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 \— \u, t, k). Moreover \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 , £ ±. We note that if 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 + (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.
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