Annales Henri Poincaré

, Volume 10, Issue 1, pp 19–34 | Cite as

Borchers’ Commutation Relations for Sectors with Braid Group Statistics in Low Dimensions

Article

Abstract.

Borchers has shown that in a translation covariant vacuum representation of a theory of local observables with positive energy the following holds: The (Tomita) modular objects associated with the observable algebra of a fixed wedge region give rise to a representation of the subgroup of the Poincaré group generated by the boosts and the reflection associated to the wedge, and the translations. We prove here that Borchers’ theorem also holds in charged sectors with (possibly non-Abelian) braid group statistics in low space-time dimensions. Our result is a crucial step towards the Bisognano–Wichmann theorem for Plektons in d = 3, namely that the mentioned modular objects generate a representation of the proper Poincaré group, including a CPT operator. Our main assumptions are Haag duality of the observable algebra, and translation covariance with positive energy as well as finite statistics of the sector under consideration.

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidade Federal de Juiz de ForaJuiz de ForaBrazil

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