Abstract
We make a rigorous computation of the relative entropy between the vacuum state and a coherent state for a free scalar in the framework of AQFT. We study the case of the Rindler Wedge. Previous calculations including path integral methods and computations from the lattice, give a result for such relative entropy which involves integrals of expectation values of the energy-momentum stress tensor along the considered region. However, the stress tensor is in general non unique. That means that if we start with some stress tensor, then we can "improve" it adding a conserved term without modifying the Poincar\'e charges. On the other hand, the presence of such improving term affects the naive expectation for the relative entropy by a non vanishing boundary contribution along the entangling surface. In other words, this means that there is an ambiguity in the usual formula for the relative entropy coming from the non uniqueness of the stress tensor. The main motivation of this work is to solve this puzzle. We first show that all choices of stress tensor except the canonical one are not allowed by positivity and monotonicity of the relative entropy. Then we fully compute the relative entropy between the vacuum and a coherent state in the framework of AQFT using the Araki formula and the techniques of Modular theory. After all, both results coincides and give the usual expression for the relative entropy calculated with the canonical stress tensor.
Highlights
The algebraic description of quantum field theory (AQFT) focuses on the local algebras of operators generated by fields in regions of the space rather than the field operators themselves
III we briefly review the algebraic formulation of the free scalar field
IV we review the basic concepts of the modular theory of von Neumann algebras
Summary
The algebraic description of quantum field theory (AQFT) focuses on the local algebras of operators generated by fields in regions of the space rather than the field operators themselves. Given a global pure state Φ ∈ H one can consider the reduced density matrix ρRΦ in a space region R of a lattice and compute its von Neumann (vN) entropy This is divergent and not well defined in the continuum due to a large amount of entanglement of UV modes between both sides of the region boundary. This is essentially the only boundary term we can add with the correct dimensions and that does not require a dimensionful coefficient with negative dimensions This can have nonzero expectation values for certain states and makes the definition of ΔhKRi ambiguous. In order to (partially) settle this issue, in this paper we analyze the relative entropy between a coherent state for a free scalar field and the vacuum in the Rindler wedge. For a better reading of the article, the proof of some theorems and some tedious but straightforward calculations were placed into the Appendixes
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
