The following text was automatically extracted from the image on this page using optical character recognition software:
The purpose of this note is to establish similar results when polynomials are re-
placed by polynomial spinors and harmonic polynomials by polynomial Dirac
spinors, i.e., polynomial solutions of Dirac equations.
To this end consider an action of the real Clifford algebra Cln := Cl(R") on
some complex space CN. Equivalently, one is presented with n skew-Hermitian
N x N complex matrices EL, E2,..., E, such that for every i, Ej = - Id and
EE + EjjE = 0, for every i - j. The Euclidean Dirac operator is then the
differential operator
p:C"(U, CN) C'(U, CN), U C " open
defined for spinors s E C (U, C"') written in column form by
$i8=
Os
where - represents component-wise differentiation of a with respect to r. It is
easily seen that $ is a self-adjoint first order elliptic differential operator satisfying
the following properties:
S$(fs)=gradf.a+fJ es, fE C"(U, C), gradfsa:= i E8 s (4)
p2 = -A, where A is the component-wise Laplacian on C"(U, C"). (5)
Denote now by Pk the subspace of CO(R1n, C) consisting in spinors with poly-
nomial components, homogeneous of degree k, and by Hk the subspace of Pk
consisting in polynomial Dirac spinors, i.e.,
Hk := {pEk P; F (Pk)= 0}.
Clearly, lD(Pk) C Pk-i (P-1 = 0). If one denotes by z. the Clifford multiplication
in CN .by x E Rn, i.e., x . v = uE XtEi, v E CN, then x PAk P+1
i=1
Lemma. Let hk e Hk be a polynomial Dirac spinor of degree k. Then
p(c * hk) = -(n + 2k)hk. (6)
Consequently, x . hk has harmonic components.
122
Nicolae Anghel
