Bell Doesn't Play Dice! The Classical Mechanical Origin of Quantum Entanglement

Bell doesn’ play dice!
S udy on he Classical Mechanical O igin o
Quan um En anglemen
Dona ello Dolce1*
1*Uni e si y o Came ino, Piazza Ca ou 19F, 62032 Came ino, I aly.
Co esponding au ho (s). E-mail(s): [email p o ec ed];
Abs ac
Wi hin he mos gene al amewo k allowed by he classical P inciple o Leas
Ac ion, we ind ha a speci ic subclass o classical dynamics, cha ac e ized by
In insic Pe iodici y o space ime, po en ially leads o iola ions o Bell’s inequal-
i ies. This iola ion o Bell’s inequali y a a classical le el is cohe en wi h he
indings o ou p e ious wo ks whe e we ha e ex ensi ely p o en he exac o mal
equi alence be ween he s a is ical desc ip ion o hese ul a- as inhe en ly clas-
sical cyclic dynamics and he p edic ions o s anda d quan um mechanics. We
ha e a scena io concealing Eins ein’s ision o a causal, local eali y wi h Bell’s
eques o non-locali y, wi hou in ol ing any hidden a iable. In pa icula , hese
cyclic classical dynamics in oduce a peculia elemen ha could be in e p e ed
as a “non-locali y” o wha which conce ns he Bell expe imen , bu i is in u h
undamen ally dis inc om he non-locali y pos ula ed in quan um mechanics.
I is mani es ly compa ible wi h classical mechanics and is also implici ly p esen
in Gene al Rela i i y, wi hou b eaking causali y. The appa en inde e minism
o quan um mechanics seems o a ise om he p esen undamen al expe imen al
limi a ion in di ec ly obse ing hese ul a- as cycles, which occu on imescales
o app oxima ely ∼10−21 seconds.
1
Con en s
1 In oduc ion 3
2 How Many Classical Mechanics? 6
2.1 S anda d Classical Mechanics ....................... 7
2.2 Cyclic Classical Mechanics ......................... 7
3 Se Up o Viola ion o Bell’s Inequali y in Classical Mechanics 11
3.1 The Classical O igin o Canonical Non-Commu a i i y ......... 12
3.1.1 Non-Commu a i i y o Cyclic Classical Mechanics (Equi alen
o s anda d QM) ......................... 13
3.1.2 Commu a i i y o S anda d Classical Mechanics ........ 14
3.1.3 Gene al Fo malism o Cyclic and S anda d Classical Mechanics 14
3.2 The Classical O igin o En angled S a es ................. 15
4 Viola ion o Bell’s Inequali y in Classical Mechanics 17
4.0.1 The Phase Shi on he Condi ion o In insic Pe iodici y
induced by he Pola ize s ..................... 17
4.1 Join P obabili y Acco ding o Gene al Classical Mechanics ...... 18
4.1.1 Join P obabili y o Cyclic Classic Mechanics: Viola ion o
Bell’s Inequali y .......................... 20
4.1.2 Join P obabili y o S anda d Classical Mechanics: Bell’s
Inequali y .............................. 21
5 Concilia ing Bell’s Non-Locali y and Eins ein’s Locali y 22
6 Conclusions 23
7 Decla a ions 24
A Explici Calcula ion o he Join P obabili y 25
2
1 In oduc ion
In 1935, Eins ein, Podolsky, and Rosen (EPR) p oposed a Gedanken expe imen o
a gue o he incomple eness o he Copenhagen in e p e a ion o Quan um Mechan-
ics (QM), [1]. S anda d QM pos ula es ha conjuga e a iables a e non-commu ing,
leading o he exis ence o en angled wo-pa icle s a es. In hese s a es, measu ing a
a iable o one pa icle de e mines he co esponding conjuga e a iable o he dis an
second pa icle. EPR sugges ed ha a classical heo y migh unde lie QM, wi h QM
eme ging as a s a is ical app oxima ion. In his pape we in es iga e he possibili y
o his scena io.
To explo e his possibili y, Bell (1964) in oduced local hidden a iables, [2,3], in
addi ion o he o dina y space ime coo dina es, assuming ha hese a iables con ain
all he possible “elemen s o eali y” sugges ed by EPR, om which QM a ises s a is-
ically. Howe e , Bell’s heo em demons a ed ha he b oad class o classical local
Hidden Va iable Theo ies (HVT) exhibi s a is ical co ela ions sligh ly weake han
hose p edic ed by QM. These co ela ions a e exp essed as Bell’s inequali ies, which
cons ain classical HVT as a signa u e o i s non-local na u e. Consequen ly, acco d-
ing o Bell, only non-local HVT can econcile wi h QM p edic ions, bu hese heo ies
con lic wi h classical- ela i is ic locali y, po en ially iola ing Lo en z in a iance.
Howe e , hidden a iables was ne e men ioned by Eins ein as solu ion o he
incomple eness o QM, [3]. Bell’s heo em does no p eclude he exis ence o a clas-
sical heo y ha comple es QM i Local HVT (LHVT) do no co e all possible
classical- ela i is ic heo ies compa ible wi h he equi emen o ela i is ic causali y
and locali y1. This pape explo es such a possibili y, applying he EPR a gumen o
a classical- ela i is ic heo y ha ex ends beyond QM, [6–16]. This heo y, o mally
p o en o be s a is ically indis inguishable om s anda d QM, a oids hidden a iables
al oge he — a basic hypo hesis o Bell’s heo em. Ins ead, i is based a non- i ial
opology o ela i is ic space ime, in oducing a no el o m o “non-locali y” ha ,
ema kably, emains consis en wi h classical- ela i is ic locali y while s ill iola ing
Bell’s inequali ies.
Classical Mechanics (CM), enamed he e o con enience Gene al Classical
Mechanics (Gen-CM), encompasses all he possible physical sys ems whose dynamics
a e go e ned by he classical P inciple o S a iona y Ac ion (PSA). This means ha
hey minimize he Ac ion bo h in he ’bulk’ and a he bounda y o a gi en ime
in e al be ween an ini ial ime and a inal ime. While all sys ems wi hin Gen-CM
obey he Eule -Lag ange equa ions (EL eqs) as hei Equa ions o Mo ion (EoMs)
coming om he minimiza ion in he ’bulk’ o a ime in e al, he speci ic solu ions
a e de e mined by he Bounda y Condi ions (BCs) chosen o minimize he Ac ion
a he bounda ies. Sa is ying hese BCs, equi ed by he PSA o he bounda y o he
ime in e al, gene ally ensu es ull locali y and causali y o he esul ing mechan-
ics, wi hou iola ing Lo en z in a iance o o he classical symme ies. Impo an ly,
1On he con a y, Eins ein appea s o poin owa d an app oach closely aligned wi h ECT. As epo ed
by A. Pais, Eins ein was con inced ha “i is necessa y o s a om classical ield heo ies and ask ha
quan um laws eme ge om cons ain s imposed o hem” [4]. Eins ein w o e [5]: “Fo su e; we mus jus
o e de e mine he a iables o he ma e wa e o ield by means o cons ain s. The dynamics o he
pa icles would be o e de e mined in such a way ha he ini ial condi ions would be subjec o es ic i e
cons ain s”. A.Eins ein (1923). He u he added a equi emen o co a iance o hese cons ain s.
3
we dis inguish wo se s o BCs wi h he abo e compa ibili y wi h he PSA, wi h wo
esul ing subclasses o Gen-CM, see ig.(1).
The i s subclass o Gen-CM is no hing bu he common S anda d Classical
Mechanics (S d-CM) — see Hamil on’s P inciple [17]. As we know, he S d-CM a e
he class o synch onous a ied mo ions ha conse e he con igu a ions o he sys-
em a he bounda y o a ime in e al. They a e he s a iona y solu ions o he EL
eqs cha ac e ized by anishing a ia ions a he ime bounda ies. These BCs will be
e med in his pape S anda d BCs (S d-BCs2).
The PSA equally allows ano he choice o BCs — o en o e looked — minimizing
he ac ion a he bounda y as much as o he S d-BCs. I in ol es combina ions
o Di ichle o Neumann BCs, as well known om s ing o Kaluza-Klein heo ies
[18–21]. Fo he sake o simplici y, among hese BCs he e we conside exclusi ely
Pe iodic Bounda y Condi ions (PBCs) — which can be w i en as combina ion o
wo Di ichle o Neumann BCs. The esul ing pa icula subclass o Gen-CM will be
e e ed o as Cyclic CM (Cyclic-CM). I is impo an o no ice ha he adhe ence
o hese PBCs o he PSA gua an ees he s ic locali y and causali y o he esul ing
Cyclic-CM, despi e he ac ha he PBCs implies an In insic Pe iodici y (IP) on he
coo dina es, [18–21]. This is because hese PBCs, and hus he IP on he coo dina es,
a e dynamical unc ions as much as he pe iodici y o ela i is ic clocks in Gene al
Rela i i y (GR) o he equency (and hus he pe iodici y) o he wa e- unc ions on
undula o y mechanics, as shown in [6–16].
The Cyclic-CM, de i ed by applying PBCs wi hin he amewo k o PSA, we e
o iginally in es iga ed in [16] and a e desc ibed by he Elemen a y Space-Time Cycles
Theo y (ECT), ex ensi ely explo ed in [6–15]. In gene al, ECT is bases on a non
i ial classical opology o cu ed space- ime (in insically cyclic space- ime a ising
om he imposed PBCs) and he esul ing cyclic dynamics a e p edic ed o be ul a-
as wi h espec ou expe imen al ime esolu ion, being de e mined by he mass
o he elemen a y pa icles. Since e e y de ec o and signal is e en ually based on
elec omagne ism, ine i ably in ol ing in e nal dynamics o he elec ons, hey can
only be di ec ly obse ed a ime esolu ions be e han he Comp on ime o he
elec on ∼10−21 seconds. I p o ides an al e na i e pe spec i e o ela i is ic space-
ime, and o ela i i y i sel , en i ely consis en wi h all known physics as igo ously
demons a ed in he p e ious wo k. C ucially, he IP inhe en in hese mechanics
seems o o e a no el solu ion o he long-s anding con lic be ween he non-locali y
inhe en in Bell’s heo em and he p inciple o locali y ad oca ed by Eins ein.
We ha e a undamen al and absolu ely gene al heo em abou Cyclic-CM, p o en
in [7]: as a ma e o ma hema ical ac s, he p obabilis ic desc ip ion, classical and
de e minis ic in he essence, esul ing om he Cyclic-CM associa ed o any classi-
cal Hamil onian (symplec ic) sys em by imposing PBCs a he han S d-BCs, is ully
equi alen o he expec a ion alues o he s anda d QM esul ing om he o dina y
quan iza ion o he classical sys em i sel . The indis inguishabili y be ween he s a-
is ical ou comes p edic ed by Cyclic-CM and hose p edic ed by s anda d QM has
been igo ously p o en o all majo quan iza ion p ocedu es, [6–16]. I includes all
he desi ed key quan um ea u es such as commu a ion ela ions, Hilbe space and
2In [6,7,13–16] hey we e named Synch onous BCs
4
Sch ödinge dynamics. They a e also na u ally desc ibed a a classical le el by he
o dina y Feynman Pa h In eg al. This comp ehensi e body o e idence p o es ha
Cyclic-CM is po en ially he classical physics om which QM eme ges. Consequen ly,
he iola ion o Bell’s inequali ies wi hin he amewo k o Cyclic-CM, he cen al
hesis o his pape , is an expec ed ou come.
Gene al Classical Mechanics
(BCs wi hin he P inciple o S a iona y Ac ion)
Solu ions o he Eule -Lag ange eqs
Fig. 1 Schema ic ep esen a ion o solu ions o he Eule -Lag ange equa ions wi hin a gene ic Hamil-
onian sys em (la ge box). They minimize he ac ion in he bulk o a ime in e al. Wi hin he
P inciple o S a iona y Ac ion which also equi es minimiza ion a he bounda y (small bold box),
di e en BCs gi e ise o dis inc subclasses o CM. S d-CM, i.e. anishing a ia ion o he solu ions
a he ime bounda ies, leads o he S anda d CM, including HVT, which do no exhibi iola ions
o Bell’s inequali ies (le ci cle). Pe iodic BCs, equally minimizing he ac ion a he bounda y, gi e
ise o Cyclic-CM, cha ac e ized by he iola ion o Bell’s inequali ies (cen al ci cle). I is p o en
[6–16] ha he s a is ical desc ip ion o he Cyclic CM exhibi a ema kable ull equi alence wi h he
s anda d QM ob ained by he quan iza ion o he Hamil on dynamics i sel , sugges ing a classical
ounda ion o quan um phenomena. The S d-CM and he Cyclic-CM can be ma hema ically ela ed
(mapping) h ough a sui able choice o he ime in e al o e which he ac ion is de ined.
In his pape we exclusi ely ocus on he undamen al p ope ies o Cyclic-CM ha
gi e ise o iola ions o Bell’s inequali y ele an o he CHSH expe imen , [22,23],
demons a ing ha such iola ions can eme ge wi hin a pu ely classical amewo k,
Pa .(3). Fo he pu pose o his analysis, i will be su icien o in oduce basic aspec s
o ECT, lea ing he de ails o he ad anced aspec s, such as local modula ions o IP
wi hin cu ed space ime, o p e ious wo k — only men ioned in Pa .(2) and Pa .(5).
In ac i will be su icien o in es iga e dynamics o “signals” eely a eling a
he speed o ligh and hei in e ac ions wi h pola ize s. This ansla es o examining
Cyclic dynamics cha ac e ized by pe sis en ecu ences in ime and space. The e ec
o he poli ize s is gi en by simple phase shi s o hese pe sis en ecu ences. To
a oid con usion wi h QM ’pho ons’, we gene ically e e o hese classical signals as
“massless clocks”, o simply “clocks”. Rema kably, hese “massless clocks” o Cyclic-CM
exhibi dynamics ha p ecisely mi o he beha io o o dina y quan um ’pho ons’ in
he CHSH expe imen , despi e hei inhe en ly classical na u e.
5

In gene al, he iola ion o Bell’s inequali y is p ima y a di ec consequence o
he non-commu a i i y pos ula ed in he ma hema ical o mula ion o QM, as also
o iginally highligh ed by Eins ein’s a gumen s conce ning he incomple eness o QM,
[1], see also [24]. This pos ula ed non-commu a i i y lies a he hea o all non-local
aspec s o quan um phenomena. Con a ily, in [7], and in [6] speci ically o he second
quan iza ion, and p e ious pape s speci ically o he Feynman Pa h In eg al, we ha e
demons a ed ha he condi ion o IP associa ed o gene al Hamil onian sys ems
di ec ly leads non-commu a i i y (iden ical o QM) a a classical le el, wi hou need
o pos ula e hem, as desc ibed in Pa .(2) and Pa .(3). This allows us o de elop a
o malism o Gen-CMs ha enables dynamical analysis wi hou p e-speci ying he
chosen BCs, Pa .(3.1). Consequen ly, by subsequen ly imposing he app op ia e BCs,
we can sepa a ely de i e esul s o bo h S d-CM and he Cyclic-CM. Essen ially,
hese esul s can be di ec ed ex ac ed om he bounda y and bulk e ms o ou Gen-
CM exp essions, espec i ely. In o he wo ds, S d-CM is he “holog aphy p ojec ion”
o he Cyclic-CM on he bounda y o he heo y (and he e o e o he s anda d QM).
In Pa .(3.2) we also de elop a model ha e ec i ely mi o he gene a ion o
quan um en angled s a es a a classical le el. This model is based on Cyclic dynamics
de ined on a wo si e la ice, essen ially an ul a- as “ ic- ac clock” mechanism. I is
de eloped by using ’ Hoo ’s o malism o Cellula Au oma a, [25,26].
In Pa .(4), by pu ing oge he he classical model o en anglemen , he dynamics
o “massless clocks" and he e ec o poli ize s, we will calcula e he join p obabili y
o obse e a posi i e e en in he wo a ms o he CHSH expe imen . In Pa .(4.1), we
will ead om he single Gen-CM exp essions bo h he esul s o S d-CM, ep oducing
he esul o o dina y LHVT, wi h no iola ion o CHSH’s inequali y and he esul
o Cyclic CM, wi h iola ion o he CHSH inequali y a a classical le el, mi o ing he
p edic ions o s anda d QM. The esul is in e p e ed in e ms o Bell’s non-locali y
and Eins ein’s locali y in Pa .(5), demons a ing ha he pe spec i es o hese wo
g ea scien is s a e no con adic o y bu a he complemen a y, ul ima ely leading
o a econcilia ion o hei iewpoin s.
This wo k, as he p e ious on he same opic, in oduces a new pa adigm o clas-
sical and quan um physics, necessi a ing he in oduc ion o no el e minology. While
we ha e s i en o use clea and concise language, some e ms may be uncon en ional
o subjec o a ying in e p e a ions. We encou age he eade o app oach hese
e ms wi h an open mind and e e back o he de ini ions p o ided wi hin he ex
and p e ious wo k o cla i ica ion.
2 How Many Classical Mechanics?
The gene al de ini ion o CM, he e named o con enience Gen-CM, is he se o
dynamics, de i ed om he PSA, equi ing he ac ion o be ex emized bo h in he
bulk and a he bounda ies o a gi en ime in e al3. In his wo k, we dis inguish wo
na u al subclasses o Gen-CM, depending on he choice o he BCs.
3The PSA de ines a comple e Di ichle p oblem: i s a ia ion yields he sys em’s EoMs om he bulk e m
( ia he EL eqs), while he BCs, which selec he speci ic solu ion, a e de e mined by he bounda y e m.
Impo an ly, consis en ly wi h he PSA, his bounda y e m can be minimized in di e en ways, leading o
dis inc subclasses o Gen-CM. See also oo no e.(1)
6
We deno e by ψ( ) he gene al solu ion o mo ion in Gen-CM, p io o speci ying
he BCs ha minimize he ac ion a he bounda y.
2.1 S anda d Classical Mechanics
As is well known, he S d-CM, he o mula ion ypically employed in classical physics,
associa ed wi h he ac ion
S=Z
i
d L(q, ˙
q),(1)
whe e Lis he Lag angian, a e he s a iona y solu ions sa is ying he null a ia ion
δS =Z
i
d (Eule Lag ange eqs)δq +∂L
∂q δq
i≡0.(2)
wi h anishing a ia ions a ime endpoin s iand . In ac , he minimiza ion o he
ac ion bounda y is commonly ob ained by imposing he amilia S d-BCs
δq( i) = δq( )≡0,(3)
oge he wi h he Eule -Lag ange equa ions anishing he ’bulk’ e m. This yields he
S d-CM solu ion o local ene gy E( ), deno ed by q( )(highligh ed in ed in he ex ).
2.2 Cyclic Classical Mechanics
Cyclic-CM (leading o S d-QM) is he subclass o Gen-CM eme ging when PBCs a e
imposed, a he han S d-BCs, en o cing in insic pe iodici y in ime (and hus in
space). These PBCs a e ully compa ible wi h he PSA, minimizing he ac ion a he
bounda y jus like S d-BCs o abo e. This compa ibili y ensu es ha , when p ope ly
implemen ed, PBCs p ese e bo h causali y and locali y, as we will see.
Le us conside he ollowing ac ion whe e he ime in e al has been ede ined as
i→ and → +T( ):
˜
S=Z +T( )
d L(˜
ψ( ),˙
˜
ψ( )) .(4)
Again, we apply he PSA, equi ing null a ia ion
δ˜
S=Z +T( )
d (Eule Lag ange eqs)δ˜
ψ+∂L
∂˜
ψδ˜
ψ
+T( )
≡0.(5)
The ac ion can be equally minimized by equi ing PBCs a he ime bounda ies,
˜
ψ( )≡˜
ψ( +T( )),(6)
oge he wi h he EL eqs coming om he ’bulk’ e m. The solu ion o Cyclic-CM
(highligh ed in blue in he ex ) is deno ed by ˜
ψ( ) o dis inguish i om he Gen-CM
solu ion ψ( ).
7
PBCs can be in e p e ed as a combina ion o wo Di ichle o Neumann condi-
ions imposed a he empo al endpoin s, simila o how such condi ions a e ea ed
in s ing heo y and Kaluza-Klein models. F om a ela i is ic s andpoin , empo al
PBCs a e as na u al and consis en as spa ial ones: due o Lo en z co a iance, spa ial
pe iodici ies in one ame ans o m in o space- ime pe iodici ies in ano he , making
empo al PBCs a di ec consequence o ela i is ic co a iance. Mo eo e , as we will
show, adhe ence o he PSA implies ha he ime pe iod T( )associa ed wi h he
PBCs acqui es a dynamical na u e: i e ol es co a ian ly along he Hamil onian low
o he sys em4. This ecu ence is de e mined locally by he sys em’s ene gy h ough
he de B oglie ela ion (which akes he o m o he S okes’ heo em in Cyclic-CM),
p o iding a ully local and causal desc ip ion o Cyclic-CM.
As u he discussed in Pa .(5), he condi ion o IP exp essed in eq.(6), hough
seemingly a sou ce o mani es non-locali y, is in ac a local and inhe en ly classical
na u e, econciling Bell’s eques o non-locali y wi h Eins ein’s p inciple o locali y.
Theo em (S a is ical Equi alence Be ween Cyclic Classical Mechanics and S an-
da d Quan um Mechanics).Fo any Hamil onian sys em, he s a is ical p edic ions
o Cyclic-CM, de i ed ia co a ian cons ain o in insic pe iodici y, a e equi alen
o hose o S d-QM.
We summa ize esul s o pape [7] ele an o he in e p e a ion o ou esul s.
The solu ion o Cyclic-CM, ˜
ψ( ), sa is ies he s anda d EL eqs, bu exhibi s an
explici ly wa e-like na u e due o he ime ecu ence imposed as a cons ain —
a ea u e ha leads o he Sch ödinge equa ion. Like a ib a ing s ing, ˜
ψ( )is a
Fou ie supe posi ion o ha monic modes selec ed by he PBCs, which in p inciple
can be ex ended o e all ∈R.
The ha monic modes a e de e mined by he local ime ecu ence T( )which,
h ough he EoMs, imply a co esponding local spa ial pe iodici y λ( ). The ollowing
analysis p o ides a sys ema ic me hod o implemen he space- ime ecu ences in in-
sic o any classical sys em, oo ed in he Hamil on-Jacobi op o-mechanical analogy
[17], whe e classical ajec o ies admi a dual wa e desc ip ion.
Wi h pa icula e e ence o he spa ial IP λ( ), cons i u ing he undamen al spa-
ial ecu ences o he ha monics se , he Cyclic-CM solu ion o m a locally comple e
and o hogonal se , de ining a Hilbe space s uc u e. The sys em hus na u ally
admi s a Hilbe space ep esen a ion:
˜
ψ(x) = ⟨x|˜
ψ⟩∈Cyclic Classical Mechanics .(7)
The s a e |˜
ψ⟩is e e ed o as on ic, a ising om a classical solu ion de e mined
by he PSA. On ologically, such s a es desc ibe eal, physically exis ing con igu a-
ions — in con as o he epis emic s a es o s anda d QM, which e lec s a is ical
unde e minism. The ‘on ic’ s a es o Cyclic-CM mus be unde s ood as s a is ical ep-
esen a ions o unde lying ul a- as cyclic dynamics (e.g., a “pa icle on a ci cle”). This
4Nai ely, one may no e ha any canonical ans o ma ion o he ac ion a iables induces a co esponding
ans o ma ion o he bounda y.
8
in e p e a ion is consis en wi h de e minis ic app oaches, such as ’ Hoo ’s Cellula
Au oma on model [25,26].
A ime , we adop he ollowing uni a y no maliza ion o he wa e unc ion o
he Cycli-CM sys em o e a single wa eleng h:
⟨˜
ψ˜
ψ⟩=Zλ( )/2
−λ( )/2
dx ˜
ψ†(x)˜
ψ(x)=1.(8)
The eason o his no maliza ion is in he ac ha , in he case o Cyclic-CM, he
’on ic’ s a e |ψ⟩can be imagined as desc ibing a a s a is ical le el a “pa icle mo ing
e y as in a ci cle”, [6–16]. I na u ally de ines an Hilbe space as also con i med
by ’ Hoo ’s in e ms o Cellula Au oma a, [25,26]. In o he wo ds, simila ly o a
cu en , he p obabili y o ind he pa icle in e e y single cycle mus be one. Thus
|ψ]has he p ope cha ac e is ic o a s a is ical dis ibu ion and Bo n’s ule. Tha
is, simila o Koopman- on Neumann mechanics, [27–29], we can associa e a uni a y
p obabili y o ind he pa icle in he “ci cle”.
We can gene alize ou analysis o any e en-dimensional symplec ic mani old
equipped wi h a closed, non-degene a e 2- o m Ω, known as he symplec ic o m, wi h
dΩ=0. The symplec ic s uc u e also de ines he Poisson b acke {·,·}P B, whe e H
is he Hamil onian o he sys em — o ins ance ha associa ed o he ac ion ˜
S. The
in eg al cu es o XHco espond o solu ions o Hamil on’s equa ions.
The condi ion o empo al IP eq. (6) can be in oduced as a Di ac cons ain
Iγ
d d
d ˜
ψ( )≡0,(9)
whe e we ha e in oduced he no a ion R +T( )
d =Hγ d and γ deno es he closed
ime o bi de e mined by he local PBCs.
I leads o a cons ained Hamil onian o he o m Hγ d ˜
H=Hγ d (H−iℏdψ
d ), wi h
iℏac ing as a mul iplie . In he Hilbe space ep esen a ion, he Hamil onian can be
p omo ed o an ope a o ˜
H→ˆ
H. Di ac’s consis ency condi ion o he cons ain o
IP eq.(9) hen becomes he o dina y Sch ödinge equa ion:
Iγ
d {·,ˆ
H−iℏd
d }P B|˜
ψ( )⟩ ≡ 0.(10)
The ime-e olu ion ope a o is o mally he same one o S d-CM: U( ) =
exp[−i
ℏR
0ˆ
H( )d ]. The mul iplie iℏis de e mined by he Planck ela ion. The local
ene gy spec um o he Cyclic solu ion is
Iγ
En( ′)d ′= (n+β)h,(11)
wi h n∈Z. We ha e included a Mo se ac o βwhich esul s om a wis ac o in he
PBCs — also allowed by he PSA. Fo ins ance, he empo al componen (µ= 0) o
9
Fig. 2 This igu e depic s he CHSH expe imen wi hin he amewo k o Cyclic-CM. The en angled
sou ce is modeled as in insically cyclic classical dynamics on a wo-si e la ice, oscilla ing be ween
wo classical “on ic” s a es, | ↑]and | ↓]a ul a- as imescales, beyond he obse e ’s empo al
esolu ion. The pola ize j, o ien ed a an angle θj, shi s he phase o he emi ed signal’s cyclic
dynamics (’pho ons’) by an angle ±θj, depending on he s a e emi ed.
In pa icula , by adop ing ’ Hoo ’s no a ion, we ind ha quan um en angled
s a es can be ep oduced a a classical le el as a sys ems swi ching pe iodically, e e y
ul a-small ime in e al TS, among wo possible classical “on ic” s a es: | ↑] = 1
0and
| ↓] = 0
1, see ig.(3.2)
C ucially, hese in insically cyclic dynamics a e assumed o occu on ul a- as
imescales TS, as e ha he Comp on ime o elec ons7, a beyond he esolu-
ion o any mode n esolu ion in ime. This apid swi ching, akin a coin ipping e y
as , o a “ ic- ac clock” mechanism ope a ing a ul a- as imescale, o a — - wo
aces — ’die’ olling oo as o obse e he unde lying de e minis ic dynamics, e ec-
i ely ende s he sys em’s s a e obse able only a a s a is ical le el. Rema kably,
he esul ing de e minis ic s a is ical desc ip ion o his IP sys em p ecisely mi o s
he p obabilis ic, inde e minis ic ou comes p edic ed by QM.
Following he esul s o ECT and acco ding o ’ Hoo ’s o malism, [25,26], he
e olu ion o his IP sys em is go e ned by he ollowing e olu ion ope a o :
ˆ
Uop(TS) = 0 1
1 0 .(26)
7The emission o pho ons om an a om — as much as all de ec ion sys ems — is go e ned by elec ody-
namics, whose empo al dynamics a e ine i ably de e mined by he in e nal cyclic dynamics o he elec ons
and hus as e o equal o he elec on Comp on ime TC=h/mec2∼10−21 s, being me he mass o he
elec on. This se s a undamen al h eshold below which he ul a- as dynamics p edic ed by Cyclic-CM
canno be di ec ly esol ed. A imescales abo e his limi , he Cyclic-CM can only be accessed h ough
s a is ical desc ip ions.
16

I s diagonaliza ion implies he ollowing eigen ec o s
|Ψ] = 1
√2(| ↑]−| ↓]),|Ψ′] = 1
√2(| ↑] + | ↓]) .(27)
In his model he e is he possibili y o in oduce a Hamil onian ope a o , Hop.
Rema kably, he sys em’s e olu ion is hen desc ibed by he o dina y Sch ödinge
equa ion and sa is ies he desi ed co espondences wi h o dina y QM, including
implici non-commu a i i y induced by he IP, while s ill main aining i s inhe en ly
classical na u e, [6–16], as also con i med by ’ Hoo wo k [25,26].
To es ablish a classical desc ip ion o a sou ce o single -s a e en anglemen , we
mus also in oduce a selec ion mechanism ha isola es only he s a e |Ψ] om he
wo eigen ec o s o eqs.(27).
4 Viola ion o Bell’s Inequali y in Classical Mechanics
We de ine each o he “on ic” s a es | ↑]and | ↓]o he ele an eigen ec o |Ψ] in
e ms o wo Gen-CM ee “massless clocks” labeled 1and 2o he kind desc ibed
in Pa .(3.1), a eling in opposi e di ec ions along he z-axis a he speed o ligh .
To model he ∆J= 0 sou ce associa ed o |Ψ], we assume ha he “clocks” ha e
ans e se oscilla ions in o hogonal di ec ions. Speci ically, he wo “on ic” s a es a e
de ined as | ↑] = |11x,01y,02x,12y]and | ↓] = |01x,11y,12x,02y], whe e |1jx] = a†
jx|0]
ep esen s “clock” joscilla ing in he x-di ec ion, as allowed by he pseudo Di ac
algeb a o he single componen s o he wo Gen-CM ’clocks’, see Pa .(2) and Pa .(3):
|Ψ] ≡1
√2|11x,01y,02x,12y]−|01x,11y,12x,02y].(28)
The e ec o a pola ize o ien ed a an angle θjon he “clock" jis o o a e he
di ec ion o ans e se oscilla ion by an angle θj om he x-axis, [22,23]. Wi hin he
amewo k o Gen-CM, his e ec can be desc ibed as ollows:
aj=ajx cos θj+ajy sin θj(29)
whe e ajand ajx o ajy a e he S d-CM dynamical a iable behind and be o e he
pola ize .
4.0.1 The Phase Shi on he Condi ion o In insic Pe iodici y
induced by he Pola ize s
We ind ha he e ec o a pola ize on a ’massless clock’ is o induce a posi i e
o nega i e phase shi in i s oscilla ion. This phase shi plays a c ucial ole in ou
analysis, as i di ec ly in luences he speci ic condi ions o IP equi ed o ob ain he
co espondence wi h s anda d QM and in u n o achie e iola ions o he CHSH
inequali y. As a ule o humb, we mus emembe ha he condi ion o IP is in ac
de e mined by he ecu ence o he wa e- unc ion, [6–16].
17
Fo simplici y, we assume ha bo h ’massless clocks’ exhibi he same spa ial ecu -
ence λ=πwhen p opaga ing away be o e and a e he pola ize s. Consequen ly,
he spa ial ecu ence o he combined s a e |Ψ] is π/2, esul ing om he p oduc o
hese wo π-pe iodic phenomena. The π/2in e als is also he e e ence leng h o he
no maliza ion o he wa e- unc ion |Ψ] o he sys em.
Fo his scope i is con enien o adop he ollowing subs i u ion in place o eq.(29),
aj=1
√2(aj+eiθj+aj−e−iθj),(30)
so ha Gen-CM “en angled s a e” eads
|Ψ] ≡1
√2|11+,01−,02+,12−]−|01+,11−,12+,02−].(31)
whe e aj+=1
√2(ajx −iajy)and aj−=1
√2(ajx +iajy)and, e.g. ,|1j±] = a†
j±|0] is he
ela ed pseudo-Di ac algeb a.
As he “clock” jin e ac s wi h a pola ize o ien ed a an angle θj, i unde goes
a sudden phase shi : ψj±(z)→ψj±(z∓θj). This phase shi necessi a es a co e-
sponding adjus men o he bounda ies o he in eg a ion by pa s in eq.(19), which
mus be displaced acco dingly. Consequen ly, he condi ion o IP equi ed o es ablish
Cyclic-CM in ol es a spa ial coo dina e shi z→z∓θj o clock jwi h espec o
he ee case eq.(18). In pa icula , we ha e he shi on he pa ame e o he dynami-
cal a iable aj±(z)→aj±(z∓θj)as he “clock” passes h ough he pola ize , see also
[30–33].
Thus, he in e ac ion wi h he pola ize modi ies he IP condi ion wi h espec o
he ee case. The bounda y o he in eg a ion by pa eq.(19) a e shi ed by an amoun
±θj, co esponding o he phase shi expe ienced by “clock”. This shi , howe e , has
no signi ican impac on s a es o he o m [1j±|a†
j±aj±|1j±], as he displacemen s in
he wo bounda ies a e in he same di ec ions. Thus hei e ec cancel each o he ou .
In his case he shi will no be epo ed in he calcula ions. In con as , o s a es like
[1j±|a†
j±aj∓|1j∓], he shi s occu in opposi e di ec ions, con ibu ing he inal esul .
As demons a ed in he p e ious sec ions and igo ously p o en in [6], all ladde
ope a o s wi hin he amewo k o Cyclic-CM sa is y he same algeb aic ela ions as
hei coun e pa s in s anda d QM, which in his case a e:
Cyclic-CM: [ˆ
aj±,ˆ
a†
i±] = δij ,[ˆ
aj±,ˆ
a†
i∓]=0,(32)
while hey anish in he case o S d-CM. The ha symbol ep esen s he ac ha
he ope a o s do no commu e in he amewo k Cyclic-CM, exac ly mimicking he
s anda d QM ope a o s.
4.1 Join P obabili y Acco ding o Gene al Classical Mechanics
We can now o mula e he join p obabili y wi hin he amewo k o Gen-CM o bo h
signals o pass hei espec i e pola ize s. This p obabili y is assumed o be gi en by,
18
[22,23],
PGen−CM
⊕,⊕(θ1, θ2) = [Ψ|a†
1a†
2a2a1|Ψ] .(33)
The no a ion is u he simpli ied by in oducing
A↑=a1+a2−, A↓=a1−a2+ .(34)
By aking in o accoun he displacemen o he bounda ies induced by he pola izes
he join p obabili y o Gen-CM, eq.(33), is
PGen−CM
⊕,⊕(θ1, θ2) = 1
8[↑ |A†
↑A↑| ↑]π
4
−π
4−[↑ |A†
↑ei2∆θA↓| ↓]|π
4−|∆θ||
−|π
4−|∆θ||
−[↓ |A†
↓e−i2∆θA↑| ↑]|π
4−|∆θ||
−|π
4−|∆θ||+ [↓ |A†
↓A↓| ↓]π
4
−π
4.(35)
The inal esul mus be in a ian unde he exchange o pola ize o ien a ions,
depending solely on he absolu e alue o hei angle di e ence: |∆θ|=|θ2−θ1|.
The bounda y o he mixed e ms is se o ±π
4−|∆θ| o ensu e ha he calcu-
la ed expec a ion alues emain wi hin he physically meaning ul ange o 0 o 1 ( he
in e sion o he o de be ween he ini ial and inal poin s mus be accompanied by a
co esponding sign in e sion in he in eg al).
Explici calcula ion o he indi idual e ms is achie ed h ough in eg a ion by
pa s, simila ly o eq.(23). In he e alua ion we also ecu si ely apply he commu a ion
ela ions, keeping in mind ha hey a e ei he he s anda d one, eq.(32), o Cyclic-
CM o hey a e sys ema ically anishing o he S d-CM. See Appendix.(A) o he
explici calcula ions.
The i s and ou h e ms a e independen o ∆θas hei associa ed bounda y
e ms a e displaced in he same di ec ion. The second and hi d e ms exhibi a
dependence on |∆θ|, a ising om he ac ha hei espec i e bounda y e ms a e
displaced in opposi e di ec ions:
[↑ |A†
↑A↑| ↑]π
4
−π
4=Ψ†
↑zΨ↑π
4
−π
4
+↑[A↑, A†
↑]↑π
4
−π
4
,
[↓ |A†
↓A↓| ↓]π
4
−π
4=Ψ†
↓zΨ↓π
4
−π
4
+↓ |[A↓, A†
↓]↓π
4
−π
4
,
[↑ |A†
↑ei2∆θA↓| ↓]|π
4−|∆θ||
−|π
4−|∆θ||=hΨ†
↑ei2∆θzΨ↓i|π
2−|∆θ||
−|π
2−|∆θ||
+h↑ei2∆θ[A↑, A†
↓]↓i|π
2−|∆θ||
−|π
2−|∆θ||,
[↓ |A†
↓e−i2∆θA↑| ↑]|π
4−|∆θ||
−|π
4−|∆θ||=hΨ†
↓e−i2∆θzΨ↑i|π
2−|∆θ||
−|π
2−|∆θ||
+h↓e−i2∆θ[A↑, A†
↓]↑i|π
2−|∆θ||
−|π
2−|∆θ||.(36)
19
Nex we will ex ac sepa a ely he Cyclic-CM and S d-CM esul s by imposing IP
( anishing he bounda y e ms), yielding esul s ha p ecisely ma ch hose o QM,
o by assuming commu a i i y o ob ain he esul s o S d-CMs, iden ical o hose o
LHVT.
4.1.1 Join P obabili y o Cyclic Classic Mechanics: Viola ion o
Bell’s Inequali y
The iola ion o he CHSH inequali y is ex ac ed om he Gen-CM join p oba-
bili y eq.(33) by simply imposing IP, which means anishing bounda y e ms. These
a e he Cyclic-CM whe e he commu a ion ela ions eq.(32) a e implici ly sa is ied a
a classical le el (wi hou pos ula ing hem). Thus,
Ψ†
↑zΨ↑π
4
−π
4
= 0 ⇔↑[ˆ
A↑,ˆ
A†
↑]↑π
4
−π
4
PBCs
≡⟨↑ |[ˆ
A↑,ˆ
A†
↑]| ↑⟩ = 1 ,
Ψ†
↓zΨ↓π
4
−π
4
= 0 ⇔↓[ˆ
A↓,ˆ
A†
↓]↓π
4
−π
4
PBCs
≡⟨↓ |[ˆ
A↓,ˆ
A†
↓]| ↓⟩ = 1 .(37)
Again ⟨ ⟩ indica es ha Cyclic-CM ep oduces he same esul o o dina y QM.
The anishing bounda y e ms, implying he commu a o s eq.(32), in u n also
yields




Ψ†
↑ei2∆θzΨ↓|π
4−|∆θ||
−|π
4−|∆θ||= 0
Ψ†
↓e−i2∆θzΨ↑|π
4−|∆θ||
−|π
4−|∆θ||= 0
(38)
and hus8







h↑ei2∆θ[A↑, A†
↓]↓i|π
2−|∆θ||
−|π
2−|∆θ||
PBCs
≡⟨↑ |ei2∆θ[ˆ
A↑,ˆ
A†
↓]| ↓⟩ =ei2∆θ.
h↓e−i2∆θ[A↑, A†
↓]↑i|π
2−|∆θ||
−|π
2−|∆θ||
PBCs
≡⟨↓ |e−i2∆θ[ˆ
A↓,ˆ
A†
↑]| ↑⟩ =e−i2∆θ
(39)
These esul s a e iden ical o hose ob ained in s anda d QM, ep esen ed by he
quan i y ⟨ ⟩, despi e being de i ed en i ely wi hin he amewo k o Gen-CM. This
ag eemen is achie ed by imposing IP on he sys em, in ull adhe ence o he PLA.
Thus we ha e local CM. The join p obabili y o Cyclic-CM is hus:
Cyclic CM: PCyclic−CM
⊕,⊕(θ1, θ2) = 1
2sin2(θ1−θ2).(40)
As well-known, [22,23], a di ec consequence o his esul is he iola ion o Bell’s
inequali ies. Rema kably, in con as o con en ional belie , we ha e jus demons a ed
8He e he co esponding e ms o eq.(37) mus be p ope ly escaled by no maliza ion ac o s in o de o
ob ain he QM esul which is no malized o a single wa e-leng h π
20
he exis ence o a class o CM, named he e Cyclic-CM, which exhibi s Bell’s inequali y
iola ions o he same ex en as obse ed in QM.
Se e al poin s equi e pa icula ca e in he abo e de i a ion. The phase ac o s
e±2∆θin he bounda y e ms adjus p ecisely he ecu ence o wa e- unc ions so ha
hey ha e iden ical alues a he shi ed bounda ies. A he same ime he p esence o
a pola ize implies he modi ica ion o he IP condi ion on he coo dina e zin o de
o ob ain anishing bounda y e ms. As al eady said, he IP acqui es a posi i e o
nega i e shi ∆θas he signal passes h ough he pola ize s. These shi s cancel in
he homogeneous e ms and sum in he non-homogeneous e ms. IP means ha z
akes he same alues a he wo bounda ies. Finally, i ’s impo an o no e ha he
no maliza ion o he p obabili ies mus be pe o med o e an unpe u bed pe iod o
π/2, as he wa e unc ion |ψ]is no malized wi h espec o his pe iod.
4.1.2 Join P obabili y o S anda d Classical Mechanics: Bell’s
Inequali y
The esul o S d-CM, cha ac e ized by S d-BCs, eme ges om he Gen-CM by
simply assuming commu a i i y among all he ladde ope a o s. The only nonze o
con ibu ion in he igh hand side o eq.(48) comes om he bounda y e ms, which,
unlike in he case o In insic Pe iodici y (IP), now do no anish:
↑A†↑,∆θA↑,∆θ↑π
4
−π
4
S d-BCs
≡Ψ†
↑zΨ↑π
4
−π
4
= 1 ,
↓A†↓,∆θA↓,∆θ↓π
4
−π
4
S d-BCs
≡Ψ†
↓zΨ↓π
4
−π
4
= 1 ,(41)
and
↑A†
↑ei2∆θA↓↓|π
4−|∆θ||
−|π
4−|∆θ||
S d-BCs
≡hΨ†
↑ei2∆θzΨ↓i|π
4−|∆θ||
−|π
4−|∆θ||=
1−2|∆θ|
π
↓A†
↓e−i2∆θA↑↑|π
4−|∆θ||
−|π
4−|∆θ||
S d-BCs
≡hΨ†
↓e−i2∆θzΨ↑i|π
4−|∆θ||
−|π
4−|∆θ||=
1−2|∆θ|
π
.(42)
The con ibu ions o he phase shi s e±2∆θplay again a c ucial ole in he e alua ion
o he wo bounda y e ms. These phase ac o s ensu e ha he wa e- unc ions assume
iden ical alues a he shi ed bounda ies as in he p e ious case. Howe e , in his
scena io, he condi ion o IP is no applicable o he coo dina e z, leading o di e en
alues a he he wo bounda ies o each exp ession, and hus o he esul s abo e.
In conclusion, he esul ob ained wi hin he amewo k o S d-CM is he p edic ion
o LHV heo ies:
S anda d CM: PS d−CM
⊕,⊕(θ1, θ2) = 1
21−
1−2|∆θ|
π.(43)
21

To ensu e p ope p obabili y no maliza ion wi hin he amewo k o S d-CM, he
p obabili y ampli udes mus be mul iplied by a ac o o 2. This no maliza ion ac-
o a ises om he di e ence in ecu ence pe iods be ween he wo-pho on en angled
s a e |Ψ] and he ’single-pho on’ s a e wi hin |ψj]. As al eady said, he ’ wo-pho on’
s a e exhibi s a ecu ence pe iod π/2while he ’single-pho on’ s a e has a ecu -
ence pe iod o π. Cyclic-CM, go e ned by he condi ion o IP, inhe en ly en o ces
p obabili y no maliza ion wi h espec o he co ec ecu ence pe iod o π/2in he
Cyclic-CM. The condi ion o IP in ac says ha he uni a y no maliza ion mus be
on he elemen a y ecu ence, analogous o he concep o a ’pa icle on a ci cle’ dis-
cussed in Pa .(3.2). The p obabili y o obse ing wo ’pho ons’ wi hin he S d-CM
amewo k is pe o med wi h espec o he o iginal pe iod π, being no a ec ed by
he condi ion o IP, leading o a double-coun ing o e en s.
5 Concilia ing Bell’s Non-Locali y and Eins ein’s
Locali y
Acco ding o ECT, [6–16] whose ounda ional aspec s a e epo ed in Pa .(2), in
Cyclic-CM, when in e ac ions a e p esen , he space- ime IP becomes a dynamical
a iable: a con a a ian ou - ec o modula ed along he sys em’s Hamil onian low.
Acco ding o S okes’ heo em, see eq.(15), IP couples o he co a ian ou -momen um
o yield he ela i is ic in a ian h.
Impo an ly, he local ans o ma ion o he IP, which mus be unde s ood as a
dynamical a iable, p e en s ha he Lo en z in a iance is b oken in ECT, as a
con- sequence o i s adhe ence o he classical PSA. The ime ecu ence T(x)and
spa ial ecu ence λ(x)a e locally de e mined by he local ene gy-momen um ia
he de B oglie ela ion, which, imposed as cons ain , akes he o m a gene alized
Boh -Somme eld quan iza ion o S oke’s heo em, eq.(15). I ep esen s quan iza ion
condi ion equi - alen o he PBCs eq.(6) — condi ion o closed space ime o bi s.
Only in e ac ions wi hin he ligh cone can in luence hese modula ions, ensu ing ha
causali y and locali y a e espec ed. Thus, he condi ion o IP is a local and dynam-
ical p inciple consis en wi h ela i is ic classical physics — and wi h g a i a ional
and gauge in e ac ions [7,15].
Despi e i s classical ounda ion, Cyclic-CM exhibi s a unique o m o co ela-
ion ha esembles Bell’s no ion o non-locali y when conside ing sys ems ha a e
supposed o ha e emained en i ely isola ed, e en a space-like sepa a ed poin s.
This non-locali y, while pe missible wi hin he classical PSA, is con ingen upon he
hypo heses o absence o in e ac ions. In he absolu e absence o in e ac ions, a phys-
ical quan i y exhibi ing pe sis en IP in ime and space will ha e iden ical alues a
all poin s sepa a ed by in ege mul iples o i s spa ial and empo al pe iods x+nλ
o +nT. This implies an inhe en non-locali y, as knowledge o he quan i y’s alue
a one poin ins an aneously e eals i s alue a dis an space-like sepa a ed poin s,
assuming ha any in e ac ion — a ia ion o ene gy — is occu ed (pe sis en IP)
o al e he ecu ences. This ype o “non-locali y” unde pins he iola ion o Bell’s
inequali y obse ed in his wo k and could be in e p e ed as a “spooky ac ion a a
dis ance” i he dynamical na u e o IP is no conside ed.
22
On he o he hand, i in e ac ion occu s a space-like dis ance, in ha space-
ime poin he ecu ences will be al e na ed wi h espec o he pe sis en ( ee)
case, and he alue o he physical quan i y a ha dis an poin will no longe be
p edic able. The appa en incompa ibili y be ween he non-local na u e o S d-QM
and local na u e o classical mechanics is esol ed by ecognizing he c ucial ole o
in e ac ions in modula ing he appa en ly “non-local” condi ion o IP.
Cyclic-CM esol es he appa en con lic be ween Eins ein’s demand o locali y
and Bell’s demons a ion o non-locali y o na u e. Bo h pe spec i es a e alid wi hin
Cyclic-CM, and his long-s anding deba e is inally econciled. QM eme ges om
undamen ally classical, de e minis ic physics oo ed in he ul a- as cyclic dynamics
associa ed o e e y massi e elemen a y pa icle. Acco ding o ECT e e y elemen a y
pa icle o mass mcan be see as an elemen a y clock, icking a a a e o TC=h/mc2
(see also de B oglie in e nal clock). These a es a e inc edibly as . As an example, an
elec on, a ligh pa icle, exhibi s an IP o he o de o he 10−21 sec — see oo no e.(7).
This is an IP on he p ope ime o he pa icle, is he in insic ecu ence o he es
pa icle. The case o pho ons is pa icula ly in iguing. As expec ed om ela i is ic
p inciples, he in e nal clock o a pho on appea s ozen (TC=∞). Obse ing i s
es IP would equi e a eling a he speed o ligh , leading o in ini e ime dila ion
o he obse e . A e all his dynamical na u e o ime ecu ences is also implici
in GR, which can be consis en ly desc ibed in e ms o clock modula ions and ule
expansions induced by he g a i a ional ield, [34].
Beyond ep oducing QM, ECT ex ends he clock-based in e p e a ion o Gene al
Rela i i y o all in e ac ions, including gauge in e ac ion. As ully p o en in [15] and
gene ally shown by eq.(13), gauge in e ac ions can be de i ed om classical P inci-
ple o Gene al In a iance, wi hou pos ula ing gauge symme y — in ealiza ion o
o iginal Weyl’s p oposal. This leads o a uni ied desc ip ion whe e all in e ac ions
(including gauge in e ac ions) modula e he in e nal clocks o pa icles — keeping
hem pe ec ly synch onized since he beginning o ime — o e ing concep ual links
o supe de e minism [25,26].
Finally, we mus no e ha Cyclic-CM doesn’ in ol e any hidden a iables, being
based exclusi ely on he geome y o he space ime coo dina es and PBCs de e -
mined by he Planck cons an which ela es he space ime ecu ences wi h he
ou -momen um o he pa icles. This poin is c ucial because Bell’s heo em, which
elies on he assump ion o local hidden a iables, canno be he e o e in oked o ule
ou Cyclic-CM as a iable classical mechanics a he base o QM.
6 Conclusions
In pas wo ks, we ha e de eloped a classical heo y, namely Elemen a y Space-Time
Cycles Theo y (ECT), [6–16], s ongly indica ing a possible de i a ion o bo h s an-
da d QM and ela i is ic mechanics in a uni ied way di ec ly om he de e minis ic
classical P inciple o S a iona y Ac ion (PSA) and wi hou in ol ing hidden a iables
o any so — mo eo e , wi hin ECT, bo h gauge heo ies and g a i y eme ge om
he Gene al P inciple o In a iance, see [15], and subsequen pape s.
23
The o mal analysis o his pape indica es ha Bell’s inequali y is iola ed a a
classical le el wi hin he subclass o Classical Mechanics, named he e Cyclic Classi-
cal Mechanics (Cyclic-CM), amewo k inhe en o ECT, o he same ex en as in
s anda d QM. Con e sely, Bell’s inequali y is sa is ied in he o he subclass which is
he S anda d Classical Mechanics (S d-CM) — he common mechanics used o s udy
classical sys ems — including LHVT. By de eloping a no el calcula ion echnique,
we show ha S d-CM eme ges as a “holog aphic p ojec ion” o he mo e undamen al
Cyclic-CM, u he es ablishing ECT as he pe ec candida e o a classical heo y
ha encompasses and ex ends beyond QM.
The c ucial elemen o a comple e classical desc ip ion o QM, in he spi i o EPR,
is he assump ion o in insically cyclic dynamics a ul a- as scales o he elemen a y
pa icles, whose ecu ence pe iods mus be unde s ood as dynamical unc ion o he
local ene gy and he e o e p ese ing causali y and locali y. This In insic Pe iodici y
(IP) na u ally gi es ise o he non-commu a i i y o conjuga e a iables obse ed in
s anda d QM, wi hou equi ing i s pos ula ion as an axiom, as p o en in he mos
gene al way in [7] o e e y possible Symplec ic sys em (including sys ems wi h in ini e
deg ees o eedom such as in ield heo y and in Riemann geome ies such as in GR).
The physical p inciple IP seems o esol e he appa en con lic be ween Eins ein’s
and Bell’s iews on QM. Like a ela i is ic clock icking pe iodically, e e y elemen-
a y pa icle exhibi s a ecu ing beha io p edic able in ime, e en beyond he ligh
cone, i o al absence o in e ac ions is assumed. This inhe en p edic abili y embodies
Bell’s non-locali y, which, while compa ible wi h Gene al Rela i i y, in e ac ions, such
as hose modeled by pola ize s in his pape , causally al e hese ecu ences, e lec -
ing Eins ein’s locali y. The e o e, Cyclic-CM is a alid candida e o he concilia ion
o bo h: Bell’s non-locali y (p edic able ecu ence o elemen a y pa icle dynamics)
and Eins ein’s locali y (local, causal modula ion o cyclic dynamics mean o dynam-
ical cha ac e simila o ela i is ic clocks in GR). An impo an con i ma ion o ou
heo e ical analysis would be he implemen a ion o a classical algo i hm, based on
he Cyclic-CM desc ibed in ou wo ks, ha exhibi s a iola ion o Bell’s inequali y,
wi hou in oking hidden a iables.
Acknowledgemen
I would like o hank P o esso s Ge a d ’ Hoo , B ian Josephson and Theophanes
Rap is o hei help ul discussions, and Mau a Pandol i o he suppo .
7 Decla a ions
No unding was ecei ed o conduc ing his s udy.
24
A Explici Calcula ion o he Join P obabili y
We ha e he ollowing use ul iden i ies o commu a o s among gene ic ope a o s A,
B,Cand D
[AB, CD] = A[B, C]D+C[A, D]B+1
2[A, C](BD +DB) + 1
2[B, D](CA +AC)(44)
I s ecu si e applica ion on he Gen-CM he le side o he expec a ion alues eq.(48)
yields
[↑ |a†
↑a↑| ↑] = 1
2[0|a1+a†
1+ +a2−a†
2−|0]
=1
2[11+,01−,02+,02−|a†
1+a1+|11+,01−,02+,02−]
+1
2[01+,01−,02+,12−|a†
2−a2−|01+,01−,02+,12−]
[↓ |a†
↓a↓| ↓] = 1
2[0|a1−a†
1−+a2+a†
2+|0]
=1
2[01+,11−,02+,02−|a†
1−a1−|01+,11−,02+,02−]
+1
2[01+,01−,12+,02−|a†
2+a2+|01+,01−,12+,02−]
[↑ |a†
↑a↓| ↓] = 1
4[0|a1+a†
1+ +a1−a†
1−+a2+a†
2+ +a†
2−a2−|0]
=1
4[11+,01−,02+,02−|a†
1+a1+|11+,01−,02+,02−]
+1
4[01+,11−,02+,02−|a†
1−a1−|01+,11−,02+,02−]
+1
4[01+,01−,12+,02−|a†
2+a2+|01+,01−,12+,02−]
+1
4[01+,01−,12+,02−|a†
2+a2+|01+,01−,12+,02−]=[↓ |a†
↓a↑| ↑](45)
In his o m i is possible o pe o m he in eg a ion by pa s analysis, , simila ly
o eq.(19, isola ing he bounda y e ms con aining he esul o S d-CM and he bulk
e ms con aining he esul s o Cyclic-CM. Fo b e i y, he bounda y alues a e no
explici ly s a ed. Thus:
[↑ |a†
↑a↑| ↑] = 1
2[Ψ†
1+zΨ1+] + 1
2Zdz؆
1+[a1+, a†
1+]Ψ1+
+1
2[Ψ†
2−zΨ2−] + 1
2Zdz؆
2−[a2−, a†
2−]Ψ2−
[↓ |a†
↓a↓| ↓] = 1
2[Ψ†
1+zΨ1+] + 1
2Zdz؆
1+[a1+, a†
1+]Ψ1+
25