Scale and Möbius Covariance in Two-Dimensional Haag–Kastler Net

Abstract

Given a two-dimensional Haag-Kastler net which is Poincaré-dilation covariant with additional properties, we prove that it can be extended to a Möbius covariant net. Additional properties are either a certain condition on modular covariance, or a variant of strong additivity. The proof relies neither on the existence of stress-energy tensor nor any assumption on scaling dimensions. We exhibit some examples of Poincaré-dilation covariant net which cannot be extended to a Möbius covariant net, and discuss the obstructions.

Spacelike deformations: higher-helicity fields from scalar fields

“…Because the spectral projection E m does not commute with the dilations, the latter are not defined on the subspace H m , and the BGL construction is not possible. Indeed, it is wellknown that in the massive case, H m (R + ) (to be identified with H m (V + ) in the net on Minkowski spacetime) has trivial symplectic complement ( [11,12]), in contrast to the duality (i H 0 (R + )) ⊥ = H 0 (R − ) in the massless case. On the other hand, we know that the massive local subspace H m (I ) of an interval I on the time axis coincides with the local subspace H m (O I ) for the doublecone O I spanned by I ; and by the work [6] of Eckmann and Fröhlich, we have a local unitary equivalence between the massive and massless time-zero algebras.…”

Ground State Representations of Some Non-Rational Conformal Nets

“…Conformal covariance implies the split property [34], and even the split property is unknown to hold in these dual nets. The split property may fail in the dual net in two-dimensional Haag-Kastler net [20] (Section 4.2), therefore, it may be worthwhile to try to (dis)prove the split property in these nets.…”

“…[18] (Theorem 5.5)) for a uniqueness theorem for Diff + (S 1 )-action). See [19] (Chapter 4) for some attempts to construct Diff + (S 1 )-covariance), and [20] Proof. The GNS representation of ω • α is given by π ω • α = α, where π ω is the vacuum representation, the identity map.…”

Covariant Homogeneous Nets of Standard Subspaces

“…As a consequence, the algebraic structure of the model, through the Tomita-Takesaki theory, contains the information about the symmetry group acting on the model. Starting with the BW property, one can enlarge the symmetry group of a QFT [GLW98,MT18], find new relations among field theories [GLW98,LMPR19,MR20], establish proper relations among spin and statistics [GL95], and compute entropy in QFT [LX18,Wi18]. For recent results on this property we refer to [Gu19,DM20].…”