MAXWELL - DIRAC EQUATIONS

5

It was proved in [28] that this is a linear map from U(p) to the vector space of differentiable

maps from E^ to E^ (see formula (1.10) of [28] and the sequel). For completeness we

recall the proof in [28]. The vector field Xx, X G p , defines a linear differential operator dx

of degree at most one operating on the space C°°(£*£,) by dxF = DF.TX, F G

C00{E00).

The fact that X H- TX is a nonlinear representation of p on E^ implies that X i— • dx is a

linear representation of p on linear differential operators of degree at most one. This linear

continuous representation has a canonical extension Y i— • dY to U(p) on linear differential

operators of arbitrary order operating on

COC(EPQ).

If e

y

, Ye U(p), is the part of dY of

degree not higher than one, then Y •-» eY is a linear map of U(p) into the space of linear

partial differential operators of degree at most one. Let Y e U(p). We write Y = Z + a,

where a G C • I and Z has no component on C • I (relative to the natural graduation of

U(p)). Then the previous definition of TY gives eYF = DF.Tz + aF, which proves that

Y t— Ty is a linear map on U(p). Moreover, it was proved in [28] that

dt

where

TY{t)(u(t))=TPoY{t)(u(t)), YeU(p), (1.10)

y(t)=exp(tadp

0

)r, adPoZ=[P0,Z], (1.11)

if

u(t)=Tfl,(u(t)). (1.12)

We note that definition (1.11) makes sense since (adp0)n, n 0, is a linear map from U(p)

to U(p) leaving invariant the subspace of elements of degree at most /, I 0, in U(p) and

since then exp(ta,dp0)Y = 2no(

n

0

- 1

(^

a

dpo)

n

^ converges absolutely. For completeness

we also recall the proof of (1.10). Since ^Y{t) = adpoy(£), it follows from (1.9) and (1.12),

that

^TY{t){u{i)) = T

Ay(t)

(u(t)) 4- (DTY(t).TPo)(u(t))

= T[Po,Y(t)]{u(t)) + TY(t)Po(u{t)) = TPoY(ty

TY, Y G U(p), denotes the n-homogeneous part of Ty and we shall identify TY with

a n-linear symmetric map and also with a continuous linear map from ® E^ into E^

where 0 is the n-fold complete tensor product endowed with the projective tensor product

topology. T¥(u), r^

n

(w) and U^{u) (resp. T$(u),

Tfn{u)

and U°(u)) is the projection

of TY(u), T$(u) and Ug(u) on M? (resp. D). We also define fY,Y G U{p) by

TY=TY+fY. (1.13)

Since it does not bring any contradiction, we shall also denote by

TM1

or

T1M

(resp.

TD1

or T1D) the linear representation of p which is the restriction of the linear representation

T1

to MP (resp. D). Similarly,

U™1

or

U]M

(resp.

U^1

or

U*D)

denotes the restriction

of U\ to MP (resp. D).