Annales Henri Poincaré

, Volume 20, Issue 8, pp 2555–2584 | Cite as

Split Property for Free Massless Finite Helicity Fields

  • Roberto Longo
  • Vincenzo MorinelliEmail author
  • Francesco Preta
  • Karl-Henning Rehren


We prove the split property for any finite helicity free quantum fields. Finite helicity Poincaré representations extend to the conformal group \({{\mathcal {C}}}\) (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 \(\mathrm {Tr}\;(e^{-\beta L_0})<\infty \) for the conformal Hamiltonian \(L_0\) 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.



V. M. thanks Gerardo Morsella and Massimo Bianchi for valuable discussions. R. L. and V. M. acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

Note Added.

The suggestions discussed in the Outlook are confirmed in [28].


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Roberto Longo
    • 1
  • Vincenzo Morinelli
    • 1
    Email author
  • Francesco Preta
    • 2
  • Karl-Henning Rehren
    • 3
  1. 1.Dipartimento di MatematicaUniversità di Roma Tor VergataRomeItaly
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.Institut für Theoretische PhysikUniversität GöttingenGöttingenGermany

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