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,£).