SOME REMARKS ON THE THEORY OF EXPANSORS

This paper is a brief discussion of the properties of expansors introduced by Dirac(1945). After transforming the expansors Anrst to ξ-representation defined by 〈ξ|〉=∑ξ0-n-1ξ1rξ2sξ3tAnrst (1) and showing that undergoes transformations identical with standardrepresentations in a Lorentz transformation, it is shown that of the two fundamental invariants J=-1/2IklIkl,I=-1/2εklmnIklImn (2) characterizing the different irreducible representations of the Lorentz group, the second one in the theory of expansors is always zero. It is also shown that the requirement of the different eigenfunctions of J in the space to behave regularly for all ratios of ξ leads to J1/2, (3) (ii)I′=±(1+J)1/2i,J′=1+J.(4) Explicit formula for such matrixes are also worked out.If we require expansors after operations by such operators remain as expansors, we must let -1≤J≤0 and confine ourselves to the selection rule I′=0,J′=1+J-2(1+J)1/2. (5) Since successive transformations of J by the above formula starting with an initial value of J, say J1, satisfying -1≤J1≤0 do not lead to values of J beyond J1 and J2 J2≡1+J1-2(1+J1)1/2 (both J1, J2 being negative), it is clear that in constructing a wave equation of the type (-irμpμ+k)ψ=0 (6) with ψ in expansor spaces, the simplest formulation is to let the space of ψ to consist of two such expansor spaces (J1 0), (J2,0). Of course, the matrixes γμ are so choosen that only the selection rule (5) is effective. It is shown that the operators ξvξv?/(?ξμ)-(1±(1+J)1/2)ξμ (7) transform the (J,0) space to the spaces (1+J±2(1+J)1/2,0) respectively. Thus γμ may be constructed easily in terms of the operator ξvξv?/(?ξμ)-(1-(1+J)1/2)ξμ. Investigations of such wave equations will be left later.