12 INTRODUCTION
where ipf = elipip, A'^ = A^ d^ip and F^u d^Au duA^y it follows from statement iii)
of Theorem 6.16 that
lim [ \t^(t,x)-t?(t,x)\dx= lim \\A'(t)\\L„\\iL(t)\\2D.
By unitarity of U1D, ||^
L
(*)IID = IWID- Using statement iii) of Theorem 6.16 for ||A(£)||Loo
and using statement ii) of Lemma 4.4 for ^ ( A ^ V ) a n d estimate (6.246) for 9M(/?
tf^W1, •)) [t follows that 11^(^)1^00 - 0, when t - oo. Therefore
lim / It^Ct, x) - ^ ( t , x)| dx = 0. (1.32)
Similarly
lim / | ^ ( t , x ) - ^ ( t , x ) | d x = 0. (1.33)
Limits (1.32) and (1.33) show that, as far as one measures energy-momentum and current,
the solutions of the M-D system are asymptotically indistinguishable from free solutions
because of gauge invariance. Condition (1.17c) is therefore natural. A similar discussion
can be made for the angular momentum tensor; however it is technically more involved,
so we omit it.
1.4.b Gauge transformations. A gauge transformation ipf = elXip, A = A^ 9MA,
respecting the Lorentz gauge condition, i.e. DA = 0, transforms a solution (A, ip) of the
M-D equations (l.la)-(l.lc) into a solution (A',rpf) of the M-D equations. Let v G UQQ.
v is the initial data for the solution (A, x/j) of the M-D equations, and if A is sufficiently
small and regular, the initial data v' for the solution (A', tp') is in Uoo, since Uoo is open.
Let u = ^(v) and v! = ft+H^')- S i n c e x(v) = A (°) + #(A,2/), with AM = 9MA, it follows
from (1.27) that
\\(A'(t),A'{t)) - {AfL{t\A'L(t))\\MP + WV) ~ e-^'^^fl/L(t)\\D - 0, (1.34)
when t -+ oo, where (A'L(t),A'L(t)^'L(t)) = U^p{tPo)uf and u' = (/',/' , a'), //,(*) =
ffi(x) ~ (dMA)(0,:r), fii(x) = ffi(x) ~ (^M^OA)(0,X), a! elX^a. Since we can choose the
constant A(0) arbitrarily, it follows that the admissible gauge transformations u \— u' of
the scattering data are given by
A'LlM = ALv-d»\, VL = ^
L
, (1.35)
where c e R and A is such that DA = 0, and such that (A(0, •), A(0, •)) G E%g, AM(t,x) =
(0MA)(t,aO,
K(^x)
= {dod»X){t,x).
We now introduce the notion of gauge-projective map. Let Q: E%£ » E^ be a C°°
map leaving Oio invariant. More general situations are possible, but to fix the ideas we
make this hypothesis. If there exists a gauge transformation G of the form (1.35) such
that
(fi+(Q(«))) - U^p{tPo)(fJ)\\M, (1.36)
+ \\u°p{tPo)(n+(Q(u)))
-^is[+)(G(u)"'~ia)pe(-id)ulg{tPa)oc\\D
- o,
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