In particular, it follows that, if
\Pf(x)\CM(l + \x\y-3'2-M-, (1.6e)
r/(^)|CM(l + |x|r-5/2-H-,
for each |i/| 0 and some e 0, then ( / , / ) G M£, 1/2 p 1. Thus M& contains
long-range potentials.
1.1.c Nonlinear Poincare covariance. A nonlinear representation T of p in ££,, in the
sense of [10] (see a detailed definition in §1.6.1.b of this introduction), is obtained by the
fact that the M-D equations are manifestly covariant:
Tx=Tk+1%, X€p, (1.7)
where X i-» Tx is linear, T 1 is given by (1.5a)-(1.5d) and, for u = (/, / , a) G .£?£,
Tp0(u) = (0,a7a,-i/
7Vc*) 7 = (70,71,72,73), (l-8a)
T%. = 0 for 1 i 3, T ^ = 0 for 1 i j 3, (1.8b)
(T2M0j{u))(x) = ^(T|
(tz))(x), 1 j 3,x = (x!,x
,X3). (1.8c)
Tx is a continuous polynomial from E^ to ££, (see Lemma 2.17) and it satisfies
[TX,TY] = T[XIY], x , Y e p , ( i . r )
where the left hand side is defined by
[A, B] = DA.B - DB.A, (1.7")
(DA.B)(f) being the Frechet derivative of the map A at the point / in the direction
B(f)}^ The gauge condition (1.1c) takes on initial data u = (/, / , a) the form
/o = X) ^
= E */* -
H2- (Llc')
lz3 1*3
The subspace (topological) V£, 1/2 p 1, of elements in ££ which satisfy these gauge
conditions, is diffeomorphie to E£g (see Theorem 6.11). The problem to integrate globally
the nonlinear Lie algebra representation T therefore consists of proving the existence of
an open neighbourhood UQQ of zero in V& and a group action U: VQ X U^ Z^oo, which is
C°° and such that Ug(0) = 0 for g G V0 and ft Uexp{tx)(u) \t=0= Tx(u), X ep.
Continuing to follow [28], we extend the linear map X 1— Tx, from p to the vector
space of all differentiable maps from E^ to ££,, to the enveloping algebra U(p)by defining
inductively: T\ = I, where I is the identity element in U(p), and
TYX = DTy.Tx, Y G *7(p),X G p. (1.9)
x) We also use the notation (DnA)(f;fi,...,fn) for the n-th derivative of A at /
in the directions / i , . . . , /
and the notation DnA.(fi,..., fn) for the function
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