MAXWELL - DIRAC EQUATIONS 7

where w(k) = (m2 + I&I2)1/2. We postulate that the asymptotic representations have the

following form:

U^(u) = (U^M(u),U^D(u)), U^M = U1M, (1.17a)

(U^D(u))A(k) = J2 c ^ ( t t - e f c ) P

e

( f c ) ( ^ a ) A ( A ; ) , g G V0, (1.17b)

e=±

where u = ( / , / , a ) G £££, the function (g,u,k) H+ ^ ( u , -efc) from VQ x E^ x R 3 to R

is C°° and if (/i^(w))(t,x) = ipg{u,mx(t2 - |a?|2)~~1/2), t 0, |x| t, then D^(w ) = 0.

Moreover growth conditions on pg should be satisfied, so that the main contribution to the

phase and its derivatives in g and k comes from UgD, the restriction of U1 to D. Finally

we impose the asymptotic condition

\\U™p{tPo)(n+(u)) - U™{m)(f, f)\\M, (1.17c)

+ Hc£p(tFb)(n+(«)) - £

^Hu^~i9)pe(-id)uZim)a\\D

- o,

e=±

when t — oc, where ^ = (/, / , a ) G £Yoo, {u,t,k) »- s^+'(-U, £, A;) is a C°° function from

ZYoo x R x R3 to R, s^ (w, £, — id) is the operator defined by inverse Fourier transform of the

multiplication operator k \— s£ \u, t, k). Moreover s£ \u, t, k) should satisfy growth con-

ditions so that its major contribution comes from U^,tp ^ (u). There are similar conditions

for U^~\ 0 _ and s£ , £ — ±. We note that if s£ is determined, then 17+ is determined

and so is U^. It was proved in [14], that on a set of asymptotic states (/, / , a) such that

/ , / G C£°(R3 - {0}) and a G C£°(R 3 ) 2 \ it is possible to choose (formula (3.33a) of [14])

4+)(M*0

= -#{AM(u),{t,-etk/v(k))), (1.18)

where A^ (u) is a certain approximate solution of the M-D equations absorbing the long-

range part of A for a solution (A, ?/), and where

0(H,y)= [ E^z)dz^ 2/GR 4 , (1-19)

JL{y)

where H:R4 - R4 and L(y) = {z G R4\z = sy,Q s 1}. The function (t,k) ^-•

Se(u,t,k) was determined by the fact, that has been proved in [14], that S£(u,t,k) =

eu)(k)t + s£ (u, t, k) has to be in a certain sense an approximate solution of the Hamilton-

Jacobi equation for a relativistic electron in an external electromagnetic potential:

{^Se(u,t,k) + A0{t,-VkS£(u,t,k)))2 (1.20a)

3

- Y, iki + Mt, -VkSe(u, t, k)))2 = m2.

2) C£(X), k 0, denotes the space of k times continuously differentiate functions on

X with compact support.