2

INTRODUCTION

1.1.b Poincare covariance: linear part and representation spaces. Since equations

(1.1a), (1.1b) and (1.1c) are manifestly covariant under the action of the universal covering

group Vo = R4(gSL(2, C) of the Poincare group, it is easy to complete the time translation

generator, formally defined by (1.2a) and (1.2b), to a nonlinear representation of the whole

Lie algebra p = R4£$[(2,C) of TV To do this we first introduce topological vector spaces

on which the representations will be defined. Let

Mp,

—1/2 p oo be the completion

of S(R3,R4) © S(R3,R4) with respect to the norm

IK/,/)HMP

=

(iiivr/u2L2(R3,R4)

+

iiivrviii2(M3,M4))1/2,

u.4a)

where

S(R3,R4)

is the Schwartz space of test functions from

R3

to

R4

and where |V| =

(-A) 1 / 2 . Let D = L2(R3, C4) and let Ep = Mp 0 £, -1/2 p oo, be the Hilbert space

with norm

ll(/,/,«)llBp =

(II(/,/)II2M,

+

IMIi)1/2-

(i.4b)

When there is no possibility of confusion we write E (resp. M) instead of Ep (resp. Mp)

for 1/2 p 1.

Mop

is the closed subspace of elements (/, / ) G

Mp,

-1/2 p oo such

that

/o= X)

di^ L4c)

li3

/o = - X

ivr20i/,

li3

where / = (/c/1,/2,/3) and / = (/o,/i,/2,/3). A solution £M of D5M = 0, 0 \i 3,

with initial conditions (/, / ) G Mp satisfies the gauge condition d^B^ = 0 if and only if

(/, / ) G

Mop.

We define

Eop

=

Mop

0 D.

Let II = {PM,Ma^|0 / i 3 , 0 a / ^ 3 } b e a standard basis of the Poincare

Lie algebra p = R43isl(2, C), where PQ is the time translation generator, Pi, 1 i 3 the

space translation generators, M^, 1 i j 3, are the space rotation generators and

M)j» 1 J 3, are the boost generators. We define Map = —Mpa for 0 /3 a 3.

There is a linear (strongly) continuous representation

U1

of VQ in

Ep,

—1/2 p 00,

(see Lemma 2.1) with space of differentiable vectors ££, the differential of which is the

following linear representation

T1

of p in ££:

(/,A/,2a), (1.5a)

»(/,/, a), l i 3 , (1.5b)

-(xi9j - Xjdi)(f,f,a)(x) + (nijf,nijf,Tija)(x) (1.5c)

1 « i 3, cr^- = l/27»7j G su(2), ra^ G so(3),

3

(xi/(x),^a7-(xt9j/(x)),XiPa(a:)) 4- (noi/,nw/,(7oia)(x), (1.5d)

i=o

1 i 3,aoi = I/2707;, n0i G 50(3,1),

T^(/,/,a )

(T^(/,/,a))(x )

(rir

0 i

(/,/,a))(x)