MAXWELL - DIRAC EQUATIONS 3

where x = (x\,X2,#3) are related to contravariant coordinates yM by x\ = yl (we use

this notation to avoid writing components of vectors in M3 with upper indices that could

be confused with powers). The explicit form of n

a

0 , 0 a j3 3, defining a vector

representation of p in M4, is not important here. We remark that U1 leaves Eop invariant.

Following closely [28], we introduce the graded sequence of Hilbert spaces E?, i

0, Eg = Ep, where E^ c Ep. c EP for i j . E? is the space of ^-vectors of the

representation U1. Suppose given an ordering X\ X2 • • • Xi0 on II. Then in the

universal enveloping algebra U(p) of p, the subset of all products X^1 • • • X^0, 0 a*, 1

i 10, is a basis IT of U{p). If Y = X?1 • • • X^ 1 0 , then we define \Y\ = \a\ = Eo*io a*-

Let E?, i G N be the completion of E^ with respect to the norm

N k = ( E H^(*) 1/2 ' (!-6a)

yen'

where T

y

, F € t/(p) is defined by the canonical extension of T 1 to the enveloping algebra

U(p) of p. Let M°p = MpnMop, E\p = EpDEop, M^ = M ^ f l M 0 " and E^ = E^nE0^

where Mp (resp. M£) is the image of EP (resp. E^) in M p under the canonical projection

of Ep on Mp. Let D* (resp. £oo) be the image of EP (resp. E^) in Z) under the canonical

projection of Ep on D. To understand better what the elements of the spaces Ep are, we

introduce the seminorms qn, n 0, on ££, where

?«(«) = (?f («)2 + ?« («) 2 ) 1 / 2 , u = («,a) eE^ve M £ , a 6 £«,,

9nM(")=(

E

l|M^||2M,)1/2,

IMIM»

«£(«)=( E

\\M^a\\l)1/2,

\fj,\n

\u\n

where // = (/ii,^2,^3), ^ = (^1,^2,^3) are multi-indices, 9M = (9i)Ml(92)M2(d3)M3 and

The norms || • \\pP and qn are equivalent (see Theorem 2.9). This

shows in particular that D^ = 5(R 3 ,C 4 ) . Moreover, if 1/2 p 1 and (/,/ ) G M&,

then (see Theorem 2.12 and Theorem 2.13)

(l + N ) 3 / 2 ^ | / ( x ) | C | | ( / , 0 ) | |

M f

, (1.6b)

(1 + | x | ) 5 / 2 - ^ " - " ( | ^ f t / ( x ) | + |0"/(s)|) C

M

| | ( / , / ) | |

W

p

| + a

, (1.6c)

for |i/| 0, 1 i 3 and if / G L«, 9 - 6/(3 - 2p), xad^dif G LP and z a ^ / G 1^,

p = 6/(5 - 2p), |a| \0\ n, 1 i 3, then

II(/,/)HA«

Cn Y, ( E H^ft/H^P + K^/||

L P

). (1.6d)

0|a||^|n 0i3