MAXWELL - DIRAC EQUATIONS 11

for g = exp(aMPM), ui = (fi, fi,oti) G E%£, i G {1,2}, and where B\ ' is the corresponding

free field given by (1.22b).

When the limit tf(£+)\ (t, -tek/u(k)) -+ r f f i W 1 , (u;(fc), -efc)) exists for t - oo,

then the infrared cocycle is in fact a coboundary. In particular, this is the case when /M, /M

are equal to zero in a neighbourhood of zero.

1.4 PHYSICAL REMARK S

1.4.a Physical motivation of the asymptotic condition. Next, we return to the physical

justification of conditions (1.17a)-(1.17c), and to be more specific, as this is the main

case of this paper, we choose Se to be given by (1.18) and A+ to be given by (1.22a)-

(1.22c) with Xo — 1- According to statement iv) of Theorem 6.16 and Theorem 6.19, the

asymptotic condition in statement iii) of Theorem II is equivalent to

\\(A(t),A(t)) - (AL(t),AL(t))\\Mp + \\m - (e-*1L)(t)\\D - 0. (1-27)

when t - oo, where (A(t),A(t)^(t)) = *7

e x p ( t P o )

(n

+

(u)), (AL(t),AL{t)^L(t)) =

UexP(tP0)u a n d where for a fixed u = (/, / , a) G G&\ (£, x) i-» p(t, x), (£, X ) G E + X M3 is a

C°° function given in Theorem 6.16. When needed we shall write (u, (t,x)) H- p(u, (t,x))

to stress the dependence of (p on u. It follows from Theorem A.l that ip in (1.27) can

be replaced by #(J4( + ),-) but for technical reasons we keep (p. In fact, from a heuris-

tic point of view, since tf(A(+), (t,x)) = -se{u,t,-ex/{t2 - |x| 2 ) 1 / 2 ), (1.27) with (p re-

placed by i?(i4^+\ •) is obtained from the leading term in the stationary phase expansion

The energy-momentum tensor for the M-D system is, in one of its forms, given by

#*" = -F»aF»a + -^Fa(3F^ + \®r(id» - A")il + ((id» - A")^^), (1.28)

where 0 / x 3 , 0 z / 3 and F^ — dilAv — dvA^. The current density vector is given

by

f = ^

7

^ , 0 fi 3. (1.29)

t^v and j , / x are invariant under gauge transformations: ipf = elXip, A!^ — A^ — d^X (not

necessarily respecting the Lorentz gauge condition that gives here DA = 0). We also

introduce the energy-momentum tensor and the current density vector for the system of

free fields AL and ipL by

t

u

=

_FmFU

+

\g^FLaiiFf + \(4Lr(idniL + (WWIWL), (1-30)

and

fL=^Ll^L, (1-31)

where

FL^

= d^A^ —

9„AL

M

.

Since t^v can be written as

t

t» = -F»aFva + ]g^Fa0Fai3 + \$-f{idr - A'v)iP' + {{id» - A'v)^)-f^),