8 INTRODUCTION
We proved and used the fact that
y»dll#(H,y) = ifHll(y), (1.20b)
to establish that S£(u, t, k) is an approximate solution of the HamiltonJacobi equation.
1.3 PRESENTATION OF THE MAIN RESULTS
1.3.a Some notations. Let u+ = (/, / , a ) G E^g and let
J^(t,x) =
(m/t)3(u(q(t,x))/m)5
(1.21)
J2((Pe(id)a)A(£q(t,x)/m))+1°^((P,(id))ar(eq(t,x)/m),
e=±
for * 0, x t, where g(i,x) =
mx/(t2

x2)1/2
and, for \x\ 0, 0 t \x\, let
4 + ) (i,x) = 0. Since a € S(E3,C4), it follows that 4 + ) € C°°((E+ x E3)  {0}) and that
its support is contained in the forward light cone. We choose \ € C°°(E), x(T) = 0 f°r
T 1, X(r) = 1 for r 2, 0 *(T) 1 for T € E,
Xo
€ C°°([0,ooD, XoM = 1 for T 2,
0 Xo(T) 1 for r G [0, oo[, and introduce A^ = A^1 + A^2 by
(AM\t))(x) =
Xo
((t2  aT2)V2)(jB+)i(t))(a.) .
(^+ 2(*))(x) =x((* 2 W 2 ) 1/2 )(S(+) 2 (i))(x) J t o r ^ Inl
and (1.22a)
(A+*(t))(x) = 0 J
iort\x\
where
BWH*) = cos(Vt)/M +
IVI"1
sin(Vt)/M, t G E, (1.22b)
/
OO
V
1
sin(V(*s))jW(5,)^, t 0 . (1.22c)
The cutoff function \ has been introduced to exclude in a Lorentz invariant way the points
(0,x), x G M3 from the support of A^2. [/+) is defined by formulas (1.17a) and (1.17b)
and by (fg = 0 for g G SX(2, C) and
pg(u, sk) = tf°° ((Xo o
p)(B(+)1(
+ «)  ^
( + ) 1
(0), M*0, ek)), (1.23a)
for # = exp(aMPM), /c G R3, where
/•OO
tf°°(fr,2/) = / y»H^sy)ds, y G
M4,
and p(t,x) =
(t2

\x\)^2.
(1.23b)
Jo
We have made the identification convention that B^x{t, x) = (B^1(t))(x). We note that
the function y i— • ^(H, y) is homogeneous of degree zero, so we could also have taken the