Join Phase Space (x-p) P obabili y and F ee Pa icle Quan um Mechanics
F ancesco R. Rugge i Hanwell, N.B. No . 21, 2025
The e seems o be much in e es in he li e a u e (1) in associa ing quan um mechanics (in
pa icula he Sch odinge equa ion) wi h a classical join phase space p obabili y F(x,p, ) such
as appea s in classical Liou ille heo y e c.
He e, we conside ha P(x)dx = dx/L o a pa icle a es and analyze he ex a physics which
appea s when one conside s a single pa icle bouncing back and o h in a box, which is wha a
pe son in a ame mo ing wi h - would see (albei wi h a Lo en z con ac ed L.)
We sugges ha he cons ain L is impo an as i is linked wi h physical o ces which appea
when he pa icle mo es. In o he wo ds, a pa icle a es is linked wi h a s a ic p obabili y, while
o mo ion, one mus ha e a p obabili y which is linked o mo e han x and , bu o o ce
conside a ions in x and , a dynamical p obabili y. (Expe imen bea s his ou as we discuss.)
In pa icula , we a gue ha in he case o a mo ing pa icle, wo physical concep s ela ed o
o ce appea , namely p essu e which is an a e age o e ime and impulse hi which is only
linked wi h space. Thus, wo di e en pa icles may ha e he same p, bu di e en and
p essu e alues. We hus sugges a physical g ouping o (p,x) and (p essu e, ). We hen no e
ha one needs o somehow p ese e he no ion o P(x)=1/L, i.e mo ion does no in oduce a
bias o x, no o ime.
In a es ame, he e is no mo ion and so he e is nei he p no p essu e, bu P(x)=1/L pe ains.
I one ies o de ine a p obabili y wi h wo g oupings, one o p essu e, and he o he o p,x,
hen i one mul iples a p1 case by a p2 case, he p,x g ouping should yield p1+p2, x.
Fu he mo e, a p,x g ouping should imply mixed e ms, e.g. px e c, o he wise one does no
eally ha e an associa ion be ween he wo a iables. This sugges s a p (x) o m linked o he
p-x g ouping. Gi en he no ion o special ela i i y, i seems ha one may eplace p essu e wi h
he no ion o ene gy and c ea e a Lo en z in a ian p obabili y. This would hen lead o he ee
pa icle quan um wa e unc ion (p obabili y) exp(-iE +ipx), bu we a gue ha he oo s a e in
classical physics and a desi e o physically desc ibe he physical in o ma ion p esen o a
pa icle bouncing back and o h in a box.
Join Phase Space (x-p) P obabili y
I seems ha he e a e many a emp s (1) o link quan um mechanics (i.e. he Sch odinge
equa ion) o a join phase space p obabili y:
F(x,p, ) ((1))
This ype o p obabili y applies o many pa icles and appea s in Liou ille classical mechanical
heo y. Quan um heo y, howe e , should apply o a single pa icle. We u he no e ha a
quan um wa e unc ion is complex exp(-iE +ipx) and no complex p obabili y appea s in Liou ille
heo y. As a esul , app oaches which conside F(x,p, ) end o ma hema ically o ce a complex
numbe in o he scena io.
He e we sugges he ollowing. Fo a pa icle a es in a cons ained leng h o L, he
p obabili y o ind he pa icle in dx is gi en by he uni o m dis ibu ion (as he maximum lack o
in o ma ion):
P(x)dx = dx/L ((2))
A pe son in a ame mo ing wi h - , howe e , would see he pa icle as mo ing and gi en he
cons ain on L (now Lo en z con ac ed, bu s ill a cons ain ), would conside his a p oblem o
a pa icle bouncing back and o h in a box. The ques ion hen becomes, How would such a
pe son analyze his si ua ion?
Pa icle Bouncing Back and Fo h in a Box
We a gued abo e ha ((2)) o mally applies o a pa icle a es in a cons ained leng h L. Fo
a mo ing pa icle, his cons ain immedia ely becomes associa ed wi h o ce because he
pa icle mus be s opped om mo ing ou side he L egion. In he s a iona y case, he L
cons ain is no linked wi h o ce as one simply places he pa icle somewhe e wi hin he L
egion. We s ess, howe e , ha in bo h cases L is a cons ain . In he mo ing case, his
cons ain is linked o o ce and so one should conside he o ces in he p oblem. We sugges
ha he e a e wo conside a ions one may make.
(A) The e is he impulse hi p opo ional o p which occu s a x.
(B) The e is a ime a e age o he hi agains he wall which is called p essu e and so linked
o ime and eloci y.
We sugges ha he e a e wo dis inc g oupings in his p oblem and bo h mus be conside ed in
o de o cap u e he ele an physical in o ma ion o he p oblem, i.e.
(A1) p mus be g ouped wi h x. Ma hema ically, his p esumably means e ms wi h mixed p and
x ac o s
(B1) p essu e should be mixed wi h ime
We nex no e ha e en hough a pa icle is mo ing in a box, he e is no special weigh gi en o
any x o alue and his no ion mus be p esen in a ma hema ical p obabili y scheme. In ac ,
his conside a ion sugges s a eal alued weigh :
F(x,p, ) =cons an ((3))
As a esul , all in o ma ion abou p and p essu e is los in such a p obabili y p esc ip ion. This
app oach is equi alen o ((2)) wi h a simila exp ession exis ing o which is consis en wi h
((2)). This begs he ques ion:
E en hough p (impulse hi ) and p essu e exis physically, is i eally necessa y o in ol e hem
in a p obabili y scheme? ((4))
We ha e no ed abo e ha L is a cons ain which appea s in ((2)) e en o a pa icle a es . I
canno be elimina ed. This same cons ain mus in ol e he no ion o o ce o a mo ing pa icle
and we sugges ha i he cons ain canno be elimina ed, hen nei he can i s accompanying
o ce. In o he wo ds, he cons ain and all o i s physical consequences mus be conside ed.
This sugges s ha he answe o ((4)) is Yes. This is su p ising because one is used o he
pa icle a es scena io.
By answe ing ((4)) wi h a yes, one is o ced o include he p and p essu e in o ma ion in a
manne which does no iola e ((2)), i.e. o p=0 , p essu e=0, and one mus ob ain he eal
alue ((2)). Fo p and p essu e no equal o 0, he same no ion o equal weigh o all and x s ill
applies, hus i seems ha one mus use a complex numbe o a p obabili y which includes
p essu e and p. We sugges ha his is how he no ion o a complex numbe appea s in he ee
pa icle si ua ion. In o he wo ds, i has o be in oduced in o de o “hide” p essu e and p wi hin
a phase which seems o be a depa u e om classical physics. Thus, one migh be wa y o
app oaches which use classical Liou ille heo y which should no ha e a complex p obabili y
and hen in oduce complex unc ions.
((2)) is a p obabili y linked wi h a pa icle a es which does no in e ac . A mo ing pa icle is a
dynamical one and one is no simply in e es ed wi h posi ion in space, bu wi h in e ac ions in
space and ime and his equi es a dynamical p obabili y. Thus, one canno sepa a e x and
om he in e ac ions wi h which hey a e associa ed, we a gue.
We no e ha in (1), a ma ginal p obabili y:
In eg al dp exp(i p dx) F(p,x, ) ((5))
is in oduced a p io i, bu i is no clea why a complex alue should be used. (1) no es ha one
may hen w i e ((5)) as W*(q-dq/2, ) W(q+dq/2, ), bu his is ma hema ics.
We sugges ha he complex alue na u e a ises because one mus include he physics o he
o ces linked o he cons ain L in a p obabili y, bu mus a he same main ain he equal weigh
in a ,x si ua ion. Thus, he p, p essu e in o ma ion is hidden (ma hema ically) in he phase o a
complex unc ion.
The Link Wi h Special Rela i i y
We s a ed abo e wi h he no ion o a pa icle a es and ((2)). We hen conside ed an
obse e in a mo ing ame as seeing he pa icle in mo ion. Such a ame scena io is linked o
special ela i i y. We sugges , as we ha e in p e ious no es, ha a p obabili y linked o p,
p essu e, , x should eally be Lo en z in a ian . We no e ha p essu e is eally linked o p and
and ha =pc/E o a pa icle wi h es mass. Thus, he Lo en z in a ian :
-E +px ((6))
is a candida e o he p obabili y exp(-iE +ipx) ((7))
((7)) has an addi ional ea u e. I one has a p1,x g ouping AND a p2,x g ouping, hen p obabili y
ma h implies mul iplica ion and he esul mus be equi alen o p1+p2. The o m o ((7)) allows
o his and also o E1, E2→ E1+E2. In o he wo ds ((7)) is consis en wi h conse a ion o
ene gy and momen um.
Link wi h Physical Reali y
We ha e a gued o a p obabili y which con ains E linked wi h and p wi h x. In ((4)), we
conside ed he possibili y o ha ing p obabili y which does no include p and E as a physical
possibili y as p and E a e no impo an in he es mass scena io. The undamen al ques ion
hen becomes: Is he e any physical eason one equi es he p esence o a p,x g ouping in a
p obabili y? We a gue ha he e is. Fi s , exp(ipx) and exp(-iE ) show ha in New onian 2-body
elas ic sca e ing, any (ei,ej) (pi,pj) (momen um ec o ) ou come has he same p oduc
p obabili y. Secondly, one dimensional pho on o pa icle e lec ion- e ac ion a an n1-n2 index
o e ac ion junc ion leads o a p obabilis ic esul and we a gue ha his is linked o he alues
o p as expe imen shows. This poin s o a p obabili y associa ed wi h p and x, i.e. a dynamical
p obabili y.
Conclusion
Ou main conclusion is ha gi en a pa icle a es , P(x)dx=dx/L, whe e L is an a bi a y leng h,
L is o en seen as linked o no maliza ion, bu we sugges i is a physical cons ain . One canno
place a pa icle a es ou side he L egion. I one iews a pa icle a es om a mo ing ame
(- ), he pa icle mo es wi h , bu i he cons ain L (o i s Lo en z con ac ed alue) is aken
se iously, he pa icle mus now bounce back and o h in a box.. This implies ha o ces now
appea such as p (impulse) linked o x and he ime a e age p essu e, linked o . These o ces,
howe e , do no change he basic idea ha no x o ca y any special weigh . No in o ma ion
abou p and p essu e is needed o each his conclusion and one migh sugges ha P(x)= 1/L
and P( ) = 1/T. In such a case, all in o ma ion abou p and p essu e would be d opped e en
hough i is physically ele an in o ma ion. The idea would be ha i has no hing o do wi h
p obabili y in x and . We sugges , howe e , ha inding a pa icle a x is associa ed wi h an
impulse hi p and ime conside a ions a e linked o o a mo ing pa icle. These no ions ollow
om a Lo en z ans o ma ion o a pa icle a es . Thus, we eplace p essu e wi h E and s ill use
p and y o c ea e a Lo en z in a ian p obabili y wi h he ex a o ce in o ma ion hidden in a
complex phase, i.e. exp(-iE +ipx). This sugges s ha one has he no ion o conse a ion o
ene gy and momen um con ained in such a p obabili y which is physical and is some hing ha
one does no need o conside in a es pa icle case.
In o he wo ds, a pa icle a es is linked wi h a non-dynamical p obabili y as no in e ac ions
occu while a pa icle in mo ion is linked wi h a dynamical p obabili y which may handle
in e ac ions/ o ces which a e linked wi h x.
Re e ences
1. Sil a Filho, O. and Fe ie a, M. A Bi d’s Eye View on a New S ochas ic In e p e a ion o
Quan um Mechanics (Oc . 2025)
Ma hema ics 2025, 13(21), 3571; h ps://doi.o g/10.3390/ma h13213571
h ps://www.mdpi.com/2227-7390/13/21/3571
