Table Of ContentNONLINEAR PROBLEMS IN CLASSICAL AND QUANTUMELECTRODYNAMICS
A.O. Barut
The Univers i ty of Colorado, Boulder, Colorado 80309
Table of Contents
Page
I . Introduct ion 2
I I . Classical R e l a t i v i s t i c Electron Theory 2
I I i o Quantum Theory of Se l f - l n te rac t ion 5
Other Remarkable Solutions of Nonlinear
Equations 10
Some Related Problems 11
References 13
NONLINEAR PROBLEMS IN CLASSICAL AND QUANTUMELECTRODYNAMICS
A. O. Barut
The Univers i ty of Colorado, Boulder, Colorado 80309
I . INTRODUCTION
We present here a discussions of the non- l inear problems ar is ing due to se l f -
f i e l d of the e lectron, both in c lassical and quantum electrodynamics. Because of
some shortcomings of the conventional quantumelectrodynamics 11 an attempt has
been made to carry over the nonperturbative rad ia t ion react ion theory of c lassical
electrodynamics to quantum theory. The goal is to have an equation for the rad ia t ing
and se l f - i n t e rac t i ng electron as a whole, in other words,an equation for the f i na l
"dressed" electron. In addi t ion the theory and renormalizat ion terms are a l l f i n i t e .
Each pa r t i c l e is described by a s ingle wave funct ion ~(x) moving under the inf luence
of the s e l f - f i e l d as well as the f i e l d of a l l other par t i c les . In pa r t i cu la r , we dis
cuss the completely covariant two-body equations in some de ta i l , and point out to
some new remarkable solut ions of the non- l inear equations: These are the resonance
states in the two-body problem due to the in terac t ion of the anomalous magnetic mo-
ment of the par t i c le which become very strong at small distances.
I I . CLASSICAL RELATIVISTIC ELECTRON THEORY
The motion of charged par t ic les are not governed by the simple set of Newton's
equations as one usual ly assumes in the theory of dynamical systems, but by rather
complicated non- l inear equations invo lv ing even th i rd order of der ivat ives. To see
th is we begin with Lorentz's fundamental postulates of the electron theory of mat-
te r :
( i ) Matter consists of a number of charged par t ic les moving under the inf luence
of the electromagnetic f i e l d produced by a l l charged par t i c les . The equation of mo-
t ion of the i th charged par t i c le is given by
m Z ( i ) : e F (x) ZV ( i ) l (1)
lx=z ( i ) '
where Z (S) is the worl 'd- l ine of the par t i c le in the Minkowski space M4 in terms of
an invar ian t time parameter S (e.g. proper time) - the der ivat ives are with respect
to S, and F is the to ta l electromagnetic f i e l d .
~v
( i i ) The to ta l electromagnetic f i e l d F obeys Maxwell's equations
P~
F '~(x) = j (x) , (2)
where j (x) is the tota l current of a l l the charges. For point charges we have
j~(x) = ~ e (k) ~(k)~ ~(x - Z (k))
(3)
We have in p r inc ip le a closed system of equations i f we have in addi t ion some
model of matter t e l l i n g us how many charged par t ic les there are.
These equations taken together give for each par t i c le i a h igh ly nonl inear
equation because due to the term k = i in (3), F (x) in (1) depends nonl inear ly
on Z ( i ) . This is the socalled s e l f - f i e l d of the U~it h pa r t i c le . Ac tua l ly th is term
is even i n f i n i t e at X = Z ( i ) due to the factor ~(X-Zki)") . ' In pract ice th is i n f i n i t e
term does not cause as much trouble as i t should-one simply drops such terms in
f i r s t approximation. The reason for th is is that a major part of the s e l f - f i e l d is
already taken into account as the i ne r t i a or mass of the par t i c le on the l e f t hand
side of e q . ( i ) : in other words, the mass m in ( I ) is the socalled renormalized
mass mR as I shal l explain in more de ta i l . Unfortunately not the whole of the s e l f -
f i e l d is an i n e r t i a l term in the presence of external forces. Otherwise the whole
electrodynamics would be a closed and consistent theory wi thout i n f i n i t e s . For a
s ingle pa r t i c l e , i t is true by d e f i n i t i o n , that a l l the s e l f - f i e l d is in the form
of an i n e r t i a l term because then the equation is mR~, = O. But the presence of
other par t ic les modifies the cont r ibut ion of the s e l f - f i e l d to an i n e r t i a l term
mRZ. And th is is r ea l l y the whole story and problem of electrodynamics, c lassical
or quantummechanical: How much of the s e l f - f i e l d is ine r t ia? . Af ter the i n e r t i a l
term has been subtracted, the remainnder gives r ise to observable ef fects which we
ca l l rad ia t ive phenomena l i ke anomalous magnetic moment, Lamb s h i f t , etc. I w i l l
now show f i r s t how th is is done in c lassical electrodynamics, and the existence of
nonl inear rad ia t ive phenomena l i ke anomalous magnetic moment and Lamb shCft even in
c lassical mechanics.
Let us separate in Eq.(1) the se l f f i e l d term:
moZ~ = e~ FeXt(x) Z~ + eFself(x)pv ~v
(4)
~v x=z
where I have introduced a parameter ~(~=1) in order to study the l i m i t ~ ÷ 0 for a
free pa r t i c l e . The f i r s t term on the r i gh t hand side of (4) is f i n i t e , but the se-
cond term becomes i n f i n i t e at X=Z. By various procedures one can however study the
st ructure of th is term 121. The resu l t is as fo l lows. The s e l f - f i e l d term in (4)
can be wr i t t en , using (3), as a sum of two terms
e 2
- ~÷olim2 --~-Z~ + ~ e2 (Z"" + ~ 2 ) (5)
Here Z~ depends on x as we l l , Zp = Z (S,X)... The f i r s t term is an i n e r t i a l part
which we wr i te as -6m Z and bring i t to the l e f t hand side of (4). We shal l now
"renormalize" eq.(4) such that for x ÷ 0 we have the free par t i c le eq. m~ Z = O. The
renormalizat ion procedure is not unambiguous: we have to know to what form we want
to b r ing our e q u a t i o n s . The above r equ i r emen t f o r x ÷ 0 g ives us the f o l l o w i n g f i n a l
equation
mR ~ = e~ Fext.(x)~v ~v +2~ e 2 (Z"" + ~ 2 ) _°LI '2e2 ('~ ÷ ~ "~2) , (6)
~ x=O
mR = mo + 6m
Had we not subtacted the las t term, a "free" pa r t i c l e (~ = O) would be governed by
a complicated equation, and that is not how mass is defined. Also, eq.(6) , shows
without the las t term the pecul iar phenomena of preaccelerat ion and socalled run
away solut ions 131whichhave bothered a l o t of people up to present time. The las t
term in (6) el iminates these problems.
The nonl inear term in (6) has a l l the correct physical and mathematical propeL
t ies :
I ) ? Z~ = O, where r = C (i" ÷ Z 22),
2) I t gives correct rad ia t ion formula and energy balance.
3) I t is a non-perturbative exact resu l t .
I t has moreover, the physical i n te rp re ta t ion as Lamb-shift and anomalous
magnetic moment. These can be seen by considering external Coulomb or magnetic
f i e lds and evaluat ing i t e r a t i v e l y the e f fec t of the rad ia t ion react ion term 141.
The c lassical theory can be extended to par t i c les with spin 151. The spin va-
r iab les are best described today using quant i t ies forming a Grassman algebra 161. The
main resu l t , except for addi t ional terms, is the same type non- l inear behaviour ra-
d ia t ion term as in eq.(6).
Some so lu t ions of the rad ia t i ve equations w i th spin are known 171. They exh i -
b i t much of the t yp ica l e r r a t i c behavior of the t r a j ec to r y around an average t r a j e ~
to ry which we know from the Dirac equation as "z i t terbewegung". Conversely, the
c lass ica l l i m i t of the Dirac equation is not a sp in less p a r t i c l e , but a p a r t i c l e
w i th a c lass i ca l spin. Thus the spin of the e lec t ron must be an essent ia l feature
of the s t ruc tu re of the e lec t ron (not j u s t an inessent ia l add i t i on ) .
I I I . QUANTUM THEORY OF SELF-INTERACTION
We see thus tha t the e lec t ron 's equation of motion is fundamental ly non- l i near .
When we go over to quantum mechanics we do not quant ize the " r ad i a t i ng , s e l f - i n t e -
rac t ing e lec t ron" but f i r s t the free e lec t ron . Let us compare the c lass ica l and
quantum equations p a r a l l e l y :
m Z = e ~ Fext ' (x=Z)Z ~ + e Fsel f (x=Z) ZU
o p ~ p~
( - i y p ~p - m)~ = e yUA (x) ¢,(x) + ? , (7)
o r , n o n - r e l a t i v i s t i c a l l y and fo r A = O,
~2
(i~i ~ - ~ A ) ~ = U ~ + ? (8)
We see that in the standard wave mechanics the non l inear terms coming from s e l f -
f i e l d have been omit ted, and a renormalized mass have been used. But th i s is only
an approximation. Hence we wish now to complete the wave equations by the inc luss ion
of the s e l f - f i e l d terms.
Nonl inear terms have been added to (7) and (8) in order to have s o l i t o n - l i k e
so lu t ions 181, 191. I should l i ke to discuss here the non- l i near terms in the stan-
dard theory, thus w i thou t in t roduc ing any new parameters.
We consider the basic framework of Lorentz, eqs . (1 ) - ( 3 ) , but when the e lec t ron
is described by a Dirac f i e l d ~(x) :
( - i y P ~ - mo) ~ : eyP~(x)A (x) (9)
F v '~ (x ) = j (x) = e~(x)y ~(x) . (10)
In the gauge Au = O, we can e l iminate A (x) from these equations and obtain the
non- l i near i n t e g r o d i f f e r e n t i a l equations
( - i y u ~ - mo ) @(x) =
I f we work with localized functions always, the theory, including renormaliza
tion procedure, is f in i te , and nonperturbative; i t describes a dressed, radiating
self-interacting particle.
We consider now in a bit detail the two-body coupled equations for ~ and n:
(y~P ml)~ = e yUA( 1)self e UA( 2)
- i ~ ~ + i Y
(~UPu - m2)q = e2~A(2)selfu n + e2~AJl)
with (15)
A(1)(x)u = e I Idy D(x-y) ~(y)yu~(y)
A(2)(x)u = e2 idy D(x-y) ~(y)~ q(y)
We shall bring these equations into a manageable radial form using the ansatz
r ( i ) t
-iL m
#(i)(x ) = ~ #~i)(~)e
(16)
n
where n labels the quantum numbers (En,J,M,K) and
l i gn( i ) ( r ) ~ i ) ( ~ i ~
, ~JiM( ~) ~ ~CJ M Y~(~)× ~ , (17)
Im;½
with these substitutions, and
d4k e- ikx
(18)
D(x) = - ~ k2 +i~
one obtains after much computations the coupled radial equations 1101
2
df s Ks-1 l e l ~ -
= dr' ( r , r ' )
dr r fs + (Es-ml)gs 2 - ~ 7 Es=Em-En+Er j VlEnEm
n.m.l
x I gr gn* gm' -T1n msr + f 'n* f 'm T~ 'm'sr + f r g'n* f 'm -T2n m' sr' f'*n gm' T~ 'msr'
+
2i7 erle 22 s>Em En+Er fd r' V1EnEm ( r . r ' ) { gr -L,e*n em,~ln 1m sr + H-n, *H-m,T-1 n 'm'sr
+ fr e~* dm' T2n m'sr' _ dn. em' Tn2 'msr'
K + 1 2 r
dgs + s 1 el > r
d--r- r gs - (Es + m) fs = - 27 r 2 dr ' VIEnE ( r , r ' )
Es=Em-En+Er m
If r gn* gm-1 + f'*n f'm Tn1 m sr - gr gn'* f'm -2Tnm's'r-fn*gm' Tn'ms2 I
1 ele2> i { ~-e,.e,Tnms'r'+d,.d,T n'm's'r'
27 r 2 dr' V1 E E ( r , r ' ) f r n m 1 n m 1
Es=Em-En+Er n m
- gr eL ~* d' mnm's'r d '* e' Tn'ms'r
m'2 n m'2 '
f ' = f ( r ' ) , g' = ( r ' ) , etc. (.19)
There are two similar equations for e r and d r .
Here the Kernels V are known integrals
f= k2dk j~ (k r ) j~ (kr ' )
( r , r ' ) = - 17 r2r'2 (12o)
V~EnEm o (En-Em)2- k2 + i
1
En ÷ Em 1 i 3/2 r< ~+~
, T2~+---- ~ (r<r>) (~)
T 1 and T2 are known functions of Clebsch-Gordon coeff icients.
The terms on the r i gh t hand side are the various in terac t ion and rad ia t ive po-
t en t i a l s . To see these more c lear ly we special ize to the s tat ionary Is -s ta te (of
positronium, for example): I = 0 , K = - I , a l l J = 1/2, etc. Then
d-fr + ~2 f + (El - ml) g = 21~ rI 2 g Id r' VOEIE(1r , r ' )
x e (g 'g ' * + f ' * f ' ) + ele 2 (e '*e ' + d ' *d ' ) - ~-~ ~- f Id r ' ( r , r ' )
× e21 f ' g ' - e l e 2 d ' e ' (21)
and s im i l a r l y the other equations. We shall refer to the f i r s t term as " e l e c t r i c " ,
to the second term as "magnetic" po ten t ia l , because they are mul t ip l ied by g and
f , respect ively.
One can see by e x p l i c i t ca lcu lat ion that the contr ibut ion of the second par-
t i c l e , also in s-state, to the f i r s t gives an e lec t r i c po ten t ia l , in the l i m i t
r ÷ ~ ,
ele 2
- - g + . . . . ( 2 2 )
4~r
as we have noted ea r l i e r , and as r ÷ 0
YZ4~-m-- ~g = const, g. (22')
The magnetic potent ia l behaves l i ke
1 2 y + l ele2 f
, as r ÷ ~
8m 3 r 2
(23)
1 (Z~) 3 m2
- 3 4~y(2¥- I ) r f , as r ÷ O
I f the second par t i c le is heavy for example, we can use the Coulomb potent ia l only
and obtain
10
2
df K-I e eI >r
dr r f + (E-m - A~--~) g = r2 g VOEE ( r , r ' )
(g'*g' + f ' * f , ) _ ~ 4K2 f I o~ d r 'V iEE ( r , r ' ) f ' g ' 1
- 2e7~ ~I F ( r ) f + G ( r ) g - Vmf + Veg (24)
s im i l a r l y for the other equations.
Here we introduced the e lec t r i c and magnetic form factors G(r) and F(r ) , res-
pect ive ly . The magnetic form factor F(r) has the form (which we shall need la te r )
~ - 2 r / r 0
F(r) = C dr' VIEE(r,r ' ) f ( r ' ) g ( r ' ) = C ' ( l - e ( l + p o l y n ( 2 r / r o ) ) , (25)
i 0
thus star ts from zero and approaches a constant for large distances, a behavior
which we know from perturbation theory.
Let us compare this resu l t with the Dirac equations for the electron with an
anomalous magnetic moment a in the Coulomb f i e l d
df K-1 f + (E - m - ele 2 ) g = a ele 2 f (26)
dr T ~ 2mr2
which is of course va l id for r ÷ ~, hence the i den t i f i ca t i on of the anomalous mag-
net ic moment in teract ion.
The anomalous magnetic moment has also an interact ion with the s e l f - f i e l d . S~
m i l a r l y , we have addi t iona l e lec t r i c potent ia ls , and, as we see from (21), a
charge renormalization due to the term (e /4~) ~ g. But before using these values,
we must renormalize the self-energy ef fects. In fac t , from the integrals evaluated
with the t r i a l Coulomb type functions, for example we must subtract the i r values
when e 2 ÷ O, the free par t ic le values.
Other Remarkable Solutions of Nonlinear Equations
The Dirac equation in Coulomb f i e l d without the rad iat ive terms on the r igh t
hand side, has the well-known discrete spectrum and the continium, the complete
set of solutions is known. We get a h int for a new class of solutions with radia-
t i ve terms corresponding to sharp resonances from eq.(24). Eliminating one of the
