Abstract
The mutual information I(A, B) of pairs of spatially separated regions satisfies, for any d-dimensional CFT, a set of structural physical properties such as positivity, monotonicity, clustering, or Poincaré invariance, among others. If one imposes the extra requirement that I(A, B) is extensive as a function of its arguments (so that the tripartite information vanishes for any set of regions, I3(A, B, C ) ≡ 0), a closed geometric formula involving integrals over ∂A and ∂B can be obtained. We explore whether this “Extensive Mutual Information” model (EMI), which in fact describes a free fermion in d = 2, may similarly correspond to an actual CFT in general dimensions. Using the long-distance behavior of IEMI(A, B) we show that, if it did, it would necessarily include a free fermion, but also that additional operators would have to be present in the model. Remarkably, we find that IEMI(A, B) for two arbitrarily boosted spheres in general d exactly matches the result for the free fermion current conformal block {G}_{Delta =left(d-1right),J=1}^d . On the other hand, a detailed analysis of the subleading contribution in the long-distance regime rules out the possibility that the EMI formula represents the mutual information of any actual CFT or even any limit of CFTs. These results make manifest the incompleteness of the set of known constraints required to describe the space of allowed entropy functions in QFT.
Highlights
Another formulation of QFT is based on the assignation of operator algebras to open regions of spacetime, rather than field operators at a point [2]
We explore whether this “Extensive Mutual Information” model (EMI), which describes a free fermion in d = 2, may correspond to an actual CFT in general dimensions
We review the fact that the EMI coincides with a free fermion in d = 2
Summary
Another formulation of QFT is based on the assignation of operator algebras to open regions of spacetime, rather than field operators at a point [2]. One inconvenience is that the mathematical description gets more complicated, involving the theory of von Neumann algebras In this sense, to simplify the description of models in this context, and in analogy to the field operator description, a natural idea is to assign numbers to these algebras using the vacuum state. To simplify the description of models in this context, and in analogy to the field operator description, a natural idea is to assign numbers to these algebras using the vacuum state These numbers could only represent statistical measures of vacuum fluctuations for the different regions. Calling Λ to a Lorentz transformation, and taking X, Y , as regions with spatial boundary in a common null plane, we have
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