We show that any Poincar6-invariant equation for particles of zero mass and of discrete spin provide a unitary representation of the conformal group, and find an explicit expression of the conformal group generators in terms of Poincar~ group generators.
It is well-known that the relativistic equations for massless particles are invariant under the conformal transformations. This was first established for the Maxwell equations [1] and then for the equations describing the massless particles of spin 1/2 [2] and of any spin [3].
L. Gross [4] has demonstrated that the solutions of the Maxwell and of the Rarita-Schwinger (with mass m = 0) equations provide a unitary representation of the conformal group C4. The proof given in [4] is rather tedious and in some sense non-constructive, since it does not give an algorithm to obtain an explicit form of Hermitian generators of the group C4 for any conforreal invariant equation.
In this note, we shall formulate a theorem, which generalizes the results [1--4] and give a simple and constructive proof of it. Without restricting ourselves by any concrete form of equations for massless particles we show that any (generally speaking, reducible) representation of the Lie algebra of Poincar~ group P (1, 3), which corresponds to zero mass and discrete spin, can be extended to provide a representation of the conformal group Lie algebra, and find the explicit expression of the generators of the group C4 through the generators of its subgroup P (1,3).
T H E O R E M 1. Any Poineard-invariant equation for particles of zero mass and of discrete spin is invariant under the conformal algebra Ca (*), basis elements of which are given by the operators eu, Juv and
_ 1 [ p o P a / p 2 , J o a ] +,