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NONLINEAR PROBLEMS IN CLASSICAL AND QUANTUMELECTRODYNAMICS
A.O. Barut The U n i v e r s i t y of Colorado, Boulder, Colorado 80309
Table of Contents Page I. II. IIio
Introduction
2
Classical R e l a t i v i s t i c Electron Theory
2
Quantum Theory of S e l f - l n t e r a c t i o n
5
Other Remarkable Solutions of Nonlinear Equations
10
Some Related Problems
11
References
13
NONLINEAR PROBLEMS IN CLASSICAL AND QUANTUMELECTRODYNAMICS A. O. Barut The U n i v e r s i t y of Colorado, Boulder, Colorado 80309
I . INTRODUCTION We present here a discussions of the n o n - l i n e a r problems a r i s i n g due to s e l f f i e l d of the electron, both in classical and quantum electrodynamics. Because of some shortcomings of the conventional quantumelectrodynamics 11[ an attempt has been made to carry over the nonperturbative r a d i a t i o n reaction theory of classical electrodynamics to quantum theory. The goal is to have an equation for the r a d i a t i n g and s e l f - i n t e r a c t i n g electron as a whole, in other words,an equation for the f i n a l "dressed" electron. In addition the theory and renormalization terms are a l l f i n i t e . Each p a r t i c l e is described by a single wave function ~(x) moving under the influence 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 p a r t i c l e s .
In p a r t i c u l a r , we dis
cuss the completely covariant two-body equations in some d e t a i l , and point out to some new remarkable solutions of the n o n - l i n e a r equations: These are the resonance states in the two-body problem due to the i n t e r a c t i o n of the anomalous magnetic moment of the p a r t i c l e which become very strong at small distances. I I . CLASSICAL RELATIVISTIC ELECTRON THEORY The motion of charged p a r t i c l e s are not governed by the simple set of Newton's equations as one usually assumes in the theory of dynamical systems, but by rather complicated n o n - l i n e a r equations i n v o l v i n g even t h i r d order of d e r i v a t i v e s . To see t h i s we begin with Lorentz's fundamental postulates of the electron theory of matter: ( i ) Matter consists of a number of charged p a r t i c l e s moving under the influence of the electromagnetic f i e l d produced by a l l charged p a r t i c l e s . The equation of mot i o n of the i th
charged p a r t i c l e is given by m Z( i ) : e F
(x) ZV ( i ) l
(1) lx=z ( i )
'
where Z (S) is the worl'd-line of the p a r t i c l e in the Minkowski space M4 in terms of
an i n v a r i a n t time parameter S (e.g. proper time) - the d e r i v a t i v e s are with respect to S, and F~v is the t o t a l electromagnetic f i e l d . (ii)
The t o t a l electromagnetic f i e l d F
P~
F
obeys Maxwell's equations
'~(x) = j (x)
,
(2)
where j (x) is the t o t a 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 i n c i p l e a closed system of equations i f we have in addition some model of matter t e l l i n g us how many charged p a r t i c l e s there are. These equations taken together give f o r each p a r t i c l e i a h i g h l y nonlinear equation on Z ( i ) .
because due to the term k = i in (3), F This is the socalled
is even i n f i n i t e
self-field
(x) in (1) depends n o n l i n e a r l y
of the Uith ~ p a r t i c l e . A c t u a l l y t h i s term
at X = Z ( i ) due to the f a c t o r ~(X-Zki)).' " In practice t h i s i n f i n i t e
term does not cause as much trouble as i t should-one simply drops first