Split Property for Free Massless Finite Helicity Fields

We prove the split property for any finite helicity free quantum fields. Finite helicity Poincaré representations extend to the conformal group C (cf. Mack in Commun Math Phys 55:1–28, 1977) and the conformal covariance plays an essential role in the argument: The split property is ensured by the trace class condition Tr(e-βL0)<∞ for the conformal Hamiltonian L of the Möbius covariant restriction of the net on the time axis. We extend the argument for the scalar case presented in Buchholz et al. (Commun Math Phys 270:267–293, 2007). We provide the direct sum decomposition into irreducible representations of the conformal extension of any helicity-h representation to the subgroup of transformations fixing the time axis. Our analysis provides new relations among finite helicity representations and suggests a new construction for representations and free quantum fields with nonzero helicity.