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,