Spin, Lo en z In a ian Equa ion o F ee P obabili y and Lo en z In a ian
Physical Equa ion
F ancesco R. Rugge i Hanwell, N.B. Oc . 11, 2025
In his no e, we sugges ha a undamen al idea o ee pa icle quan um mechanics is he
wa e unc ion exp(-iE +i p do ) which we call a complex p obabili y. This p obabili y, howe e ,
does no desc ibe he ull physics o a ee objec . Fo example, i does no ell one i he objec
has es mass o no , o whe he i has elec ic and magne ic ields as in he case o a pho on.
Thus, we sugges ha he e mus exis a Lo en z scala equa ion desc ibing he ull physics o
he p oblem, bu ha his equa ion makes use o he p obabili y exp(-iE +ip do ), bu no
necessa ily in a linea o m (as in he con inui y equa ion o ene gy densi y and Poyn ing
momen um o a pho on).
A he same ime exp(-iE +ip do ) is an eigen unc ion o linea ope a o s i d/d pa ial and -i
g ad pa ial. Thus, we pos ula e ha a he oo o any physical Lo en z in a ian equa ion using
unc ions based on exp(-iE +i p do ), he e should be a co esponding Lo en z scala equa ion
in exp(-iE +i p do ). Such an equa ion mus be based on he ope a o s id/d pa ial and -i g ad
pa ial, bu as a Lo en z scala , i mus con ain an ope a o which o ms a do p oduc wi h - i
g ad. I px, py, pz and E a e he pieces o in o ma ion o in e es , hen his new ec o is no
based on x-space e c, bu may be a se o ma ices which allow his in a ian base equa ion o
become he Lo en z scala physical equa ion. In o he wo ds, one needs o add a kind o
geome y o exp(-iE +ip do ) o allow i o desc ibe he ull physics o a p oblem. This geome y
seems o be hidden in he way he E, px, py and pz a iables a e in e ela ed. We conside bo h
he spin ½ and spin 1 pho on cases.
Quan um F ee Pa icle and exp(-iE + i p do )
In p e ious no es, we a gued ha gi en a collision o E1,E2 and p1,p2 wi h conse a ion o
bo h, one does no know he ou come alues Ei, Ek and pk,pl (one dimension). I one assumes
equal p obabili y o any pai which main ains conse a ion, hen one would expec a p obabili y
o he o m:
exp(i E) and exp(i p) one dimension ((1))
“I” is used because he p obabili y o a ee pa icle canno be eal, i.e. one does ha e any
weigh as in an ideal gas. exp(ip) and exp(-ip), howe e , mus ha e he same alue and so ((1))
canno be co ec . One mus combine p wi h a a iable which also changes when one changes
he di ec ion o he x axis. This sugges s hen:
exp(-iE ) and exp(ipx) o ha e an o e all Lo en z in a ian scala p obabili y ((2))
A undamen al issue now a ises. The p obabili y exp(-iE +i p do ) does no desc ibe he ull
p ope ies o a ee objec . Fo example, exp(-iE +i p do ) could apply o a pa icle wi h es
mass o a pho on wi h no es mass.
A pa icle wi h es mass is desc ibed by he Lo en z in a ian scala equa ion:
-EE = cc p do p + momo cccc ((3))
A pho on is desc ibed by bo h: E = c |p| ((4))
as well as an equa ion using elec ic and magne ic ields, i.e.
d/d pa ial( .5eo E do E + .5/uo B do B) + 1/uo g ad do E x B = 0 ((5))
((5)) depends one exp(-iE +i p do ) h ough E = C1 Real (exp(-iE +i p do ) ec o along
and B = C2 Real (exp(-iE +i p do ) ec o along ((6))
((3)) is nonlinea in he ope a o s id/d pa ial and -i g ad pa ial, bu linea ac ing on exp(-iE +ip
do ), whe eas ((5)) is linea in he ope a o s id/d and -ig ad, bu quad a ic in he p obabili y
exp(-iE +i p do ).
We pos ula e he ollowing. We a gue ha he ee objec p obabili y exp(-iE +i p do ) is a key
unde lying pa ial desc ip ion o a ee objec . I is an eigen unc ion o id/d pa ial and -i g ad
pa ial, bu is missing in o ma ion con ained in equa ions such as ((3)) and ((5)). We sugges
ha jus as ((3)) and ((5)) a e Lo en z in a ian equa ions which make use o he E, px,py,pz,
he e mus exis a Lo en z scala equa ion, linea in id/d pa ial, -i g ad pa ial o exp(-iE +i p
do ). Fu he mo e, his equa ion o he p obabili y mus be he ounda ion upon which one may
c ea e he mo e gene al physical equa ions such as ((3)) o ((5)). The wo equa ions canno be
independen .
This sugges s ha one mus ind a ec o o use in a do p oduc wi h p (o -i g ad). This
ec o , howe e , does no seem o be he ec o o any o he physical ec o . Ins ead, i seems
ha i is equi ed o desc ibe a kind o hidden geome y in he p oblem. To see his speci ically,
we conside an abs ac ma h ec o consis ing o ma ices. In such a case, one should ha e:
-mo Ma ix A + Ma ix0 (id/d pa ial) + Sum i=1,3 Ma ix-i (id/dx) } exp(-iE +ipx) objec = 0
((7))
((7)) mus be a building block equa ion o ((3)), wi h he ob ious ope a ion being he
mul iplica ion o bo h sides o ((7)). This yields a ious condi ions o he ma ices. Di ac has
al eady sol ed his p oblem and ound ha he:
Ma ix-i (i=1,2,3) a e 4x4 ma ices wi h he Pauli 2x2 spin ma ices on he an i-diagonal, wi h he
second one appea ing wi h a minus sign. ((8))
The poin is ha we sugges ha ((7)) should exis a p io i because he e mus be a Lo en z
scala equa ion in id/d and -ig ad ac ing on exp(-iE +ip do ) and ((3)) canno be independen o
his equa ion.
I his is he case, hen one should be able o use ((7)) wi h mo=0 o a pho on.The same basic
building block idea should hold. In a p e ious no e, we ha e shown ha one may ake ((5)) and
linea ize i by ac o ing ou Elec ic ield + i B ield o ob ain:
d/d (El+iB) + eijk d/dxk (El+iB) ((9))
He e eijk is he Le i-Ci i a symbol and ij a e he ma ix en ies, while k=1,2,3 desc ibes he
ma ix i sel . As a esul , he geome y is di e en in ((9)) because he physical Lo en z scala
equa ion ((5)) di e s om ((3)).
In o he wo ds, spin desc ibes in a geome ical manne how a basic building block equa ion
linea in id/d and -i g ad ac ing on exp(-iE +ip do ) wi h a ec o o ma ix M-i do ed wi h d/dxi
may be used o c ea e he Lo en z scala physical equa ion as he wo canno be independen .
Conclusion
In conclusion, we y o unde s and how spin a ises. We sugges ha wi hou conside ing any
quan um mechanical ee pa icle ea u es, he e exis physical equa ions such as ((3)) and ((5))
o a pa icle wi h es mass and a pho on desc ibed in e ms o elec ic and magne ic ields.
These a e Lo en z in a ian equa ions.
We nex sugges ha he quan um mechanical ea u e o a ee objec ollows om
pos ula ing ha a gi en E1,E2, p1, p2 (one dimension) can c ea e any Ei, Ej, pl, pk ou come
pai s wi h equal p obabili y as long as ene gy and momen um a e conse ed. We hen a gue
ha he p obabili y mus be desc ibed by exp(-iE +i p do ) which is a Lo en z in a ian . Thus,
he physical Lo en z scala equa ions ((3)) and ((5)) ac ually depend on exp(-iE +i p do ), bu
his does no mean ha he ope a o s id/d and -ig ad appea linea ly ( hey do in ((5)) bu no in
((3))). No does i mean ha exp(-iE +i p do ) appea s linea ly (i does in ((3)), bu no in ((5))).
We sugges ha exp(-iE +i p do ) is an eigen unc ion o -id/d pa ial and i g ad pa ial and so
he e should exis a Lo en z scala equa ion o exp(-iE - i p do ). We u he sugges ha he
ec o -i g ad should o m a do p oduc wi h a ma h ec o o ma ices. This Lo en z scala
equa ion o exp(-iE +i p do ) would hen se e as he building block o he physical Lo en z
scala equa ion, as he wo canno be independen . Thus, he ec o o ma ices de e mines
wha physical Lo en z scala equa ion esul s om a building block exp(-iE +i p do ) wi h
ma ices equa ion. In he case o ((3)), he ma ices linked wi h p a e 4x4 one wi h 2x2 Pauli
ma ices along he diagonal wi h he second ma ix ha ing a ac o o -1. In he case o ((5)), eijk
is he ma ix wi h i,j ep esen ing he ma ix elemen s. In o he wo ds, he spin o m desc ibes a
hidden geome y equi ed o con e he building block Lo en z scala equa ion o exp(-iE +i p
do ) in o he physical Lo en z scala equa ion.
