Covariant Sectors with Infinite Dimension and Positivity of the Energy

Abstract

We consider a Moebius covariant sector, possibly with infinite dimension, of a local conformal net of von Neumann algebras on the circle. If the sector has finite index, it has automatically positive energy. In the infinite index case, we show the spectrum of the energy always to contain the positive real line, but, as seen by an example, it may contain negative values. We then consider nets with Haag duality on the real line, or equivalently sectors with non-solitonic extension to the dual net; we give a criterion for irreducible sectors to have positive energy, namely this is the case iff there exists an unbounded Moebius covariant left inverse. As a consequence the class of sectors with positive energy is stable under composition, conjugation and direct integral decomposition.

25 pages, Latex2e

Cited by in corpus (8)

• Modular localization and Wigner particles
• Topological Sectors and a Dichotomy in Conformal Field Theory
• Implementation of conformal covariance by diffeomorphism symmetry
• The Principle of Locality. Effectiveness, fate and challenges
• Solitons and nonsmooth diffeomorphisms in conformal nets
• Spin and Statistics and First Principles
• Bisognano-Wichmann property for rigid categorical extensions and non-local extensions of conformal nets
• On a Connes Fusion Approach to Finite Index Extensions of Conformal Nets