An -Be nouilli Random Va iable, Time Re e sal In a iance and he Quan um exp(-E )
F ancesco R. Rugge i Hanwell, N.B. No . 6, 2025
In (1), a ype o p obabili y conse a ion o an an i-Be nouilli a iable linked o diagonal mo ion
along box diagonals in an x- space wi h 1-md being he p obabili y ha x di ec ion mo ion
changes leads o a p obabili y o exp(im ). This app oach seems o in ol e a kind o ime
e e sal symme y as (1) p oposes a -x space consis ing o g ids o spacing e and possible
mo ion along diagonals wi h bo h +/- e mo ion in and x. In pa icula , (1) conside s mo ion in he
posi i e di ec ion om he o igin o a some T (wi h +/-e mo ion possible) and hen e aces he
di ec ion o e u n o he o igin. In o he wo ds, (1)’s m is no an a bi a y numbe , bu is linked o
a conse ed quan i y, ene gy, i.e. p obabili y s eps a e in e ms o ene gy i one compa es wi h
he quan um exp(-E ), which (1) does. (1) ob ains a kind o Di ac equa ion, bu al hough we
p o ide analysis o his, we only conside he o m exp(im ) as ele an . Fu he mo e, exp(iE ) is
consis en wi h conse a ion o ene gy and conse a ion o momen um ollows om he x o m
o he solu ion exp(ipx) and so he p obabili y app oach o (1) is somehow linked o a change in
di ec ion, i.e. collision, which ul ima ely is associa ed wi h a conse ed m, i.e. E (ene gy).
In p e ious no es, we sugges ed ha one may ob ain he ee pa icle quan um wa e unc ion
exp(-iE +ipx) by sugges ing ha a p obabili y exis s in New onian wo-body elas ic sca e ing.
We a gued ha gi en ini ial (e1,e2) and (p1,p2) (ene gy, momen um) pa icles, any (ei,ej) (pi,pj)
ec o s has he same p oduc p obabili y (equal o he p oduc p obabili y o he (e1,e2) (p1,p2)
case) has he same alue. In (2), we showed ha one canno use exp(ip) o he momen um
p obabili y because i is no unique and iola es ime e e sal symme y. We sugges ed using a
Lo en z in a ian esul exp(-iE +ipx).
The a gumen we make in his no e is ha he app oach o (1) and (2) seem o be he same.
A i s his seems o be su p ising. (1) ob ains exp(-iE ) s ic ly h ough a “special” an i-Be nouilli
somewha - andom walk app oach. The e is no concep o ene gy o momen um, hence no
concep o hei conse a ion and no concep o special ela i i y. (2) on he o he hand is based
on all o hese ideas. We sugges ha gi en ha p obabili y is in oduced in (2), he no ion o
physical pa hs and changes is common o bo h app oaches. Fu he mo e, we sugges ha bo h
a e based on ime e e sal symme y. We ha e al eady no ed ha he numbe m in (1) is
equi alen o ene gy. Thus, ene gy is linked wi h he an i-Be nouilli p obabili y mo ion in ime
(and p wi h mo ion in x). This, in a sense, is he same as a conse a ion o ene gy and
momen um app oach, we a gue. I one conside s ha ene gy and momen um ollow om
special ela i i y by Lo en z boos ing a es mass, hen one eally has he concep o
conse a ion o he es mass which is he same as conse a ion o a ce ain kind o p obabili y.
Thus, we a gue ha (1) is an in e es ing app oach o lushing ou he p obabilis ic na u e o he
Di ac equa ion which we a gue is also di ec ly linked o in oducing a dynamic di ec ional
p obabili y which desc ibes ene gy and momen um conse ing in e ac ions.
An i-Be nouilli Quasi-Random Walk o (1)
In (1) a special p obabilis ic quasi- andom walk which leads o he de i a ion o an exp(im )
p obabili y is in oduced. The key poin we make is ha his is a “special” ype o p obabili y. In
(1), a s anda d Be noulli app oach is used and i does no lead o exp(im ), bu o (cosh(m ),
sinh(m )). As a esul , we sugges one mus analyze he assump ions which a e used in (1)’s
p obabilis ic model ca e ully, we sugges .
A -x g id is conside ed wi h spacing o e. A pa icle may mo e o he igh o le , bu only
along a diagonal. I a pa icle mo es o he igh in x, his is associa ed wi h a unc ion phi+(x,y),
o he wise wi h phi-(x,y). A pa icle which has jus mo ed o he igh has
P obabili y 1-em o mo e o he igh on he nex mo e ((1))
P obabili y em o mo e o he le ((1b)) and ice e sa
Thus, a key idea is in oduced. The numbe m go e ns he p obabili y o mo ion in x.
An an i-Be nouilli a iable is in oduced as:
Value = 1 i he pa icle mo es om o +d ((2))
0 i i doesn’ mo e
-1 i i mo es backwa ds in ime om o -d
In (1) i is s a ed ha a pa icle begins a he o igin and mo es only in he posi i e ime ( )
di ec ion un il i eaches a ime T. A each poin , i may mo e in he posi i e o nega i e x
di ec ion acco ding o ((1)). A e e y o he poin whe e em applies, i.e. i should change
di ec ion, i doesn’ , bu places a ma ke which may be used on he e u n ip.
As a esul , (1) p o ides zigzag diag ams o mo ion in he posi i e ime di ec ion and he in he
nega i e ime e u n ip. These wo zigzags c ea e ec angles. I one chooses a pa h along he
ou side walls o he ec angles, one ob ains a single pa h which con ains bo h o wa d and
nega i e imes. I is hese pa hs which (1) conside s and he weigh s nega i e ime pa hs wi h a
-1. This is his an i-Be nouilli a iable.
In (1) i is s a ed ha one may de ine wo unc ions, phi+(x, ) and phi-(x, ) o accoun o he
a e age o he an i-Be nouilli a iable ((2)) in ei he he posi i e o nega i e x di ec ion. Then:
Phi+( ) = Sum o e x phi+(x, ) and Phi-( ) = Sum o e x phi-(x, ) ((3))
In (1) i is s a ed ha o conse e p obabili y one has:
Phi+( +e) = (1-em) Phi+( ) - em Phi-( ) ((4a))
Phi-( +e) = (1-em) Phi-( ) + em Phi+( ) ((4b))
We wish o examine ((4)) in mo e de ail. Fi s , one no ices ha ((4a)) con ains a minus sign and
((4b)) does no . Thus, he e is symme y b eaking. To y o see his esul , we ollow diag ams
p o ided by (1). We i s conside zigzag mo ion in x, bu only an inc ease in o a ce ain T. This
zigzag is colou ed blue and ep esen s o wa d mo ion in ime. Then, one c ea es a ed zigzag
making ec angles wi h he blue zigzag. This ep esen s e u n mo ion which is nega i e in ime
(i.e. -1 Be nouilli alue). I one conside s wo blue ou e ec angle lines, hen an ou e ed line
emana es pe pendicula o he second. In o he wo ds one has a zigzag o blue (posi i e x
di ec ion)-blue (nega i e x di ec ion)- ed (posi i e x di ec ion).
We s ess ha he zigzag is a combina ion o o wa d and nega i e ime mo ion, i.e. i is a
ma hema ical pa h which consis s o pieces o a o wa d in ime andom ype walk and a
nega i e.
The second junc ion shows blue x-neg and ed-x posi i e o
Ph-( +e) = (1-em) Phi-( ) -1 Phi+( )
The i s wo junc ions o he zigzag (blue posi i e x) - (blue nega i e x) yield:
Phi+( +e) = Phi+( ) (1-em) + em Phi-( )
This is he opposi e o ((4)), bu ha is no a p oblem. The poin is ha one equa ion should
ha e a posi i e sign and he o he , no . This leads o:
Phi= (Phi+, Phi-) and d/d Phi = -m Phi -m i Pauli sigma(y) ((5))
Wi h Phi+(0)= Phi-(0)=0 one inds ha Phi+( )= exp(-m ) (cos(m ), sin(m )) ((6))
The key poin o ((6)) is ha no one has he o m exp(im ) which is a quan um ee pa icle
esul linked wi h conse a ion o m. Thus, conse a ion o m na u ally appea s. In ac , his
mimics he quan um exp(-iE ) and o =d , exp(-iEd ) = 1-Ed which is he p obabili y (1) uses o
a di ec ion change in x no o occu . In o he wo ds exp(-iEd ) means ha he pa icle con inues
in i s same gene al x di ec ion o a ee non-in e ac ing pa icle.
One mus , howe e , accoun o x as well . In (1), his is done h ough pu ing in he x de ail in
he wo zigzags desc ibed abo e. (No e: The zigzags desc ibe in e change Phi+ and Phi-, bu
ha is i ele an .) Gi en ha x is p esen , we now use he no a ion phi+(x, ), phi-(x, ) as Phi+
means one has summed o e x.
phi+(x, +e) = (1-em) phi+(x-e, ) - em phi-(x, ) ((7a))
We no e ha he second e m on he RHS is phi- which changes i s di ec ion.
phi-(x-e, +e) = (1-em) phi-(x, ) + em phi+(x-e, ) ((7b))
((7a)) and ((7b)) a e expanded in a Taylo se ies. We ob ain a di e en sign in one e m han (1)
and so w i e ((8)) (see below) as -sigma Pauli z no Sigma-z Pauli. ((8)) is w i en in he o m o
((1)).
Then (1) calls phi+/- = exp(-m ) W+/- and ob ains he key equa ion:
d/d W = sigma Pauli z d/dx W -im sigma Pauli y W ((8))
In (1), ((8)) is called a Di ac equa ion al hough o mally he Di ac equa ion in ol es 4x4 ma ices
and 4- ec o s. I one wishes o w i e i in e ms o a 2- ec o , hen one has a mixing o he wo
2- ec o s comp ising he 4- ec o , so we a e unce ain as o how ((8)) cons i u es a Di ac
equa ion.
A second p oblem is ha sigma-z and sigma-y a ise na u ally, bu i is no clea wha o call he
a iable x. I could be x,y,z. Finally, we sugges ed ha m ac s as ene gy in exp(i m ), bu in ((8))
i has he appea ance o a kind o es mass which is con using. We hus limi ou sel es o he
o m exp(im ) =exp(iE ) om he analysis o (1). This also has an issue because he e is a
damping exp(-m ) e m as well as he wo- ec o cos(m )+i sin(m ), bu we ocus on he la e .
Compa ison wi h New onian Sca e ing P obabili y
In p e ious no es (2), we a gued ha one may in oduce a ee pa icle p obabili y in o
New onian 2-body sca e ing. In such a case, an ini ial (e1,e2) ene gy and (p1,p2) momen um
ec o se s should gi e ise o possible ou comes (ei,ej) (pi,pj) each o which ha e he same
p obabili y i momen um and ene gy a e conse ed. Gi en ime e e sal in a iance, we a gued
ha one canno use exp(ip), bu should use exp(ipx). This is also necessa y because exp(ip) is
no unique because p1=p+ 2n*3.14 has he same p obabili y. Thus, as no ed in (1), i is a
wa eleng h (i.e. a spa ial equency) which gi es uniqueness). We inally se led o :
exp(-iE +ipx) ((9))
which is Lo en z in a ian . One may a gue ha he abo e app oach equi es knowing abou
momen um and ene gy as well as ime- e e sal and Lo en z in a iances and conse a ion o
ene gy and momen um. We sugges ha ime- e e sal in a iance o exp(ipx) i..e p→-p x→-x
is con ained in Lo en z in a iance and Lo en z in a iance also espec s conse a ion o
momen um in he sense. Fi s , gi en a es mass mo a x, he no ion o momen um and ene gy
ollow om obse ing his objec om a ame mo ing wi h cons an - . This is an al e na i e
app oach o New on’s conside a ion o a o ce ac ing on mo. In a cen e -o -mass ame, all
momen a add o ze o and so one in a sense has an o e all mo wi h p o al=0. I he
cen e -o -mass ame is no accele a ed, i.e. all o ces a e ac ion- eac ion in e nal ones, hen
one is gua an eed momen um conse a ion. Thus, we a gue ha i is no acciden ha ((9))
which is Lo en z in a ian has buil -in momen um and ene gy conse a ion. We ob ained i he e
by using conse a ion a p io i, bu in o he no es we de i ed i wi hou any conside a ion o
conse a ion, bu only by examining he Lo en z in a ian :
-E +px = cons an ((10))
The poin we make is ha one uses an in a iance idea (Lo en z in a iance) and inds ha i
gi es ise o conse a ion o ene gy and momen um. We hus sugges ha assump ions in he
p obabili y model used by (1) may also lead o conse a ion o ene gy, which (1) inds. In
pa icula , he app oach o (1) seems o be linked o a kind o ime e e sal symme y as (1)
allows o a zigzag pa h in x in posi i e ime and hen e e ses he pa h o e u n o he o igin.
He hen analyzes he wo pa hs in e ms o ec angles, c ea ing a new pa h based on he
ou side ec angula pieces o each and applies a conse a ion o p obabili y o his new
“c ea ed” pa h. He gi es a weigh o -1 o pieces which ep esen a el backwa ds in ime (i.e.
he e u n ip). Fu he mo e, he gi es a weigh o a change in di ec ion in x as:
m d ((11))
He e m is some numbe . The p obabili y o con inue in a ce ain di ec ion is hen:
1- md ((12))
This is iden ical o exp(-E ) o =d . In o he wo ds, conse a ion o p obabili y is linked o a
a iable m which ac s like E (ene gy). The app oach o (1) does no only conse e p obabili y,
bu inds a p obabili y exp ession which conse es m, he a iable linked o changes in di ec ion
o x, i.e. pa o he p obabili y conse a ion equa ion. Thus, we hink ha he ideas o (1)’s
p obabili y app oach a e e y simila o hose used in c ea ing an exp(-iE +ipx) p obabili y based
on ime e e sal in a iance o exp(ipx) and o e all Lo en z in a iance and we sugges ha he
app oach o (1) may be used as an example o p obabili y used o ob ain exp(iE ). We no e ha
changing he di ec ion o mo ion in x is physically equi alen o sca e ing which is wha he
exp(-iE +ipx) app oach is all abou -sca e ing wi h conse a ion o E and p and a eal weigh o 1
o any ee pa icle.
Conclusion
In conclusion, we sugges ed in p e ious no es (2) ha one may ob ain he quan um ee
pa icle p obabili y exp(-iE +ipx) by conside ing ha a gi en se (e1,e2) ene gy (p1,p2)
momen um ec o s in New onian mechanics has equal weigh o any ou come se (ei,ej) (pi,pj)
i ene gy and momen um a e conse ed. A i s , one migh hink his leads o exp(-E C1) exp(p
C2) (cons an s o uni s), bu nei he o hese a e unique as p1=p+2*3.14*n (and a simila
exp ession) a e he same. One wishes o ha e a unique p obabili y. Fu he mo e, exp(pC2) is
no ime e e sal in a ian . To make i so, one migh in oduce exp(ipx) and inally exp(-iE +ipx)
which is Lo en z in a ian . This gi es a p obabili y linked o change in ime as exp(-iE ) which
becomes 1-Ed o =d . Thus, he e is a ee pa icle ene gy conse ing p obabili y o iny d
s eps linked o 1-Ed .
In (1), an an i-Be nouilli a iable so -o andom walk is in oduced which also akes in o
accoun ime e e sal. A pa icle s a s a he o igin and mo es along he diagonal o squa es in
and x- space. I may mo e o he igh o le in x, bu only posi i e in d un il T is eached and
hen i e e ses i s mo ion (based on ma ke s pu down). (1) de elops a conse a ion o
p obabili y scheme o pa hs which include bo h mo ion posi i e in and nega i e and bases a
change in x di ec ion o a p obabili y md wi h 1-md being he p obabili y ha he e is no
change. Such a change in x is linked wi h a collision and we no e he simila i y be ween 1-md
and exp(iEd ). In ac , (1) inds ha his an i-Be nouilli p obabili y is p opo ional o exp(im ) and
we sugges ha his may be linked o ou app oach o in oducing exp(-iE +ipx) o a ee pa icle
p obabili y which conse es ene gy and momen um. We no e ha a p io i one does no need o
know abou conse a ion o ene gy and momen um because hese ollow om Lo en z
in a iance. In o he wo ds, one migh ind exp(-iE +ipx) om he Lo en z in a ian -E +px =
cons an . A ee pa icle can only ha e a uni eal weigh and so i s p obabili y should be
complex, bu exp(iEC1), exp(ipC2) a e no unique (i.e. p1=p+2*3.14*n gi es he same alue).
Thus, one equi es wo a iables o ha e uniqueness and ime e e sal in a iance. As a esul ,
we sugges ha some symme y p ope ies ( ime e e sal o Lo en z in a iance) may “b ing in”
physical conse a ion o ene gy and momen um. We sugges ha he p obabili y model o (1)
does some hing e y simila .
Re e ences
1. O d, G. Quan um Mechanics in a2-Dimensional SpaceTime-Wha is a wa e unc ion?
(2009 o la e )
h ps://ma h. o on omu.ca/~go d/abs ac s/QWa e/QWa e4Elsa R1.pd
2. Rugge i, F ancesco R. Classical Elas ic Sca e ing P obabili ies and he Necessi y o a
Wa eleng h (p ep in , zenodo, 2025)
3.h ps://en.wikipedia.o g/wiki/Di ac_equa ion
h ps://en.wikipedia.o g/wiki/Pauli_ma ices
