4 INTRODUCTION

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/

M

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|

o

(tz))(x), 1 j 3,x = (x!,x

2

,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) ^

A/o

= 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 , . . . , /

n

and the notation DnA.(fi,..., fn) for the function

f~(DnA)(f;f1,...,fn).