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)
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