e, o in o ma ion g ain?
The ex book ques ion “Is he elec on a pa icle o a wa e?” becomes less mys e ious unde
blu .
In his pic u e:
•An elec on is a g ain o in o ma ion in a space o possible con igu a ions.
•
When we model i as a pa icle, we a e looking a i wi h a blu ha keeps only a na ow
se o deg ees o eedom (posi ion, momen um) and commi s o sha p e en s (hi s on a
sc een).
•
When we model i as a wa e, we a e e ec i ely allowing a wide blu in phase space: we
in eg a e o e many possible sha p ajec o ies and keep ack o in e e ence be ween hem.
In a double–sli expe imen :
•
I he blu is wild enough o allow cohe ence be ween pa hs, we see an in e e ence pa e n:
s a is ically, we sample he ensemble o his o ies ha would be compa ible wi h a ixed blu
and an un esol ed which–pa h in o ma ion.
•
I we na ow he blu so ha which–pa h in o ma ion is esol able (by adding a de ec o ,
say), we e ec i ely collapse o one o he pa icle–like his o ies; he in e e ence ades.
The elec on, in his iew, is nei he a classical pa icle no a classical wa e; i is a locus
whe e di e en blu –compa ible his o ies can in e e e. Wha we see depends on which blu we
insis on keeping cons an h oughou he expe imen .
6 The obse e and cons ancy o blu
6.1 Obse e s as channels wi h ixed blu
An obse e is no an ex e nal ideal eye; i is a physical pa o he Uni e se wi h:
•a ini e numbe o senso s and deg ees o eedom,
• ini e ene gy dedica ed o in o ma ion p ocessing,
•buil –in noise and ini e esolu ion a e e y s age.
This means each obse e comes wi h a na i e blu :
•a cha ac e is ic esolu ion in space, ime, and ene gy,
5
•a cha ac e is ic bandwid h and la ency,
•and a cha ac e is ic way o in eg a ing signals in o cohe en s o ies (models).
Fo an obse e o make sense o a p ocess as one his o y, i mus oughly keep:
•i s blu scale ixed o e he ele an ime and space;
•i s coding scheme (how i bins measu emen s) s able.
O he wise i s own pic u e o he wo ld would be in e nally inconsis en .
6.2 Equi alence o obse e s and blu
Eins ein’s p inciple o ela i i y says, oughly, ha he laws o physics ake he same o m in all
ine ial ames. In he blu language, we add a complemen a y equi emen :
To pa icipa e in he same his o y o he Uni e se, obse e s mus sha e, up o
equi alence, he same blu s uc u e.
I wo obse e s di e wildly in hei blu — o example, i one can access a bi a ily ine
in o ma ion and he o he canno — hen hey do no simply see he same his o y di e en ly;
hey inhabi and desc ibe di e en e ec i e uni e ses. Thei ca alogues o possible his o ies,
hei quan a, and e en hei no ion o “laws” may no be compa able.
In p ac ice, ou physical cons an s (such as he speed o ligh , Planck’s cons an , e c.) can
be ead as in a ian s o he sha ed blu :
•They ix how as in o ma ion can a el (ligh speed),
•how inely ac ion can be esol ed (Planck’s cons an ),
•how s ongly di e en scales a e coupled (coupling cons an s).
These cons an s make i possible o di e en obse e s o ag ee on he same se o e ec i e
his o ies.
7 Cosmic his o ies and al e na i e blu
7.1 Uni e se as a blu – ixed model
F om his pe spec i e, a Uni e se is no jus a mani old wi h a me ic and ields; i is:
•a se o dynamical laws, plus
•a globally consis en blu s uc u e ha :
–limi s how much in o ma ion any subsys em can ca y,
–and en o ces he same quan iza ion pa e n e e ywhe e.
The cosmic his o y we in e is hen one pa icula blu –his o y:
•one among many ha could ha e been compa ible wi h o he blu egimes,
•
bu p i ileged o us because i ma ches he blu buil in o ou bodies, ins umen s, and
en i onmen .
6
7.2 Local weaks s. global cohe ence
Quan um expe imen s and ex eme g a i a ional phenomena show ha :
•locally, wi h mode a e ene gy, we can manipula e blu sligh ly:
– une cohe ence imes,
–na ow o widen ene gy esolu ion,
–enginee in e e ence and en anglemen .
•
his e eals mic o–phenomena ha a e ou side nai e classical expec a ions, bu do no
dis up he mac oscopic consis ency o he Uni e se.
In he blu language:
•we a e b ie ly ouching al e na i e his o ies compa ible wi h sligh ly di e en local blu ;
•we can see shadows o hese his o ies (in e e ence pa e ns, nonlocal co ela ions);
•
bu we canno globally adop hei blu wi hou ew i ing he whole appa a us ha keeps
ou Uni e se s able.
Thus:
Mic o–phenomena can legi ima ely de ia e om classical in ui ion wi hou h ea ening
he g and pic u e, because hey li e in small pa ches whe e blu is locally modi ied.
Blu o e s a calm explana ion o why we obse e egula , ep oducible anomalies a small scales:
he anomalies a e egula because he local blu manipula ions a e egula .
8 A me a– heo em: ini e blu – amilies o well–posed lows
The ini eness esul s o he h ee–body p oblem and Na ie –S okes can be abs ac ed in o a
gene al heo em abou well–posed lows on me ic spaces equipped wi h blu ope a o s. This
sec ion eco ds a clean e sion o ha s a emen ; de ailed wo ked examples appea in [4,5].
Le (
X, dX
)and (
P, dP
)be sepa able me ic spaces (s a e and pa ame e spaces). Fo
T >
0,
suppose ha o each p∈Pwe ha e a low
Φ
p:X→X, 0≤ ≤T,
such ha he map (p, x, )7→ Φ
p(x)is con inuous on P×X×[0, T ].
Le
BX
:
X→X
and
BP
σ
:
P→P
be blu ope a o s o s a e and pa ame e spaces,
depending on scales > 0and σ > 0, wi h:
•∥BX
∥≤1,∥BP
σ∥≤1 o all , σ;
•BX
x→xand BP
σp→pas , σ ↓0, o each ixed x∈X,p∈P.
Fo compac se s
K⊂X
,
P0⊂P
, de ine he blu ed ajec o y associa ed wi h (
p, x
)
∈P0×K
as
7−→ BX
Φ
p(x),0≤ ≤T,
and equip he se o such cu es wi h he sup emum me ic
d ,T (p, x),(p′, x′):= sup
0≤ ≤T
dXBX
Φ
p(x), BX
Φ
p′(x′).
7
Theo em (Me a– heo em: ini e blu – amilies on compac se s).Le (
X, dX
),(
P, dP
),Φ
p
,
BX
,
BP
σ
be as abo e, and le
K⊂X
,
P0⊂P
be compac . Fix a ime ho izon
T >
0, blu scales
> 0,σ > 0, and ole ance ε > 0. Then he e exis ini ely many ep esen a i es
(pj, xj)∈P0×K, j = 1, . . . , N,
such ha o e e y (p, x)∈P0×K he e is some jwi h
d ,T (p, x),(pj, xj)≤ε.
Equi alen ly: a ha blu scale and on [0
, T
], he blu ed ajec o ies a ising om
P0×K
o m
a mos Ndis inc blu – amilies, each ep esen ed by one o he (pj, xj).
P oo ske ch. Conside he map
F: (p, x, )7→ BX
Φ
p(x)
om
P0×K×
[0
, T
] o
X
. By hypo hesis, Φ
p
(
x
)is con inuous in (
p, x,
), and
BX
is con inuous,
so
F
is con inuous. Since
P0×K×
[0
, T
]is compac ,
F
is uni o mly con inuous: o e e y
ε >
0
he e exis s δ > 0such ha
dP(p, p′)+dX(x, x′)≤δ=⇒sup
0≤ ≤T
dXBX
Φ
p(x), BX
Φ
p′(x′)≤ε.
Choose a ini e
δ
–ne
{
(
pj, xj
)
}N
j=1
in
P0×K
(possible because
P0×K
is compac ). Then
o any (p, x)∈P0×K he e exis s jwi h dP(p, pj)+dX(x, xj)≤δ, which gi es
d ,T (p, x),(pj, xj)≤ε.
Thus he se o blu ed ajec o ies spli s in o a mos
N ε
–balls, each cen ed a a ep esen a i e
cu e.
One can easily ex end Theo em 8 o include:
•an explici blu on he pa ame e space (using BP
σ),
•an obse e map M:P×X→Yin o a ini e–dimensional da a space Y,
•and a blu ope a o BY
ρon Y,
leading o ini eness o obse able blu – amilies o any ixed obse e and blu scales, as wo ked
ou conc e ely o he h ee–body p oblem in [
4
]. The heo em abo e is he co e ma hema ical
eason why he “ ini e his o ies unde blu ” language is no jus me apho ical: a each ixed scale,
he e eally a e only ini ely many e ec i e s a es on compac egions and ini e ime windows.
9 Wha his poin o iew cla i ies
Summa izing he main cla i ica ions his blu pic u e o e s:
•
Quan a as epis emic necessi y. Disc e eness appea s whene e we combine a ini e
in o ma ion channel wi h a ixed blu , independen ly o he unde lying con inuum. The
quan um you see is he g ain o in o ma ion you blu can eliably ack.
•
Cons an s as blu in a ian s. Physical cons an s can be ead as in a ian s o he global
blu s uc u e—cons ain s equi ed o di e en obse e s o pa icipa e in he same his o y.
8
•
Chaos wi hou despai . In chao ic sys ems, ini eness o blu – amilies shows ha
unp edic abili y does no come om an explosion o s a es a a gi en blu , bu om sensi i e
swi ching be ween s a es and om ying o push blu o ze o.
•
Quan um wei dness as blu misma ch. Quan um phenomena look s ange om a
classical pe spec i e because hey expose beha iou s a blu scales di e en om he de aul
mac oscopic one. In e e ence and supe posi ion a e shadows o many blu –compa ible
his o ies, seen when we do no commi o a single sha p one.
•
Cosmology as a blu –locked s o y. Ou pic u e o he Uni e se is one cohe en s o y
w i en a a pa icula blu . O he blu egimes migh suppo di e en e ec i e his o ies,
bu hey a e no accessible wi h ou cu en in o ma ion budge .
P inciple (Blu –locked quan iza ion).Whene e we insis on a globally consis en desc ip ion o
a sys em a ini e in o ma ion cos , we mus commi o a ixed blu s uc u e. Tha commi men
o ces us o see he wo ld in quan a: disc e e s a es o le els whose spacing e lec s he chosen
blu , no necessa ily he unde lying con inuum. Quan um beha iou , in his iew, is he uni e sal
signa u e o ying o unde s and Blank h ough ini e, blu –locked channels.
Final ema ks
The pic u e p esen ed he e is in en ionally modes . I does no claim o eplace quan um heo y,
gene al ela i i y, o any es ablished amewo k. I o e s:
•a uni ied language o alking abou disc e eness, chaos, and obse a ion;
•a way o ead quan iza ion and cons an s as consequences o blu and in o ma ion limi s;
•
and a concep ual b idge be ween mac oscopic and mic oscopic phenomena ha does no
equi e mi acles.
Whe he his can be sha pened in o p ecise heo ems in speci ic models (beyond he dynamical
examples al eady ea ed wi h blu ) emains an open p og amme. Bu e en a he p esen
heu is ic le el, he mo al is simple and, pe haps, eassu ing:
We do no see he wo ld in quan a because he wo ld likes o be chopped in o pieces.
We see i in quan a because ha is he only way a ini e obse e wi h ixed blu can
ell a cohe en s o y abou i .
Re e ences
[1] A. Pe išić, A Ca ego y o Blu and he G and Lemma, Zenodo, 2025.
[2] A. Pe išić, Blu Be ween Addi ion and Mul iplica ion, Zenodo, 2025.
[3] A. Pe išić, Blu as a Uni e sal P inciple, Zenodo, 2025.
[4] A. Pe išić, Blu o he Th ee–Body P oblem, Zenodo, 2025.
[5] A. Pe išić, Na ie –S okes, Blu , and Blu iche sky Geome y, Zenodo, 2025.
[6] A. Pe išić, F om Hadele–Hidele Sys ems o To ic F obenioids, Zenodo, 2025.
[7] A. Pe išić, Small Theo ies on P imes, Zenodo, 2025.
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