MAXWELL - DIRAC EQUATIONS 33
|jx| = \v\ n + 1, rj£ = r^(x , —iV). Let t = (/, / ) M ^ and let r be one of the partial
differential operators r ^ , |/x| = |i/| = ra + 1. Definition (1.5) of
T1
restricted to M ^ give
for « = (/,/)
£ UPxVHU (2-44)
xen
C^m^Mu+^Uxidj-Xj^rMM + |||V|'-1[A)r]/||La
i ij
+ El||V|"[xi,r]/||L2+E|||Vr1[E^^^]/ll^)' r = r(a:,-tV).
i i I
By the same argument which led to (2.37) it follows that there exist polynomials p(X, x,£),
for X = PM and for X = M^-, 1 i j 3, and polynomials p^(X,x,£), for X = M0j,
i = l,2, and 1 j 3, such that
[di, r(x, -tV)] = p(Pi? x, -tV), (2.45)
[A,r(s,-tV)]=p(P
0
,x,-*V),
[Xi9j Xj^,r(x, zV)] = p(Mij,x, iV), 1 i j 3,
[x»,r(o;,-iV)] =p{1){M0i,x, -iV),
[]T ftxift, r(x, -tV)] -
p(2)(M0i,
x, -tV).
z
Since, according to (2.41) r satisfies r(a~1x,a^) = r(x,£) and degr 2n, it follows from
(2.45) that
p(Pi,
a~xx,
a£) = ap(Pi, x, f), degp(P*) 2n - 1, (2.46a)
p(Po,a
-1
*,aO =
a2p(P0,;c,£),
degp(P0) 2n, (2.466)
p(Mtj,a_1a;,a^)
=p(Mij,x,f), deg p(Mij) 2n, (2.46c)
p^{M^a-xxM)
= o-V^CMw^.O, degp^(Moi) 2n - 1, (2.46d)
pW(Mw,o-1a;,a^)=op2)(M(M,x,0, degp(2)(M0i) 2n + 1. (2.46c)
It follows from (2.45), (2.46a) and (2.46c) that
E
Wl°irMM
+ £ HMi - *A'MI ^ Cng{v). (2.47)
Due to (2.46b) and (2.46e) we can write:
a) p(P0,x, -iV) =^2djpj(P0,x, -»V), degp,(P0) 2n - 1, (2.48)
Pj^Po^
- 1
^,^ ) =apJ-(Po,x,0,
b)
p(2)(M0j,x,
-tV) =
£ftpz(2)(M0i,a:,
-tV),
degft(2)(M0i)
2n, (2.49)
i
p[2\Moj,a-1x,aZ)=p[2\Moj,x,£).
Previous Page Next Page