Join Phase Space (x-p) P obabili y and F ee Pa icle Quan um Mechanics Pa 3
F ancesco R. Rugge i Hanwell, N.B. No . 23, 2025
In Pa 1, we conside ed New onian elas ic sca e ing and p oposed a p obabili y exp(-iE +ip
do ) o allow o equal p oduc p obabili y weigh s o any (ei, ej) (ene gy) and (pi, pj)
(momen um ec o ) ou come pai . We insis ed on he p obabili y being a Lo en z scala and
a gued ha all ee pa icles ha e he same eal alue weigh , hence he modulus o 1. This
no ion is based on p obabili y as i is applied o possible ou comes. This means ha one does
no ha e a ully de e minis ic si ua ion because o he wise one would know he exac ei,ej, pi,pj
ou come and would no equi e p obabili ies in he i s place.
In Pa 2, we a gued ha a de e minis ic app oach based on x= o a ee pa icle con ains
no no ion o p obabili y and so his seems o con adic exp(-iE +ipx) wi h i s physical in e als:
del a = hba /E and del a x = hba /p. We hen no ed ha hese in e als and hence p obabili y
a e ound in he Ac ion= Lag angian * = -E +px wi h =x/ . We concluded ha x= , a solu ion
o d/d dL/d pa ial - dL/d = 0 loses some o he in o ma ion p esen and ha examining he
ac ion e eals mo e in o ma ion because one is o ced o deal wi h x, , E and p a he same
ime.
He e, we e u n o New onian elas ic wo-body sca e ing and assume ha one has he
ollowing de e minis ic case: e1,e2, p1, p2 → e3,e4, p3,p4. One would hen assume ha no
p obabili y o any kind is p esen . We sugges ha momen um is conse ed due o
ac ion- eac ion, some hing ha New on al eady p oposed wi hou any no ion o p obabili y
equi ed. Ene gy is conse ed because i canno be ul ima ely c ea ed o des oyed and we
conside special elas ic collisions.
We hen conside he New onian idea o a pa icle being ac ed upon by a o ce -dV(x)/dx,
whe e V(x) is a po en ial. The in e ac ion occu s o a d , dx, bu New onian physics assumes
ha dx→0, d →0. Thus, New onian mechanics allows o a minimum dx->0 o conse a ion o
momen um. By going in o he cen e -o -mass ame o wo pa icles which collide elas ically, one
has p and -p in his ame. When he wo pa icles come oge he a x, , o a iny ime one has a
join pa icle wi h no ne momen um which is in e ac ing and b eaking apa . Then, in he 1D
case, one pa icle mo es ou wi h p1 and he o he wi h -p1 in he cm- ame.
I one conside s A= -E +px o one pa icle, hen New on’s d ->0, dx->0 as well as x=0, =0
yield A=0. I is known om special ela i i y, howe e , ha he e is a jump in x o a pa icle
iewed om a mo ing ame. In pa icula , i he pa icle is a x=0 a in he es ame (cm
ame), hen in he mo ing ame: x’ = g( ) whe e g( )=1/sq (1- /cc).This is linked o he
pe son in he mo ing ame (- ) measu ing he pa icle which seems o mo e o him/he . This
measu emen akes some minimum amoun o ime.
Thus, i seems ha one mus ejec dx→0 on he g ounds o special ela i i y. The ques ion
hen becomes: Wha is he smalles dx and d one may ha e based on special ela i i y? We
sugges ha gi en he A=0 case, one may ha e del a x =hba /p and del a = hba /E. E en in a
de e minis ic elas ic in e ac ion, he no ion ha d and dx canno be 0 leads o minimal alues o
hese which a e p obabilis ic, i.e. hey a e based on special ela i i y and no on he ac ual
in e ac ion. I one has ini ial unce ain y in x and , e en i he pa icle hen mo es wi h x= , his
unce ain y should also be p opaga ed along as well, because he dx, d unce ain y canno
simply anish. We sugges ha he no ion o exp(-iE +ipx) eme ges e en i one emo es he
ini ial p obabilis ic assump ion o elas ic sca e ing wi h unknown ou comes. Known ou comes
s ill lead o unce ain dx=hba /p and d =hba /E in keeping wi h he unce ain y con ained in L =
-E +px, which is used o de e minis ic calcula ions. Thus, i is special ela i i y which places a
cons ain on how small dx and d may be in a physical eac ion and one canno simply use he
New onian dx→0, d →0.
exp(-iE +ipx) Eme ging om A P io i P obabili y
In Pa 1, we a gued ha one may ob ain he quan um ee pa icle wa e unc ion (which we
call a p obabili y) exp(-iE +ipx) (o exp(-iE +ip do )) by in oducing an a p io i p obabili y in o
New onian elas ic sca e ing. In o he wo ds, p obabili y exis s because we in oduce i in o his
physical sca e ing p oblem by s a ing ha one does no eally know he ou come and a guing
ha any ei,ej (ene gy), pi,pj (momen um ec o ) se s which conse e ene gy and momen um
ha e equal p obabili y o occu ing. We hen a gue ha one equi es a Lo en z in a ian
p obabili y and a i e a exp(-iE +ipx) which is complex because e e y ee pa icle has he
same eal alue weigh , i.e. he same modulus o 1. Ex a de ails o he p obabili y a e hen
o ced o exis in he phase o he complex numbe . We poin ou , as we did in Pa 2, ha one
migh objec o he in oduc ion o a p io i p obabili y and a gue ha in a de e minis ic p oblem
would ac ually know he exac ou come o a speci ic elas ic sca e ing case and no need o
eso o p obabilis ic a gumen s.
exp(-iE +ipx) om he Classical Ac ion -E +px
In Pa 2, we igno ed a p io i p obabili y and conside ed he solu ion:
x= ((1))
((1)) shows no signs o p obabili y wha soe e which is a di ec con adic ion o he physical
in e als:
dx = hba /p and d = hba /E ((2))
p edic ed by exp(-iE +ipx). We hen no ed ha x= is ul ima ely a solu ion o :
d/d dL/d pa ial - dL/dx = 0 ((3))
whe e L is he Lag angian and is gi en by -mo sq (1- /cc) in he ee pa icle ela i is ic case.
We hen no ed ha he Lag angian is linked wi h momen um p h ough dL/d pa ial and wi h
ene gy E=Hamil onian = p - L. In o he wo ds, x, ,p,E a e he a iables o in e es , no jus x
and in ((1)). We sugges ed ha ((3)) does no ep esen he ull in o ma ion p esen and a gued
ha he Lo en z in a ian (also he ac ion L )
-E + px ((4))
shows dx=hba /p and d =hba /E. In o he wo ds, hese in e als also appea i one conside s
de e minis ic physics as long as one examines he ull ac ion, L , and no simply ((1)).
We nex conside de e minis ic elas ic wo body sca e ing.
De e minis ic Elas ic Two-Body Sca e ing
In Pa 1, we in oduced p obabili y a p io i in o wo body New onian elas ic sca e ing. He e
we sugges ha i New onian mechanics is eally de e minis ic, one should conclude ha one
could know he ou come o an e1, e2 p1, p2 (ene gy and momen um ec o s) elas ic collision,
i.e. he e would be a single e3,e4, p3,p4 ou come and all p obabili y would seemingly disappea .
This would hen nulli y he a gumen s made in Pa 1. Thus, we conside his example in mo e
de ail.
I one goes in o he cen e -o -mass ame o he e1,e2, p1,p2 collision, hen one sees a p and
-p. A some ins an o ime he wo pa icles will be a x, and he o e all momen um will be 0.
This is like ha ing a pa icle a es which is b eaking apa o yield a di e en p1, -p1. Du ing
his change om p,-p o p1,-p1 ac ion- eac ion o ces occu and momen um and ene gy a e
conse ed as New on al eady no ed o an elas ic collision.
In New onian mechanics, an in e ac ion occu s in dx, d wi h o ce = -dV(x)/dx, whe e V(x) is
he po en ial (i in ac i depends only on x). New onian mechanics allows o dx→0 and d →0.
We, howe e , conside special ela i i y and no e ha o a pa icle a es a x=0 a , he e is a
shi in x. In pa icula , x=0, =0 and x’=0, ’=0 apply, bu when he pa icle is seen o mo e due o
a ame mo ing wi h - , one has:
x’ = g( ) and ’= g( ) whe e g( ) = 1/sq (1- /cc) ((5))
Thus, e en hough x’ is conside ed o ini ially be a x=0 a =0, when a i es, x=0, bu x’=
g( ) . Thus, he e is a ini e change in dx’, i d ’ is ini e. One may, howe e , a gue o being
a bi a ily small. The poin we make is ha his shi in x’ is linked wi h a eloci y measu emen
made by a pe son in he mo ing ame and so is echnically no 0.
We nex no e ha o a mo ing pa icle:
A = -E’ ’ + p’ x’ ((6))
I one conside s x’=0, ’=0 and an in e ac ion occu ing in a iny in e al ending o 0, hen A=0
o x’=0, ’=0, bu also o d ’ and dx’ =0. This bypasses, howe e , he no ion o a shi in x’ as
a gued abo e. We no e ha he e is in ac a second solu ion in ((6))
d = hba /E and dx= hba /p ((7))
This allows A=0, bu o ces a minimal alue o d and dx based on E and p and does no allow
o minimal dx→0 and d →0. As a esul , i one accep s dx=0 and d =0, one would ha e o
disca d a legi ima e solu ion o A=0 which seems o be mo e in keeping wi h special ela i i y
han dx’=0, d ’=0. A e all, he pe son in he ame mo ing wi h - mus measu e he eloci y o
he pa icle and his should no occu in dx’=0, d ’=0. I is possible ha ((7)) a e e y iny alues
which app oxima e 0, as in he usual classical wo ld scena io, bu he e may be leng h scales
(a oms e c) o which ((7)) a e no close o 0.
We sugges ha e en in he case o a de e minis ic wo body elas ic collision, one has ini ial
unce ain y in dx and d gi en by ((7)) o each pa icle. This unce ain y does no simply anish,
bu mus p opaga e, e en hough on a e age one has x= . This unce ain y mus be di ec ly
linked o p obabili y and we sugges once again ha :
exp(-iE +ipx) ((8))
Is he ele an p obabili y which appea s due o special ela i is ic cons ain s e en in a
de e minis ic p oblem.
Conclusion
In conclusion, we no e ha in Pa 1, we conside ed elas ic wo-body New onian sca e ing in
e ms o a p io i p obabili y. In pa icula , we a gued ha gi en an e1,e2 (ene gies) and p1,p2
(momen um ec o s), any ou come ei,ej, pi,pj has he same p oduc p obabili y. This led o a
p obabili y exp(-iE +ipx) which is Lo en z in a ian only because one assumed a p obabilis ic
p oblem in he i s place. One may a gue ha a pa icula wo body elas ic in e ac ion is
de e minis ic and s a e ha a speci ic e3,e4, p3,p4 is he ou come. Then, all p obabili y seems
o disappea . We a gue he e, howe e , ha i does no due o special ela i i y.
We conside he de e minis ic case in he cen e -o -mass ame. The e, one sees a p, -p
change in o a p1, -p1 due o an in e ac ion. In New onian mechanics, an in e ac ion may occu
in dx→0, d →0 as is he usual ea men which applies o a o ce = -dV(x)/dx whe e V(x) is a
po en ial. I one conside s he special ela i is ic in a ian A = -E ’+px’, which is also he
classical ac ion, hen x’=0, ’=0 yields A=0 as does dx’=0, d ’=0. The e is, howe e , ano he
solu ion, dx=hba /p and d =hba /E which also yields A=0 and we a gue ha his canno be
disca ded. We also no e ha i akes ime o a pe son in a mo ing ame o measu e ha a
pa icle is mo ing and so sugges ha d →0 canno eally be 0. I is possible o hba /p and
hba /E o be iny as in he classical case in which dx→0 and d →0 hen seem o apply. Fo o he
scales (a omic e c), howe e , his does no hold and special ela i i y limi s one o he ac ual
alues o dx’ and d ’ ha one may use. We no e ha hba /p and hba /E do no depend on he
ac ual in e ac ion, bu a e unce ain ies associa ed wi h ini ial mo ion. These unce ain ies
canno simply disappea and so a e p esen oge he wi h x= , he de e minis ic esul , in he
o m o exp(-iE +ipx), we a gue. Thus, we sugges ha i is special ela i i y which ul ima ely
b eaks New onian de e minism, o cing unce ain y egions in x and and yielding exp((-iE +ipx),
i.e ee pa icle quan um mechanics.
