Abstract
We construct and study solitonic representations of the conformal net associated to some vacuum Positive Energy Representation (PER) of a loop group LG. For the corresponding solitonic states, we prove the Quantum Null Energy Condition (QNEC) and the Bekenstein Bound. As an intermediate result, we show that a Positive Energy Representation of a loop group LG can be extended to a PER of H^{s}(S^1,G) for s>3/2, where G is any compact, simple and simply connected Lie group. We also show the existence of the exponential map of the semidirect product LG rtimes R, with R a one-parameter subgroup of mathrm{Diff}_+(S^1), and we compute the adjoint action of H^{s+1}(S^1,G) on the stress energy tensor.
Highlights
Much attention has been focused on quantum information aspects of Quantum Field Theory, which naturally takes place in the framework of quantum black holes thermodynamics [20,21]
We construct and study solitonic representations of the conformal net associated to some vacuum Positive Energy Representation (PER) of a loop group LG
We show that a Positive Energy Representation of a loop group LG can be extended to a PER of H s(S1, G) for s > 3/2, where G is any compact, simple and connected Lie group
Summary
Much attention has been focused on quantum information aspects of Quantum Field Theory, which naturally takes place in the framework of quantum black holes thermodynamics [20,21]. Given a locally normal state ψ of A represented on A(W u) by some vector η, consider the relative entropy S(t) = SMut (ψ ω) and the averaged stress-energy density tensor Tuu(t) = (η|Tuu(x + tu)η). Determining the Connes cocycle (5) for each deformation parameter t is a not trivial issue For this reason, the proof of the QNEC (1) on loop group models led us to the study of Sobolev extensions of Positive Energy Representations of loop groups. In the real line picture, the von Neumann algebras associated to halflines define a hsm inclusion Since on these local algebras the solitonic states ωγ of above have a representing vector satisfying (2), the convexity of the relative entropy is satisfied by Theorem 17. The exhibited proof is not just an application of Theorem 1 or of [24] and it is thought in such a way to make this work as self contained as possible
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