Wave equations for zero-mass particle with invariant helicities

Abstract

We present a brief discussion of the difficulties associated with taking the limit m→0 in massive-particle equations of the form i(∂ψ/∂t)=Hψ, in which ψ transforms according to theD(0,s) ⊕ D(s,0) representation of the homogeneous Lorentz group. It is pointed out that, though the HamiltonianH tends to a well-defined form as m→0, the resulting wave equation is still unsatisfactory as the invariant scalar product (in the space of solutions of the wave equation) has an unacceptable form. We observe that this difficulty is due to the indecomposability of the representation of the Poincare group over wave functions transforming asD(0,s) ⊕ D(s,0) in the massless case. Making due allowance for this fact we obtain, in close parallel to the treatment for massive particles, wave equations for zero-mass particles with invariant helicities ±s. The relevant invariant scalar products are also determined.