NONLINEAR 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