A formula for the relative entropy in chiral CFT
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A formula for the relative entropy in chiral CFT Lorenzo Panebianco1 Received: 28 December 2019 / Revised: 28 December 2019 / Accepted: 29 May 2020 © Springer Nature B.V. 2020
Abstract We prove the QNEC on the Virasoro nets for a class of unitary states extending the coherent states, that is states obtained by applying an exponentiated stress energy tensor to the vacuum. We also verify the Bekenstein Bound by computing the relative entropy on a bounded interval. Keywords Conformal field theory · Relative entropy · Schwarzian action Mathematics Subject Classification 81T05 · 81T40
1 Introduction In this work, we extend the recent results of [11] and we prove the Quantum Null Energy Condition (QNEC) for coherent states in (1 + 1)-dimensional chiral Conformal Field Theory (CFT) by explicitly computing the vacuum relative entropy. The first non-commutative entropy notion, von Neumann’s quantum entropy, was originally designed as a Quantum Mechanics version of Shannon’s entropy: if a state ψ has density matrix ρψ then the von Neumann entropy is given by Sψ = − tr(ρψ log ρψ ). However, in Quantum Field Theory local von Neumann algebras are typically factors of type I I I1 (see [13]), no trace or density matrix exists and the von Neumann entropy is undefined. Nonetheless, the Tomita–Takesaki modular theory applies and one may consider the Araki relative entropy [1] S(ϕψ) = − (ξ | log η,ξ ξ ).
B 1
Lorenzo Panebianco [email protected] Dipartimento di Matematica, Universitá di Roma “La Sapienza”, Piazzale Aldo Moro 5, 00185 Rome, Italy
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L. Panebianco
Here ξ and η are standard vectors of a von Neumann algebra M and η,ξ is the relative modular operator. The quantity S(ϕψ) measures how ϕ deviates from ψ. From the information theoretical viewpoint, S(ϕψ) is the mean value in the state ϕ of the difference between the information carried by the state ψ and the state ϕ. In [11], the relative entropy is applied in (1+1)-dimensional chiral CFT as follows: M is the local algebra A(0, +∞), ϕ is the vacuum state ω given by the vacuum vector and ψ is the coherent state ω f given by the vector ei T ( f ) , with f a real smooth vector field with compact support on the real line and T ( f ) the stress-energy tensor. If V is the unitary projective representation of Diff + (S 1 ) and ρ is the exponential of the vector field on the circle C∗ f , with C the Cayley transform, then one has that ω f = ωV (ρ) , that is ω f is represented by the vector V (ρ) . In [11] it is proved that if f (0) = 0 then we have c S(0,+∞) (ωV (ρ) ω) = 24
+∞
u 0
η (u) η (u)
2 du,
where η is the inverse of the diffeomorphism ρ. In this work, we remove the condition f (0) = 0. By doing this, we are able to prove that c S(t,+∞) (ωV (ρ) ω) = 24
t
+∞
η (u) (u − t) η (u)
2 du.
More in general, we notice that the same expression holds if ρ is a generic diffeomorphism of the circle fixing −1 and with unitary derivative in such point. This expression implies the QNEC for these unitary states, namely S(t) = S(t,+∞) (ωV (ρ)
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